 Research
 Open Access
Global dynamics of a statedependent feedback control system
 Sanyi Tang^{1}Email author,
 Wenhong Pang^{1},
 Robert A Cheke^{2} and
 Jianhong Wu^{3}
https://doi.org/10.1186/s136620150661x
© Tang et al. 2015
 Received: 6 February 2015
 Accepted: 7 October 2015
 Published: 16 October 2015
Abstract
The main purpose is to develop novel analytical techniques and provide a comprehensive qualitative analysis of global dynamics for a statedependent feedback control system arising from biological applications including integrated pest management. The model considered consists of a planar system of differential equations with statedependent impulsive control. We characterize the impulsive and phase sets, using the phase portraits of the planar system and the Lambert W function to define the Poincaré map for impulsive point series defined in the phase set. The existence, local and global stability of an order1 limit cycle and obtain sharp sufficient conditions for the global stability of the boundary order1 limit cycle have been provided. We further examine the flip bifurcation related to the existence of an order2 limit cycle. We show that the existence of an order2 limit cycle implies the existence of an order1 limit cycle. We derive sufficient conditions under which any trajectory initiating from a phase set will be free from impulsive effects after finite statedependent feedback control actions, and we also prove that orderk (\(k\geq3\)) limit cycles do not exist, providing a solution to an open problem in the integrated pest management community. We then investigate multiple attractors and their basins of attraction, as well as the interior structure of a horseshoelike attractor. We also discuss implications of the global dynamics for integrated pest management strategy. The analytical techniques and qualitative methods developed in the present paper could be widely used in many fields concerning statedependent feedback control.
Keywords
 planar impulsive semidynamical system
 integrated pest management
 Poincaré map
 impulsive set
 phase set
 global stability
MSC
 34A37
 34C23
 92B05
 93B52
1 Introduction
This study concerns the global dynamics of semidynamical systems with statedependent feedback arising from modeling integrated pest management (IPM) [1–4]. The challenge for the study of the system’s global dynamics is due to the statedependent impulsive control.
Impulsive semidynamical systems arise from many important applications in the life sciences including population dynamics (biological resource and pest management programs, and chemostat cultures) [1–10], virus dynamics (HIV) [11–17], medicine and pharmacokinetics (diabetes mellitus and tumor control) [18–22], epidemiology (vaccination strategies, the control of epidemics and plant epidemiology) [23–32], and neuroscience [33–40]. In some applications such as spraying pesticides and releasing natural enemies for pest control and impulse vaccinations and drug administrations for disease treatment [1–5, 8, 41, 42], the impulsive control is implemented at fixed moments to reflect how human actions are taken at fixed periods. In some applications, however, impulsive differential equations with statedependent feedback control have to be used to model densitydependent control strategies [1, 3, 4, 19, 31, 43]. In particular, in an integrated pest management (IPM) strategy, actions are taken only when the density of pests reaches an economic threshold [44, 45]. Feedback control strategies have also been applied in different fields in quite different ways [46–49].
There has also been substantial theoretical development for impulsive semidynamical systems [50–55]. Techniques including the Lyapunov method have been developed to study the stability and boundedness of solutions for impulsive differential equations with fixed moments, with applications in many important areas [1–5, 8]. Despite a few interesting studies on more complicated dynamics such as limit cycles [56–58], invariant and limiting sets [59–64], LaSalle’s invariance principle [65] and the PoincaréBendixson theorem [58, 60], much remains to be done for the qualitative theory, and especially the global dynamics, of impulsive semidynamical systems. This is particularly so for impulsive differential equations with statedependent feedback control.
Some prototype models with biological motivation are needed to guide the development of a general qualitative theory of semidynamical systems with statedependent control. A good example in the series of models motivated by integrated pest management (IPM) [1–4], where the classical LotkaVolterra model with statedependent feedback control is used and some novel techniques for the existence and stability of an order1 limit cycle, nonexistence of limit cycles with order no less than 3, the coexistence of multiple attractors and their basins of attraction are developed. The modeling framework and the developed analytical techniques have been used in a number of recent studies. For example, Huang et al. [19] proposed mathematical models depicting impulsive injection of insulin for type 1 and type 2 diabetes mellitus, and considered the existence and local stability of an order1 limit cycle. Based on biomass concentrationdependent impulsive perturbations, the studies [6, 66] proposed and analyzed chemostat models with statedependent feedback control, again focusing on the existence and stability of an order1 limit cycle. These studies also found that the models have no limit cycles with order no less than 3. The work [30, 67] also considered the existence and stability of limit cycles with different orders, in relation to the biological issue of maintaining the density of an infected plant population below a certain threshold level. See also similar work on population dynamics [10, 58, 68–73] and epidemiology [31]. These studies, however, focused on the existence and local stability of an order1 limit cycle for specific cases.

the precise information as regards the domains of impulsive sets and the phase sets, and the domains for the Poincaré map of impulsive point series;

the global stability of order1 limit cycles (including boundary order1 limit cycles);

the existence of order2 limit cycles and nonexistence of limit cycles with order no less than 3, an open problem listed in [1];

the necessary condition for the existence of order2 limit cycles, and the relation between the existence of order2 limit cycles and order1 limit cycles;

the precise information on parameter space for the finite statedependent feedback control actions, crucial for designing threshold control strategies;

the description of smaller attractors, their basins of attraction and how they are related to phase sets and interior structures of horseshoelike attractors.
2 The model with statedependent feedback control
For the model without control strategy in (2.1), r represents the intrinsic growth rate of the pest population, k represents the carrying capacity. The pest population dies at a rate ax and is predated by the predator population at a rate \(pxy\). The predator response expands at a rate \(\frac{cxy}{1+\omega x}\), which is a saturating function of the amount of pest present. The prey population also inhibits the predator response at a rate \(qxy\), which is the socalled antipredator behavior, and in the absence of the pest declines at a rate δy. Note that all parameters shown in model (2.1) are nonnegative constants.
Many experiments show that the predator and prey populations can reverse their roles, whereby adult prey attack vulnerable young predators [89–92], the so called antipredator behavior. If the variables x and y in model (2.1) describe the prey and predator populations, then the term \(qxy\) represents the effects of the prey population on the predator population, i.e. the prey can kill their predators. Simple predatorprey models with antipredator behavior have been studied [90, 93].
In model (2.1) \(0\leq\theta<1\) is the proportion by which the pest density is reduced by killing or trapping once the number of pests reaches \(V_{L}\), while τ is the constant number of natural enemies released at this time t. Different releasing methods including a proportion for the release rate rather than a constant number can be used in model (2.1) [3, 5, 8]. In order to control the pest we assume, throughout the paper, that \(\tau\geq\frac{b }{p}\) if \(\theta=0\) (from a biological point of view, sufficient of the natural enemies must be released to prevent the pest population exceeding \(V_{L}\), i.e., by maintaining \(\frac{dx(t)}{dt}<0\) (for some time) and \(\theta>0\) if \(\tau=0\). Such a strategy ensures that \(x(t)\) is a decreasing function of time once the pest population reaches the \(V_{L}\).
It is interesting to note that this model can be commonly used in depicting (i) the antipredator behavior of the interaction between pest and its natural enemies, as shown above; (ii) the interaction between the virus population (such as HIV) and its immune cells [94]; (iii) the cytotoxic T lymphocyte response to the growth of an immunogenic tumor [95]; and (iv) the interaction between a toxic phytoplankton population and a zooplankton population [96, 97].
We use this widely used model (2.1) to illustrate systematic methods for investigating global dynamics, and address the basic problems related to models with statedependent feedback control (i.e. statedependent impulsive effects). Of most interest, are questions of how the instant killing rate θ, releasing constant τ and threshold parameter \(V_{L}\) affect the dynamics of model (2.1)? To address this question completely, we choose those three parameters as bifurcation parameters and fix all others aiming to comprehensively investigate the qualitative behavior of model (2.1), of particular interest in the dynamics listed in the Introduction.
3 The ODE model and its main properties

If \(h_{1}< h< h_{2}\), then there are three intersect points between two functions \(F_{1}(x)\) and \(F_{2}(x)\), denoted by \(x_{\min}\), \(x_{\mathrm{mid}}\), and \(x_{\max}\), as shown in Figure 2. For this case, the two branches of \(y_{L}\) and \(y_{U}\) are well defined for all \(x\in[x_{\min}, x_{\mathrm{mid}}]\cup[x_{\max}, +\infty)\) with \(y_{L}\leq\frac{b}{p}\leq y_{U}\), as shown in Figure 3.

If \(h\leq h_{1}\) or \(h\geq h_{2}\), then there exists a unique intersect point between two functions \(F_{1}(x)\) and \(F_{2}(x)\), denoted by \(x_{\min}\). For this case, the two branches of \(y_{L}\) and \(y_{U}\) with \(y_{L}\leq \frac{b}{p}\leq y_{U}\) are well defined for all \(x\in[x_{\min}, +\infty)\), as shown in Figure 3.
The following theorem is useful for discussing the existence of multiple attractors of models with statedependent feedback control proposed in this work.
Theorem 3.1
Let straight line \(L_{1}\) through point \((x_{1}^{*}, y_{e}^{*})\) be parallel to the x axis, as shown in Figure 3. Take any point \(P_{0}\) (or \(Q_{0}\)) in L, draw the line L through \(P_{0}\) (or \(Q_{0}\)), perpendicular to \(L_{1}\). Choose a point \(P_{1}\) (or \(Q_{1}\)) in L such that \(P_{0}P_{1}=\ell>0\) (or \(Q_{0}Q_{1}=\ell>0\)), and then there exists a unique trajectory of system (3.1) through point \(P_{1}\) (or \(Q_{1}\)) and it intersects another point \(P_{2}\) (or \(Q_{2}\)) in L. Then we must have \(P_{0}P_{1}=\ell\geqP_{0}P_{2}\) (or \(Q_{0}Q_{1}=\ell\geqQ_{0}Q_{2}\)), where \(\cdot\) denotes the length of the line segment. Similar results can be had for the trajectory through point \(P_{3}\) (or \(Q_{3}\)), as shown in Figure 3.
Proof
4 Impulsive set, phase set, and Poincaré map
In order to employ the ideas of the Poincaré map or its successor function to address the existence and stability of orderk limit cycles, we must know the exact conditions under which the solution of model (2.2) initiating from \((x_{0}^{+}, y_{0}^{+})\in{\mathcal{N}}\) is free from impulsive effects, i.e. the more exact phase set \({\mathcal{N}}\) should be provided. Moreover, for the impulsive set \({\mathcal{M}}\), \(0\leq y\leq\frac{b}{p}\) is the maximum interval for the vertical coordinates of \({\mathcal{M}}\). Thus, we also want to know the exact interval, i.e. in which part of \(0\leq y\leq\frac{b}{p}\) the solution of model (2.2) cannot reach and then the exact domains of the impulsive set can be obtained.
4.1 Impulsive set
Lemma 4.1
For case (C_{1}), if \((1\theta)V_{L}< x_{3}^{*}\) or \((1\theta)V_{L}>x_{1}^{*}\), then the impulsive set is defined by \({\mathcal{M}}_{1}\); if \(x_{3}^{*}\leq (1\theta)V_{L}\leq x_{1}^{*}\) then the impulsive set is defined by \({\mathcal{M}}_{2}\). For case (C_{2}), if \((1\theta)V_{L}\leq x_{4}^{*}\), then the impulsive set is defined as \({\mathcal{M}}_{1}\); if \((1\theta)V_{L}> x_{4}^{*}\), then the impulsive set is defined by \({\mathcal{M}}\). For case (C_{3}), the impulsive set is defined by \({\mathcal{M}}_{1}\).
Proof
If \((1\theta)V_{L}>x_{1}^{*}\), then by using the same methods as subcase \((1\theta)V_{L}< x_{3}^{*}\) the impulsive set is defined by \({\mathcal{M}}_{1}\). Similarly, we can prove the results for case (C_{2}) and case (C_{3}) are true. □
4.2 Phase set
Lemma 4.2
Proof
Lemma 4.3
Proof
Further, the line \(L_{5}\) must intersect with \(\Gamma_{h}\) at two points, denoted by \(P_{1}=((1\theta)V_{L}, Y_{\max}^{h})\) and \(P_{2}=((1\theta)V_{L}, Y_{\min}^{h})\), which are the two roots of (4.19) with respect to y for \(x=(1\theta)V_{L}\) and can be obtained by using the same methods as those in the proof of Lemma 4.2. Moreover, both \(P_{1}\) and \(P_{2}\) are well defined due to \(A_{h}=F_{1}((1\theta)V_{L})\geq0\) for all \(x_{4}^{*}\leq (1\theta)V_{L}\). Therefore, any trajectory initiating from \((x_{0}^{+}, y_{0}^{+})\in{\mathcal{N}}\) with \(Y_{\min}^{h}< y_{0}^{+}< Y_{\max}^{h}\) will be free from impulsive effects. □
Exact domains of the impulsive set and phase set of model ( 2.2 )
Cases  \(\boldsymbol {(1\theta)V_{L}}\)  Impulsive set  Phase set 

(C_{1})  \((1\theta)V_{L}< x_{3}^{*}\), \((1\theta)V_{L}>x_{1}^{*}\)  \({\mathcal{M}}_{1}\)  \({\mathcal{N}}_{1}\) 
\(x_{3}^{*}\leq(1\theta)V_{L}\leq x_{1}^{*}\)  \({\mathcal{M}}_{2}\)  \({\mathcal{N}}_{2}^{h_{1}}\)  
(C_{2})  \((1\theta)V_{L}\leq x_{4}^{*}\)  \({\mathcal{M}}_{1}\)  \({\mathcal{N}}_{1}\) 
\((1\theta)V_{L}> x_{4}^{*}\)  \({\mathcal{M}}\)  \({\mathcal{N}}_{2}^{h}\)  
(C_{3})  \((1\theta)V_{L}< x_{2}^{*}\)  \({\mathcal{M}}_{1}\)  \({\mathcal{N}}_{1}\) 
In the following, if we consider both \(A_{h_{1}}\) and \(A_{h}\) as functions of \(V_{L}\), then we have the following results.
Lemma 4.4
\(A_{h_{1}}=A_{h}\) at \(V_{L}=x_{1}^{*}\) and \(A_{h_{1}}>A_{h}\) if \(V_{L}>x_{1}^{*}\).
Proof
Thus, by using the same methods as those in the proof of Lemma 4.3 we have the following results for model (2.3).
Lemma 4.5
Remark 4.1
It follows from Remark 4.1 that the relations among τ, \(Y_{\min}^{h}\), and \(Y_{\max}^{h}\) are crucial for the exact domains of the phase set, which will be addressed later.
4.3 Poincaré map
Theorem 4.1
Proof
In order to show the exact domains of the Poincaré map, we first need to know under what conditions the trajectory initiating from \(P_{i}^{+}\in{\mathcal{N}}\) cannot reach the point \(P_{i+1}\in {\mathcal{M}}\). There are two cases:
Case (i): \(V_{L}\geq x_{1}^{*}\) and \(x_{3}^{*}\leq(1\theta)V_{L}\leq x_{1}^{*}\). It follows from Lemma 4.2 that if the initial point \(P_{i}^{+}=((1\theta)V_{L}, y_{i}^{+})\) lies in the homoclinic cycle \(\Gamma_{h_{1}}\) or its interior, then although the two points \(P_{i}^{+}\) and \(P_{i+1}\) could satisfy (4.36), the trajectory cannot reach the line \(L_{4}\) forever, which indicates that both points \(P_{i}^{+}\) and \(P_{i+1}\) cannot lie in the same trajectory, as shown in Figure 5(A). It follows from Lemma 4.2 and Table 1 that in this case we have \(A_{h_{1}}\geq0\) and we require \(P_{i}^{+}\in{\mathcal{N}}_{2}^{h_{1}}\).
Case (ii): \(x_{2}^{*}< V_{L}< x_{1}^{*}\) and \(x_{4}^{*}<(1\theta)V_{L}\). It follows from Lemma 4.3 that if the initial point \(P_{i}^{+}=((1\theta)V_{L}, y_{i}^{+})\) lies in the interior of the closed cycle \(\Gamma_{h}\), then the trajectory cannot reach the line \(L_{4}\), which shows that both points \(P_{i}^{+}\) and \(P_{i+1}\) cannot lie in the same trajectory, as shown in Figure 5(B). It follows from Lemma 4.3 and Table 1 again that in this case we have \(A_{h}>0\) and we require \(P_{i}^{+}\in{\mathcal{N}}_{2}^{h}\).
Therefore, for case (C_{1}), if \(x_{3}^{*}\leq(1\theta)V_{L}\leq x_{1}^{*}\), then it follows from Lemma 4.4 that \(A_{h_{1}}>A_{h}\) and according to the monotonicity of the Lambert W function we have \([Y_{\min}^{h}, Y_{\max}^{h} ]\subset [Y_{\min}^{h_{1}}, Y_{\max}^{h_{1}} ]\). So no matter what \(A_{h_{1}}>A_{h}>0\) and \(A_{h_{1}}>0\geq A_{h}\) (as shown in Figure 5) the Poincaré map is given by the first case of (4.32) if \(x_{3}^{*}\leq(1\theta)V_{L}\leq x_{1}^{*}\). If \((1\theta)V_{L}< x_{3}^{*}\) or \((1\theta)V_{L}>x_{1}^{*}\), then it follows from the proofs of Lemma 4.1 and Lemma 4.2 that we must have \(A_{h}<0\), consequently the Poincaré map is given by the second case of (4.32).
The other two cases (C_{2}) and (C_{3}) of Theorem 4.1 can be obtained directly from the domains of the Poincaré map and the proof of Lemma 4.3. This completes the proof. □
It follows from Lemma 4.5 that we have the main results for the Poincaré map of the impulsive points of model (2.3).
Corollary 4.1
Compared with published definitions of the Poincaré map for model (2.3) [1, 4], we can see that more accurate domains have been provided in formula (4.39).
Based on the proofs of Lemmas 4.14.5 and Theorem 4.1 we can see that the signs of \(A_{h_{1}}\) and \(A_{h}\) play the key roles in determining the domains of the impulsive set and phase set, and in defining the Poincaré map \({\mathcal{P}}(y_{i}^{+})\). Therefore, the relations among the key parameters (i.e. θ, \(V_{L}\), and τ), the signs of \(A_{h_{1}}\) and \(A_{h}\) and the domains of the Poincaré map \({\mathcal{P}}(y_{i}^{+})\) will be discussed briefly before we address the existence and stability of the limit cycle of model (2.2), which are also important in the rest of this work.
The relations among the key parameters ( i.e. θ , \(\pmb{V_{L}}\) , and τ ), the signs of \(\pmb{A_{h_{1}}}\) and \(\pmb{A_{h}}\) and the domains of the Poincaré map \(\pmb{{\mathcal{P}}(y_{i}^{+})}\)
Cases  \(\boldsymbol {V_{L}}\)  \(\boldsymbol {\theta_{1}V_{L}}\)  \(\boldsymbol {A_{h}}\) and \(\boldsymbol {A_{h_{1}}}\)  \(\boldsymbol {{\mathcal{P}}(y_{i}^{+})}\) 

(C_{1})  \(V_{L}< x_{\min}^{h_{2}}\)  \(x_{3}^{*}\leq\theta_{1}V_{L}\leq x_{\min}\)  \(A_{h}\leq0\), \(A_{h_{1}}\geq 0\)  \(y_{i}^{+} \in Y_{D}^{h_{1}}\) 
\(x_{\min}< \theta_{1}V_{L}< x_{\mathrm{mid}}\)  \(A_{h}> 0\), \(A_{h_{1}}\geq0\)  
\(x_{\mathrm{mid}}\leq\theta_{1}V_{L}\leq x_{1}^{*}\)  \(A_{h}\leq0\), \(A_{h_{1}}\geq0\)  
\(\theta_{1}V_{L}< x_{3}^{*}\)  \(A_{h}\leq0\), ×  \(y_{i}^{+} \in Y_{D}^{1}\)  
\(x_{1}^{*}<\theta_{1}V_{L}\)  
\(x_{\min}^{h_{2}}\leq V_{L}\)  \(x_{3}^{*}\leq \theta_{1}V_{L}\leq x_{1}^{*}\)  \(A_{h}\leq0\), \(A_{h_{1}}\geq0\)  \(y_{i}^{+} \in Y_{D}^{h_{1}}\)  
\(\theta_{1}V_{L}< x_{3}^{*}\)  \(A_{h}\leq0\), ×  \(y_{i}^{+} \in Y_{D}^{1}\)  
\(x_{1}^{*}<\theta_{1}V_{L}\)  
(C_{2})  \(x_{4}^{*}<\theta_{1}V_{L}\)  \(A_{h}>0\), ×  \(y_{i}^{+} \in Y_{D}^{h}\)  
\(\theta_{1}V_{L}\leq x_{4}^{*}\)  \(A_{h}\leq0\), ×  \(y_{i}^{+} \in Y_{D}^{1}\)  
(C_{3})  \(A_{h}\leq0\), ×  \(y_{i}^{+} \in Y_{D}^{1}\) 
5 Existence of order1 limit cycles and some important relations
Investigations of the existence and stability of order1 limit cycles of system (2.2) for the whole parameter space are quite challenging, and are similar to the study of the existence and stability of limit cycles of continuous semidynamical systems. Fortunately, the analytical formula of the Poincaré map defined by the impulsive points in the phase set has been obtained, which allows us to employ it to study the existence and stability of order1 limit cycles of model (2.2).
The necessary condition for the existence of a fixed point of the Poincaré map \({\mathcal{P}}(y_{i}^{+})\) in the phase set is \(y^{*}\in Y_{D}\). Thus, it is interesting to show under what conditions the \(y^{*}\in(\tau, b/p+\tau]\) first. To do this, we consider the following two cases: (i) \(A_{h}\leq0\); and (ii) \(A_{h}>0\).
Solving the above inequality with respect to \(\tau+\frac{b}{p}\) yields \(\tau+\frac{b}{p}\leq Y_{\min}^{h}\) (which is impossible due to \(Y_{\min}^{h}<\frac{b}{p}\)) or \(\tau+\frac{b}{p}\geq Y_{\max}^{h}\). This indicates that if \(\tau+\frac{b}{p}\geq Y_{\max}^{h}\), then \(y^{*}\leq \frac{b}{p}+\tau\) when \(0< A_{h}< p\tau\).
5.1 Some important relations
Note that the key parameters θ and \(V_{L}\) determine the domains of the Poincaré map \({\mathcal{P}}(y_{i}^{+})\), and the third key parameter τ will play a crucial role in determining the dynamics of model (2.2). Thus, the parameter τ related to statedependent feedback control has been chosen to address the relations, i.e. we consider \(y^{*}\), \(y_{2}^{*}\), \(\tau+b/p\), \(Y_{\min}^{i}\), \(Y_{\max}^{i}\) for \(i=h, h_{1}\) and \(\tau+Y_{is}^{h}\) as functions of τ. As the first step, we discuss the monotonicity of the \(y^{*}\), where \(y^{*}\) is given by (5.2), and we have the following results.
Lemma 5.1
If \(0< A_{h}< p\tau\), then \(y^{*}\) reaches its minimal value (denoted by \(y_{\min}^{*}\) and \(y_{\min}^{*}=Y_{\max}^{h}\)) at \(\tau_{M}=Y_{\max}^{h}\frac{b}{p}\).
Proof
Lemma 5.2
If \(A_{h}\leq0\), then the inequality \(y*< y_{2}^{*}\) holds true naturally.
Proof
For the first equation \(\Im_{\tau}^{1}\doteq\tau+\frac{b}{p}y^{*}=0\), substituting \(y^{*}\) into it and arranging the items we can see which is equivalent to the equation \(\Im_{\tau}=0\) (defined by (5.5)). This indicates that the equation \(\Im_{\tau}=0\) has a unique positive root \(\tau_{M}\), i.e. the two curves \(y^{*}\) and \(\tau+b/p\) with respect to τ intersect at \(\tau=\tau_{M}\), as shown in Figure 6.
Note that \(A_{h_{1}}\geq0\) indicates that \(A_{h_{1}}\geq A_{h}>0\) or \(A_{h_{1}}>0\geq A_{h}\), which means that both \(\tau_{1}^{h_{1}}\) and \(\tau_{2}^{h_{1}}\) are well defined. Moreover, if \(A_{h}\leq0\), then the small root \(\tau_{1}^{h_{1}}\) disappears and \(y^{*}\) will intersect with \(Y_{\min}^{h_{1}}\) at another point, which will be discussed later.
Now we discuss the relations between \(y^{*}\) and \(\tau+Y_{is}^{h_{1}}\) when \(A_{1}\geq0\), and the relations between \(y^{*}\) and \(\tau+Y_{is}^{h}\) when \(A_{h}\leq0\). That is, we have the following main results.
Lemma 5.3
If \(A_{1}\geq0\), then \(y^{*}<\tau+Y_{is}^{h_{1}}\) for all \(\tau>\tau_{2}^{h_{1}}\) and \(y^{*}=\tau+Y_{is}^{h_{1}}\) at \(\tau=\tau_{2}^{h_{1}}\). If \(A_{h}\leq0\), then \(y^{*}\leq\tau+Y_{is}^{h}\) for all \(\tau> 0\).
Proof
5.2 Existence of order1 limit cycle
Based on the important relations discussed before, for the existence of a fixed point of the Poincaré map \({\mathcal{P}}(y_{i}^{+})\) of model (2.2) and consequently the existence of the order1 limit cycle we have the following main results.
Theorem 5.1
If \(\tau=0\) and \(A_{h}=0\) (here \(\theta>0\)), then any \(y^{*}\) in the phase set is a fixed point of the Poincaré map \({\mathcal{P}}(y_{i}^{+})\). If \(\tau=0\) and \(A_{h}\neq0\), then \(y^{*}=0\) is a unique fixed point of the Poincaré map \({\mathcal{P}}(y_{i}^{+})\).
If \(\tau>0\), then the fixed point \(y^{*}\) of the Poincaré map \({\mathcal{P}}(y_{i}^{+})\) is always well defined for (SC_{123}) with \(y^{*}\in Y_{D}^{1}\). If \(\tau>\tau_{2}^{h_{1}}\), then the fixed point \(y^{*}\) of the Poincaré map \({\mathcal{P}}(y_{i}^{+})\) exists for (SC_{11}) and \(y^{*}\in (Y_{\max}^{h_{1}}, Y_{is}^{h_{1}}+\tau ]\). If \(0<\tau<\tau_{3}^{h_{1}}\) (or \(\tau>\tau_{2}^{h_{1}}\)), then the fixed point \(y^{*}\) of the Poincaré map \({\mathcal{P}}(y_{i}^{+})\) exists for (SC_{12}) and \(y^{*}\in (0, Y_{\min}^{h_{1}} )\) (or \(y^{*}\in (Y_{\max}^{h_{1}}, Y_{is}^{h_{1}}+\tau ]\)). If \(\tau \geq \tau_{M}\), then the fixed point \(y^{*}\) of the Poincaré map \({\mathcal{P}}(y_{i}^{+})\) exists for (SC_{2}) and \(y^{*}\in [Y_{\max}^{h}, \frac{b}{p}+\tau ]\).
Proof
The results for \(\tau=0\) are true obviously. Since \(A_{h}\leq 0\) for (SC_{123}), it follows from Lemma 5.3 that \(y^{*}\leq\tau+Y_{is}^{h}\) for all \(\tau> 0\), which indicates that \(y^{*}\) exists in the phase set, i.e. \(y^{*}\in Y_{D}^{1}\).
If \(\tau>\tau_{2}^{h_{1}}\), then it follows from the relations between \(y^{*}\) and \(Y_{\max}^{h_{1}}\) that \(y^{*}>Y_{\max}^{h_{1}}\). Further, according to Lemma 5.3 we have \(y^{*}< Y_{is}^{h_{1}}+\tau\) for all \(\tau>\tau_{2}^{h_{1}}\) due to \(A_{1}\geq0\) in case (SC_{11}). Thus the fixed point \(y^{*}\) of the Poincaré map \({\mathcal{P}}(y_{i}^{+})\) exists for (SC_{11}) and \(y^{*}\in (Y_{\max}^{h_{1}}, Y_{is}^{h_{1}}+\tau ]\).
If \(0<\tau<\tau_{3}^{h_{1}}\), then it follows from the relations between \(y^{*}\) and \(Y_{\min}^{h_{1}}\) that \(y^{*}< Y_{\min}^{h_{1}}\), which means that the fixed point \(y^{*}\) of the Poincaré map \({\mathcal{P}}(y_{i}^{+})\) exists for (SC_{12}) and \(y^{*}\in (0, Y_{\min}^{h_{1}} )\). If \(\tau>\tau_{2}^{h_{1}}\), then the result can be proved by using the same methods as those for case (SC_{11}).
If \(\tau\geq\tau_{M}\), then it follows from the relations between \(y^{*}\) and \(Y_{\max}^{h}\) and the relations between \(y^{*}\) and \(\frac{b}{p}+\tau\) that \(y^{*}\in [Y_{\max}^{h}, \frac{b}{p}+\tau ]\) and consequently the last part of the results shown in Theorem 5.1 are true. □
Based on the relations discussed before and Theorem 5.1, we have the following main results for the nonexistence of a fixed point of the Poincaré map \({\mathcal{P}}(y_{i}^{+})\) of model (2.2).
Corollary 5.1
Assume \(\tau>0\). The Poincaré map \({\mathcal{P}}(y_{i}^{+})\) does not have a fixed point for case (SC_{11}) provided \(\frac{A_{h}}{p}<\tau\leq\tau_{2}^{h_{1}}\); The Poincaré map \({\mathcal{P}}(y_{i}^{+})\) does not have a fixed point for case (SC_{12}) provided \(\tau_{3}^{h_{1}}\leq\tau\leq\tau_{2}^{h_{1}}\); The Poincaré map \({\mathcal{P}}(y_{i}^{+})\) does not have a fixed point for case (SC_{2}) provided \(\frac{A_{h}}{p}<\tau< \tau_{M}\).
Theorem 5.1 and Corollary 5.1 provide the detailed conditions for the existence and nonexistence of a fixed point of the Poincaré map \({\mathcal{P}}(y_{i}^{+})\) of model (2.2), consequently the existence and nonexistence of order1 limit cycles of model (2.2) can be obtained directly. For the existence and nonexistence of a fixed point of model (2.3) we have the following results.
Corollary 5.2
If \(\tau=0\) and \(A_{0}=0\) (here \(\theta>0\)), then any \(y^{*}\) in the phase set is a fixed point of the Poincaré map \({\mathcal{P}}(y_{i}^{+})\) of model (2.3). If \(\tau=0\) and \(A_{0}\neq0\), then \(y^{*}=0\) is a unique fixed point of Poincaré map \({\mathcal{P}}(y_{i}^{+})\). If \(\tau>0\) and \(A_{0}\leq0\), then for the Poincaré map defined in the phase set there exists a unique fixed point \(y^{*}\in Y_{D}^{0}\). If \(A_{0}> 0\) and \(\tau\geq\tau_{M}\), then for the Poincaré map \({\mathcal{P}}(y_{i}^{+})\) there exists a unique fixed point \(y^{*}\) with \(Y_{\max}^{0}\leq y^{*}\leq\tau+\frac{b}{p}\). The Poincaré map \({\mathcal{P}}(y_{i}^{+})\) does not have a fixed point provided \(0<\frac{A_{0}}{p}<\tau< \tau_{M}\).
6 Local and global stability of order1 limit cycle
To address the stability of \(y^{*}\), we note that if \(\tau=0\) and \(A_{h}=0\) (here \(\theta>0\)), then \(y^{*}\) is stable but not asymptotically stable. For the case \(\tau=0\) and \(A_{h}\neq0\) (i.e. \(y^{*}=0\)) we will address it as a special case later in more detail. Thus, we first assume that \(\tau>0\) and \(y^{*}\) exists, and we provide the sufficient conditions for the local stability and global stability of the fixed point \(y^{*}\). Consequently, the global stability of the order1 limit cycle of model (2.2) can be obtained, which improved on previous results on models with statedependent feedback control [1, 4].
6.1 Local stability of order1 limit cycle
Theorem 6.1
Proof
Corollary 6.1
Assume that \(\tau>0\), \(y^{*}\) exists, and \(A_{h}>0\). If \(y^{*}\in (y_{2}^{*}, \tau+\frac{b}{p} ]\), then the fixed point \(y^{*}\) of the Poincaré map \({\mathcal{P}}(y_{i}^{+})\) of model (2.2) is unstable.
Corollary 6.2
Assume that \(\tau>0\) and \(y^{*}\) exists. If \(A_{0}\leq0\), then the fixed point \(y^{*}\) of the Poincaré map \({\mathcal{P}}(y_{i}^{+})\) of model (2.3) is locally stable; If \(A_{0}>0\), then the fixed point \(y^{*}\) of Poincaré map \({\mathcal{P}}(y_{i}^{+})\) is locally stable provided \(y^{*}\in(\tau, y_{2}^{*})\), and it is unstable when \(y^{*}\in (y_{2}^{*}, \tau+\frac{b}{p} ]\).
Existence and stability of the fixed point \(\pmb{y^{*}}\) of Poincaré map \(\pmb{{\mathcal{P}}(y_{i} ^{+})}\)
Cases  \(\boldsymbol {A_{h}}\) and \(\boldsymbol {A_{h_{1}}}\)  τ  \(\boldsymbol {y^{*}}\)  Interval of \(\boldsymbol {y^{*}}\) 

(SC_{123})  \(A_{h}\leq0\), ×  τ>0  EG  \(Y_{D}^{1}= [\tau, Y_{is}^{h}+\tau ]\) 
\(A_{h}<0\), ×  τ = 0  EG  \(y^{*}=0\)  
\(A_{h}=0\), ×  ENS  \(\forall y^{*}\in [0, Y_{is}^{h} ]\)  
(SC_{11})  \(A_{h}> 0\), \(A_{h_{1}}\geq0\)  \(\frac{A_{h}}{p}<\tau\leq\tau_{2}^{h_{1}}\)  NE  
\(\tau_{2}^{h_{1}}<\tau\)  ES  \((Y_{\max}^{h_{1}}, Y_{is}^{h_{1}}+\tau ]\)  
τ = 0  EU  \(y^{*}=0\)  
(SC_{12})  \(A_{h}\leq0\), \(A_{h_{1}}\geq0\)  \(\tau_{3}^{h_{1}}\leq\tau\leq\tau_{2}^{h_{1}}\)  NE  
\(0<\tau<\tau_{3}^{h_{1}}\)  ES  \((0, Y_{\min}^{h_{1}} )\)  
\(\tau>\tau_{2}^{h_{1}}\)  ES  \((Y_{\max}^{h_{1}}, Y_{is}^{h_{1}}+\tau ]\)  
\(A_{h}<0\), ×  τ = 0  ES  \(y^{*}=0\)  
\(A_{h}=0\), ×  ENS  \(\forall y^{*}\in [0, Y_{\min }^{h_{1}} )\)  
(SC_{2})  \(A_{h}>0\), ×  \(\frac{A_{h}}{p}<\tau<\tau_{M}\)  NE  
\(\tau_{M}\leq \tau\leq\tau_{2}\)  EU  \([Y_{\max}^{h}, \frac{b}{p}+\tau ]\)  
\(\tau_{2}<\tau\)  ES  \([Y_{\max}^{h}, \frac{b}{p}+\tau ]\)  
τ = 0  EU  \(y^{*}=0\) 
Here, × means the sign of \(A_{h_{1}}\) is not necessary for that subcase, NE denotes the nonexistence of a fixed point, EU represents the existence of a fixed point which is unstable, ES shows the existence of a fixed point which is stable, EG denotes the existence of a fixed point which is globally stable, and ENS represents the existence of a fixed point which is neutrally stable. Note that if \(\tau=0\), then for case (SC_{12}) we have \(Y_{\min}^{h_{1}}=Y_{is}^{h_{1}}\) once \(A_{h}=0\). Thus, in this subcase, any \(y^{*}\in [0, Y_{\min}^{h_{1}} )= [0, Y_{is}^{h_{1}} )\) is a fixed point of the Poincaré map \({\mathcal{P}}(y_{i}^{+})\) of model (2.2), i.e. for any solution initiating from \(((1\theta)V_{L}, y^{*})\) is an order1 periodic solution which is neutrally stable.
So far, all cases shown in Table 3 have been proved except for the global stability of the fixed point \(y^{*}\) in subcase (SC_{123}) and the stability of \(y^{*}=0\) for \(\tau=0\), which are our main purposes in the following subsections.
6.2 Global stability of the order1 limit cycle
For the global stability of the fixed point \(y^{*}\) as well as the order1 limit cycle of system (2.2), we first focus on the case \(\tau>0\) for (SC_{123}) based on the domains of Poincaré map \({\mathcal{P}}(y_{i}^{+})\) and the existence of \(y^{*}\), and we have the following main result.
Theorem 6.2
Assuming that \(\tau>0\) in case (SC_{123}), then the fixed point \(y^{*}\) of Poincaré map \({\mathcal{P}}(y_{i}^{+})\) exists and satisfies \(\tau< y^{*}< y_{2}^{*}\). Moreover, it is globally stable once it exists. Consequently, the order1 limit cycle of system (2.2) is globally stable.
Proof
Case 1 \(\tau\geq b/p\).
Case 2 \(\tau< b/p\).
Finally, if \(\tau< b/p\) and \(A_{h}=0\), then it is easy to see that \(y^{*}\in (\frac{b}{p}, y_{2}^{*} )\) and \(g(y)=1\) for all \(y\in (\tau, \frac{b}{p})\). Moreover, by simple calculation we have \(W(f(y))=\frac{py}{b}\) for all \(y\in(\tau, \frac{b}{p})\), which means that for any solution initiating from \(((1\theta)V_{L}, y_{0}^{+})\) with \(y_{0}^{+}< b/p\) we have \(y_{i+1}^{+}=y_{i}^{+}+\tau\) if \(y_{i}^{+}\in(\tau, \frac{b}{p})\). Therefore, there exists a positive integer \(k_{1}\) such that \(y_{k_{1}}^{+}\in(b/p, \tau+b/p]\) and \(y_{i}^{+}\in(\tau, b/p)\) for all \(i< k_{1}\). The result follows if we can prove that \(y_{i}^{+}\in (b/p, \tau+b/p]\) for all \(i\geq k_{1}\). To do this, we need the following result.
Claim If \(y_{k_{1}}^{+}\in(b/p, \tau+b/p]\), then we must have \(y_{k_{1}+1}^{+}\in(b/p, \tau+b/p]\).
Proof
We employ the following two methods to prove the above claim, which are useful later.
Therefore, the fixed point \(y^{*}\) is globally stable when \(A_{h}=0\) and \(\tau< b/p\). Based on results shown in Cases 1 and 2, we can see that if the conditions of Theorem 6.2 are true, then the fixed point \(y^{*}\) is globally stable. This completes the proof. □
Remark 6.1
The above two theorems (Theorem 6.1 and Theorem 6.2) have provided the detailed analyses for the existence and stability of fixed point \(y^{*}\) of the Poincaré map \({\mathcal{P}}(y_{i}^{+})\) and consequently the order1 limit cycle. Further, we note that the period of the order1 limit cycle can be analytically determined by using similar methods as those developed in reference [1].
Corollary 6.3
Assuming that \(\tau>0\) and \(A_{0}\leq0\), then the fixed point \(y^{*}\) of Poincaré map \({\mathcal{P}}(y_{i}^{+})\) for model (2.3) exists and satisfies \(\tau< y^{*}< y_{2}^{*}\). Moreover, it is globally stable once it exists. Consequently, the order1 limit cycle of system (2.3) is globally stable.
Before finishing this subsection, we would like to address some special cases of the order1 limit cycle including the existence of an order1 homoclinic cycle, and long or short order1 limit cycles.
Order1 long or short limit cycle. Based on the existence of the order1 homoclinic cycle, we see that if the fixed point \(y^{*}\) of Poincaré map is less than the \(y_{h}^{*}\) and \((1\theta)V_{L}>x_{1}^{*}\), then we say that model (2.2) has an order1 short limit cycle \(\Gamma_{s}\), as shown in Figure 7. While, if the fixed point \(y^{*}\) of Poincaré map is larger than the \(y_{h}^{*}\) and \((1\theta)V_{L}>x_{1}^{*}\), then we say that model (2.2) has an order1 long limit cycle \(\Gamma_{l}\), as shown in Figure 7. The order1 short or long limit cycle may play a key role in real problems with statedependent feedback control actions, which tells us how frequently the control tactics should be applied or how to design the control tactics to adjust the period of control actions.
6.3 Boundary order1 limit cycle and its stability
It follows from Theorem 5.1 that if \(\tau=0\) and \(A_{h}\neq0\), then \(y^{*}=0\) is a unique fixed point of Poincaré map \({\mathcal{P}}(y_{i}^{+})\) (please see Table 3 for details), which indicates that for model (2.2) there exists a unique boundary order1 limit cycle with initial condition \(((1\theta)V_{L}, 0)\). Therefore, in this subsection, we address its analytical formula and stability. Note that, if \(\tau=0\) and \(A_{h}\neq0\), then the derivative of the Poincaré map at \(y^{*}=0\) is one, which indicates that the stability of \(y^{*}=0\), which in this case cannot be determined directly.
Similarly, if \(A_{h}<0\), then \(y_{2}^{+}< y_{1}^{+}\) and \(y_{2}< y_{1}\) must hold true. In conclusion, we have the following main results for the boundary order1 limit cycle.
Theorem 6.3
Let \(\tau=0\) and \(A_{h}\neq0\). The boundary order1 limit cycle \((x^{T}(t), 0)\) is globally asymptotically stable for (SC_{123}), and it is locally asymptotically stable for (SC_{12}). The boundary order1 limit cycle \((x^{T}(t), 0)\) is unstable for (SC_{11}) and (SC_{2}).
Proof
For case (SC_{123}), we assume, without loss of generality, that any solution initiating from phase set \({\mathcal{N}}_{1}\) experience infinite impulsive effects, i.e. we have \(y_{k}^{+}\in (0, Y_{is}^{h} ]\) for all \(k\geq0\). Since \(A_{h}<0\), it follows from the above discussion that by induction we conclude that \(y_{k}^{+}\) is a strictly decreasing sequence with \(\lim_{k\rightarrow\infty}y_{k}^{+}=y^{*}\). Moreover, \(y^{*}=0\) must hold, otherwise it contradicts the uniqueness of \(y^{*}=0\) in this case. Thus, the boundary order1 limit cycle \((x^{T}(t), 0)\) is globally attractive.
The local stability of the boundary order1 limit cycle for (SC_{12}) is obvious due to the domain of the phase set. The instability of the boundary order1 limit cycle for (SC_{11}) and (SC_{2}), is shown since \(A_{h}>0\), \(y_{k}^{+}\) is a strictly increasing sequence and the solution will be free from impulsive effects after finite statedependent feedback control actions, as shown in Figure 8(C). Thus the results are true. □
Remark 6.2
To confirm the main results obtained in Theorem 6.3, we fixed the parameter values as those in Figure 8, and we can see that if \(A_{h}>0\), then the impulsive points and its phase points of trajectory shown in Figure 8(C) are two monotonically increasing sequences, and eventually the trajectory approaches a closed orbit which frees it from impulsive effects. While if \(A_{h}<0\), then the impulsive points and its phase points of trajectory shown in Figure 8(F) are two monotonically decreasing sequences, and eventually the trajectory tends to the boundary order1 limit cycle \((x^{T}(t),0)\).
Corollary 6.4
If \(\tau=0\) and \(A_{0}\neq0\), then there exists a unique boundary order1 limit cycle \((x^{T}(t), 0)\) for model (2.3). Furthermore, if \(A_{0}>0\), then the order1 limit cycle \((x^{T}(t), 0)\) is unstable; if \(A_{0}<0\), then the order1 limit cycle \((x^{T}(t), 0)\) is globally asymptotically stable.
7 Flip bifurcation and existence of order2 limit cycle
Investigating the existence or nonexistence of the limit cycle with order no less than 1 for models with statedependent feedback control is challenging, but this problem has been addressed for some special cases [1]. Thus, in the following two sections we will focus on the existence and nonexistence of order2 limit cycles for model (2.2) and provide some sufficient conditions or necessary conditions on this topic.
Lemma 7.1
Let \(V_{L}^{2}=\frac{2q+q\theta+\sqrt {B}}{2(1\theta)q\omega}\) with \(B=\theta^{2}q^{2}+4qc4\theta qc\). If \(A_{h}>0\), then there are two positive roots of the equation \(F_{A}(V_{L})=0\), denoted by \(V_{L}^{1*}\) and \(V_{L}^{2*}\), such that \(F_{A}(V_{L})>0\) for all \(V_{L}\in(V_{L}^{1*}, V_{L}^{2*})\). Further, if \(F_{A}(V_{L}^{2})>p\taub\ln (\frac{y_{2}^{*}}{y_{2}^{*}\tau} )\), then the equation \(\mu(V_{L})=0\) exists with two positive roots, denoted by \(V_{L}^{3*}\) and \(V_{L}^{4^{*}}\) (as shown in Figure 9), and \(V_{L}^{1*}< V_{L}^{3*}< V_{L}^{4^{*}}< V_{L}^{2*}\). Moreover, \(F_{A}'(V_{L}^{3*})>0\) and \(F_{A}'(V_{L}^{4*})<0\).
Proof
Theorem 7.1
Assuming that \(\tau>0\), \(A_{h}>0\), \(y^{*}\) exists and \(F_{A}(V_{L}^{2})>p\taub\ln (\frac{y_{2}^{*}}{y_{2}^{*}\tau} )\), then the family \(G(y, V_{L})\) undergoes a flip bifurcation at \((y_{2}^{*}, V_{L}^{3*})\), while the family \(G(y, V_{L})\) cannot undergo a flip bifurcation at \((y_{2}^{*}, V_{L}^{4*})\).
Proof
It follows from the relations \(\tau< y_{2}^{*}<\tau+b/p\) that \(y_{2}^{*}\tau>0\) and \(bp(y_{2}^{*}\tau)>0\). Therefore, according to the signs of \(F_{A}'(V_{L}^{3*})\) and \(F_{A}'(V_{L}^{4*})\) we have \(\frac{\partial^{2} G(y, V_{L})}{\partial y\,\partial V_{L}}_{(y,V_{L})=(y_{2}^{*}, V_{L}^{3*})}<0\) provided \(y_{2}^{*}>b/p\) and \(\frac{\partial G^{2}(y, V_{L})}{\partial y\,\partial V_{L}}_{(y,V_{L})=(y_{2}^{*}, V_{L}^{4*})}<0\) provided \(y_{2}^{*}< b/p\). Further, if \(A_{h}>0\), then \(y^{*}=y_{2}^{*}>\frac{b}{p}\), and it follows from Lemmas A.2A.3 that the family \(G(y, V_{L})\) undergoes a flip bifurcation at \((y_{2}^{*}, V_{L}^{3*})\). In contrast, the family \(G(y, V_{L})\) cannot undergo a flip bifurcation at \((y_{2}^{*}, V_{L}^{4*})\). This completes the proof. □
Corollary 7.1
(Flip bifurcation of model (2.3))
Proof
8 The necessary condition for the existence of an order2 limit cycle
Evidence for the existence of an order2 limit cycle, as discussed in Section 7, which can bifurcate from an order1 limit cycle through a subcritical flip bifurcation, and some special cases for the existence of an order2 limit cycle will be discussed in Section 11. Moreover, we note that the order2 limit cycles can only appear in cases (SC_{11}) and (SC_{2}), because \(g(y)<1\) for all y lying in the domains of Poincaré map \({\mathcal{P}}\) if \(A_{h}\leq0\). Therefore, for the necessary condition of existence of an order2 limit cycle we only need to focus on cases (SC_{11}) and (SC_{2}), which will be addressed later. So we would like to discuss the relations between order2 and order1 limit cycles first.
8.1 The relations between order2 limit cycle and order1 limit cycle
In this section, we assume that for model (2.2) there exists an order2 limit cycle, as shown in Figure 10 with \(P_{0}^{+}=((1\theta)V_{L}, y_{0}^{+})\), \(P_{1}^{+}=((1\theta)V_{L}, y_{1}^{+})\) and \(y_{0}^{+}\neq y_{1}^{+}\), and we denote the corresponding points lying in impulsive set \({\mathcal{M}}\) as \(Q_{0}=(V_{L}, y_{1})\) and \(Q_{1}=(V_{L}, y_{2})\) with \(y_{2}^{+}=y_{0}^{+}\). Without loss of generality, we let \(y_{1}^{+}>y_{0}^{+}\) and focus on case (SC_{2}), i.e. \(V_{L}< x_{1}^{*}\) and \(x_{4}^{*}<(1\theta)V_{L}\), as shown in Table 3. For case (SC_{11}), we can obtain the same results by using the methods developed in this section. Therefore, for case (SC_{2}) there are three possibilities: (i) \(y_{1}^{+}>y_{0}^{+}\geq Y_{\max}^{h}>b/p\); (ii) \(y_{1}^{+}\geq Y_{\max}^{h}>b/p>Y_{\min}^{h}\geq y_{0}^{+}\); (iii) \(b/p>Y_{\min}^{h}\geq y_{1}^{+}>y_{0}^{+}\).
Lemma 8.1
Assuming (SC_{2}) (i.e. \(V_{L}< x_{1}^{*}\) and \(x_{4}^{*}<(1\theta)V_{L}\)) and model (2.2) has an order2 limit cycle, then Cases (ii) and (iii) cannot occur.
Proof
Here we first prove that case (iii) cannot hold true and case (ii) will be proved in Section 8.2. Assume \(b/p>Y_{\min}^{h}\geq y_{1}^{+}>y_{0}^{+}\). If model (2.2) has an order2 limit cycle \(O_{2}\) with initiating value \(P_{0}^{+}\), then the two line segments \(\overline{Q_{2}P_{0}^{+}}\) and \(\overline{Q_{1}P_{1}^{+}}\) satisfy \(\overline{Q_{2}P_{0}^{+}}\parallel\overline{Q_{1}P_{1}^{+}}\), which is impossible due to \(y_{2}^{+}=y_{0}^{+}\). Thus, we conclude that case (iii) cannot appear if for model (2.2) there exists an order2 limit cycle under condition (SC_{2}). □
The following theorem shows the relations between the existence of an order2 limit cycle and the existence of an order1 limit cycle. Similar results and proofs have already been published [1].
Theorem 8.1
Assuming (SC_{2}) (i.e. \(V_{L}< x_{1}^{*}\) and \(x_{4}^{*}<(1\theta)V_{L}\)), then the existence of an order2 limit cycle of model (2.2) indicates the existence of an order1 limit cycle of model (2.2).
Proof
Remark 8.1
If \(y_{0}^{+}, y_{1}^{+}\in [Y_{\max}^{h}, \tau+b/p ]\) (i.e. case (i)), as shown in Figure 10, then it is easy to prove that the existence of an order2 limit cycle indicates the existence of an order1 limit cycle. That is, the region Ω shown in Figure 10 satisfies all the conditions of the PoincaréBendixson theorem of impulsive semidynamic systems [60]. However, this method cannot be applied when case (ii) occurs, because the domains of the Poincaré map are separated into two segments, i.e. \(y_{i}^{+}\in [\tau, Y_{\min}^{h} ]\cup [Y_{\max}^{h}, b/p+\tau ]\). Therefore, if we want to employ the PoincaréBendixson theorem of impulsive semidynamic systems, then we must exclude case (ii), as mentioned before which will be proved later by using the necessary condition of existence of an order2 limit cycle.
8.2 The necessary condition for the existence of an order2 limit cycle
Although we cannot provide the simple sufficient conditions for the existence of an order2 limit cycle as those for the existence of an order1 limit cycle, the necessary conditions shown in the following theorem are quite useful.
Theorem 8.2
Proof
In order to show the necessary condition of the existence of an order2 limit cycle, we plot the second iteration of Poincaré map \({\mathcal{P}}(y)\) with the parameter set as those shown in Figure 11(A). Obviously, with the given parameter values, for the Poincaré map \({\mathcal{P}}(y)\) there exists a period two solution, as shown in Figure 11(A). At the same time, we plot the function \(f_{2}(y)\) in Figure 11(B) and we have \(f_{2}(y_{0}^{+})=f_{2}(y_{1}^{+})\) with \(Y_{\max}^{h}< y_{0}^{+}< y_{1}^{+}<\frac{b}{p}+\tau\), which indicates the necessary condition of the existence of an order2 limit cycle holds true.
Remark 8.2
Note that the existence of an order2 limit cycle strictly depends on the \(y_{2}^{*}\), once the two roots \(y_{0}^{+}\) and \(y_{1}^{+}\) of \(f_{2}(y)=c\) coincide, i.e. \(y_{0}^{+}=y_{1}^{+}=y_{2}^{*}\), then we have \(y^{*}=y_{2}^{*}\) at which point the flip bifurcation occurs. All these results confirm that the existence of an order2 limit cycle associates with the flip bifurcation at \(y_{2}^{*}\). Moreover, the function \(f_{2}(y)\) only depends on the three parameters b, p, and τ, which is independent of control parameters \(V_{L}\) and θ.
Based on the necessary condition of existence of an order2 limit cycle, we can prove case (ii) in Lemma 8.1.
Proof of case (ii) in Lemma 8.1
Corollary 8.1
If for model (2.2) there exists an order2 limit cycle, then the order2 and order1 limit cycles coexist. Moreover, the nonexistence of the order1 limit cycle implies the nonexistence of the order2 limit cycle.
Proof
According to the proof of Lemma 8.1 that only case (i), i.e. \(y_{1}^{+}>y_{0}^{+}\geq Y_{\max}^{h}>b/p\), is feasible if for model (2.2) there exists an order2 limit cycle. Consequently, it follows from Remark 8.1 that the region Ω indicated in Figure 10 satisfies the PoincaréBendixson theorem of impulsive semidynamic systems [58]. Thus, the existence of the order2 limit cycle indicates the existence of the order1 limit cycle. □
The necessary condition also tells us that the order2 limit cycle will disappear as \(V_{L}\) is decreasing or τ is increasing. Moreover, we can obtain similar results to those shown in this section for model (2.3) and we do not repeat them here.
9 Finite statedependent feedback control actions
To address the global dynamic behavior of model (2.2) completely, for cases (SC_{1}) and (SC_{2}) we need to know under which conditions the solution initiating from \(((1\theta)V_{L}, y_{0}^{+})\), where \(y_{0}^{+}\in Y_{D}^{h}\) or \(Y_{D}^{h_{1}}\), will be free from impulsive effects after finite statedependent feedback control actions. That is, whether there exists a positive integer \(k_{1}\), such that \(y_{k_{1}}^{+}\in [Y_{\min}^{h_{1}}, Y_{\max}^{h_{1}} ]\) for case (SC_{1}) or \(y_{k_{1}}^{+}\in (Y_{\min}^{h}, Y_{\max}^{h} )\) for case (SC_{2}). This is not only important for determining the global dynamics, but also it is crucial for our real life problems considered in the present work.
Therefore, in this section we will focus on finding the conditions under which all solutions of model (2.2) with initial value \(((1\theta)V_{L}, y_{0}^{+})\) will be free from impulsive effects after finite statedependent feedback control actions. For convenience, we denote the boundary of closed trajectory \(\Gamma_{h}\) (or homoclinic cycle \(\Gamma_{h_{1}}\)) as \(\partial\Omega_{h}\) (or \(\partial\Omega_{h_{1}}\)) and its interior as \(\operatorname{Int}\Omega_{h}\) (or \(\operatorname{Int}\Omega_{h_{1}}\)).
9.1 Finite statedependent feedback control actions for case (SC_{2})
Based on the results shown in Section 6, in particular the results shown in Table 3, we have the following main theorem with respect to finite statedependent feedback control actions for model (2.2) under case (SC_{2}). Note that all trajectories from \(\operatorname{Int}\Omega_{h}\) are free from impulsive effects, and \(\operatorname{Int}\Omega_{h}\) is an invariant set of system (2.2) under case (SC_{2}).
Theorem 9.1
For case (SC_{2}), if \(\frac{A_{h}}{p}<\tau <\tau_{M}\) then any solution initiating from \(((1\theta)V_{L}, y_{0}^{+})\) with \(y_{0}^{+}>0\) will experience finite statedependent feedback control actions and enter into \(\operatorname{Int}\Omega_{h}\) eventually.
Proof
For any solution initiating from \(((1\theta)V_{L}, y_{0}^{+})\) with \(y_{0}^{+}\leq\tau\) or \(y_{0}^{+}>\tau+\frac{b}{p}\) will enter into the region \({\mathcal{N}}_{2}^{h}\) after a single impulsive effect, i.e. \(y_{1}^{+}\in Y_{D}\). It follows from \(\tau<\tau_{M}=Y_{\max}^{h}\frac{b}{p}\) that there are two possibilities: (a) \(y_{1}^{+}\in (Y_{\min}^{h}, Y_{\max}^{h} )\), and (b) \(y_{1}^{+}\in (\tau, Y_{\min}^{h} ]\). For case (a), it is easy to see the results shown in Theorem 9.1 are true, and the solution initiating from \(((1\theta)V_{L}, y_{0}^{+})\) at most experiences an impulsive effect once only before entering into \(\operatorname{Int}\Omega_{h}\).
Corollary 9.1
For case (SC_{2}), if \(0<\tau<\tau_{M}\) then any solution initiating from \(((1\theta)V_{L}, y_{0}^{+})\) with \(y_{0}^{+}>0\) will experience finite statedependent feedback control actions and enter into \(\operatorname{Int}\Omega_{h}\) eventually.
In Figure 13, we show the effects of different values of τ on the finite impulsive effects of solutions. It follows from Figure 13(A)(C) that if the solution initiating from the same initial point \(((1\theta)V_{L}, 0.2)\), then the smaller τ is, the greater the number of impulsive effects that it has. Note that not all solutions will enter into \(\operatorname{Int}\Omega_{h}\) after finite impulsive effects once the τ increases and exceeds the \(\tau_{M}\), because there exists an order1 limit cycle which could be stable or unstable, as shown in Figure 13(D) and Table 3. If so, for model (2.2) there may exist multiple attractors including a stable order1 limit cycle (indicated as \(O_{1}\) in Figure 13(D)) and \(\operatorname{Int}\Omega_{h}\). Thus, the question is what are their regions of attraction, and we will address this question in the following sections.
9.2 Finite statedependent feedback control actions for case (SC_{1})
Note that all trajectories from \(\operatorname{Int}\Omega_{h_{1}}\cup\partial \Omega_{h_{1}}\) are free from impulsive effects for case (SC_{1}), and \(\operatorname{Int}\Omega_{h_{1}} \cup\partial\Omega_{h_{1}}\) is an invariant set of system (2.2) under case (SC_{1}). In the following we provide the results for the two subcases of (SC_{1}) separately.
Theorem 9.2
For case (SC_{11}), if \(\frac{A_{h}}{p}<\tau \leq\tau_{2}^{h_{1}}\) then any solution initiating from \(((1\theta)V_{L}, y_{0}^{+})\) with \(y_{0}^{+}>0\) will experience finite statedependent feedback control actions and enter into \(\operatorname{Int}\Omega_{h_{1}} \cup \partial\Omega_{h_{1}}\) eventually.
Proof
Corollary 9.2
For case (SC_{11}), if \(0<\tau\leq\tau_{2}^{h_{1}}\) then any solution initiating from \(((1\theta)V_{L}, y_{0}^{+})\) with \(y_{0}^{+}>0\) will experience finite statedependent feedback control actions and enter into \(\operatorname{Int}\Omega_{h_{1}} \cup \partial\Omega_{h_{1}}\) eventually.
In Figure 14(A), we show that if \(\frac{A_{h}}{p}<\tau\leq \tau_{2}^{h_{1}}\), then all solutions initiating from \(((1\theta)V_{L}, y_{0}^{+})\) will be free from impulsive effects and enter into the invariant set \(\operatorname{Int}\Omega_{h_{1}} \cup\partial\Omega_{h_{1}}\) after finite statedependent feedback control actions. However, once the τ is increasing and exceeds the threshold value \(\tau_{2}^{h_{1}}\), then multiple attractors may exist (as shown in Figure 14(B)) and their regions of attraction will also be addressed later.
Similarly, for subcase (SC_{12}) we have the following main results on the finite statedependent feedback control actions.
Theorem 9.3
For case (SC_{12}), if \(\tau_{3}^{h_{1}}\leq\tau\leq\tau_{2}^{h_{1}}\) then any solution initiating from \(((1\theta)V_{L}, y_{0}^{+})\) with \(y_{0}^{+}>0\) will experience finite statedependent feedback control actions and enter into \(\operatorname{Int}\Omega_{h_{1}} \cup\partial\Omega_{h_{1}}\) eventually.
By using the same methods as those in the proof of Theorem 9.1 and Theorem 9.2 we can prove Theorem 9.3. Note that for subcase (SC_{12}) multiple attractors can exist for \(\tau<\tau_{3}^{h_{1}}\) and \(\tau>\tau_{2}^{h_{1}}\).
10 Nonexistence of orderk (\(k\geq3\)) limit cycles
It follows from Table 3 and the results shown in Section 9 that the dynamical behavior for case (SC_{11}) with \(\tau>\tau_{2}^{h_{1}}\), case (SC_{2}) with \(\tau\geq\tau_{M}\) and case (SC_{12}) with \(\tau<\tau_{3}^{h_{1}}\) or \(\tau>\tau_{2}^{h_{1}}\) could be complex. In order to address the possible complex dynamics in more detail, the nonexistence of orderk (\(k\geq3\)) limit cycles of model (2.2) will be investigated in this section. Thus, without loss of generality, we assume that \(y_{0}^{+}\neq y_{1}^{+}\neq y_{2}^{+}\) and the solution of system (2.2) with initial value \(((1\theta)V_{L},y_{0}^{+})\) experiences impulses k (\(k\geq3\)) times. Then there exists a positive integer n such that \(k=2n\) or \(k=2n+1\).
For convenience we denote the set \({\mathcal{K}}=\{0, 1, 2, 3, \ldots\}={\mathcal{K}}_{1}\cup{\mathcal{K}}_{2}\), where \({\mathcal{K}}_{1}=\{l_{1}, l_{2}, \ldots \}\) and \({\mathcal{K}}_{2}=\{m_{1}, m_{2}, \ldots \}\) are two real subsets of set \({\mathcal{K}}\). Further, we denote \({\mathcal{Y}}=\{y_{k}^{+} k\in{\mathcal{K}}\}\), \(\overline{{\mathcal{Y}}}=\{y_{l}^{+} y_{l}^{+}\geq b/p, l\in{\mathcal{K}}_{1}\}\) and \(\underline{{\mathcal{Y}}}=\{y_{m}^{+} y_{m}^{+}< b/p, m\in{\mathcal{K}}_{2}\}\), respectively.
10.1 Generalized results
We first prove the following generalized results before giving the main results in this section. Note that results similar to those shown in the following first two Lemmas have been proved in [1]. For completeness and independence we briefly provide details of the proofs here.
Lemma 10.1
 (a)\(y_{1}^{+}< y_{0}^{+}< y_{2}^{+}\). Then$$\frac{b}{p}< y_{2n+1}^{+}< y_{2n1}^{+}< \cdots < y_{3}^{+}< y_{1}^{+}< y_{0}^{+}< y_{2}^{+}< y_{4}^{+}< \cdots< y_{2n}^{+}. $$
 (b)\(y_{1}^{+}< y_{2}^{+}< y_{0}^{+}\). Then$$\frac{b}{p}< y_{1}^{+}< y_{3}^{+}< \cdots < y_{2n+1}^{+}< y_{2n}^{+}< y_{2(n1)}^{+}< \cdots< y_{4}^{+}< y_{2}^{+}< y_{0}^{+}. $$
 (c)\(y_{2}^{+}< y_{0}^{+}< y_{1}^{+}\). Then$$\frac{b}{p}< y_{2n}^{+}< y_{2(n1)}^{+}< \cdots < y_{4}^{+}< y_{2}^{+}< y_{0}^{+}< y_{1}^{+}< y_{3}^{+}< \cdots< y_{2n+1}^{+}. $$
 (d)\(y_{0}^{+}< y_{2}^{+}< y_{1}^{+}\). Then$$\frac{b}{p}< y_{0}^{+}< y_{2}^{+}< \cdots < y_{2(n1)}^{+}< y_{2n}^{+}< y_{2n+1}^{+}< y_{2n1}^{+}< \cdots< y_{3}^{+}< y_{1}^{+}. $$
Proof
It follows from the monotonicity of the Lambert W function and \(f(y)\) that we have \(y_{2}^{+}>y_{1}^{+}\) if \(\frac{b}{p}< y_{1}^{+}< y_{0}^{+}\). For the relations between \(y_{2}^{+}\) and \(y_{0}^{+}\) there are two possibilities.
If \(\frac{b}{p}< y_{0}^{+}< y_{1}^{+}\) then there are two cases corresponding to cases (c) and (d) which can be proved similarly. According to the proof of Theorem 6.2 we have \(1< g(y)<0\) for all \(y\in (\tau, \tau+b/p]\), which indicates that if \(A_{h}\leq0\), then only the cases (b) and (d) can occur. □
Lemma 10.2
If \(\tau\geq\frac{b}{p}\), then for model (2.2) there does not exist an orderk \((k\geq3)\) limit cycle other than the order1 and order2 limit cycles.
Proof
Lemma 10.3
If \(\tau<\frac{b}{p}\) and inequality \(y_{1}^{+}< y_{0}^{+}<\frac{b}{p}\) holds, then for model (2.2) a limit cycle with order no less than 2 does not exist.
Proof
Remark 10.1
(Open problem proposed in [1])
Lemma 10.4
If \(\tau<\frac{b}{p}\), \(y_{k_{1}}^{+}\in(b/p, \tau+b/p]\) and \(A_{h}\geq0\), then we must have \(y_{k_{1}+1}^{+}\in(b/p, \tau+b/p]\).
Proof
Lemma 10.5
If \(A_{h}\geq0\) then \(y^{*}>\frac{b}{p}+\frac {\tau}{2}\).
Proof
10.2 Nonexistence of a limit cycle with order no less than 3
Now we assume that the solution of model (2.2) experiences infinite pulse effects and we have the following main results.
Theorem 10.1
For model (2.2) a limit cycle with order no less than 3 does not exist.
Proof
It follows from Theorem 5.1 and Theorem 6.3 that if \(\tau=0\) and \(A_{h}=0\), then any solution initiating from \(((1\theta)V_{L}, y_{0}^{+})\) with \(y_{0}^{+}\in Y_{D}\) (or \(Y_{D}^{h}\) or \(Y_{D}^{h_{1}}\)) and \(y_{0}^{+}< b/p\) is an order1 periodic solution; if \(A_{h}\neq0\) then the unique boundary order1 limit cycle is either stable (locally or globally) or unstable. Further, according to Lemma 10.3 and Remark 10.1 it is easy to see that if \(\tau=0\) then for model (2.2) a limit cycle with order no less than 2 does not exist.
If \(\tau>0\) then it follows from Theorem 6.2 that the unique order1 limit cycle is globally stable under condition (SC_{123}), which indicates that for model (2.2) an orderk (\(k\geq2\)) limit cycle does not exist in this case.
For case (SC_{2}), if \(0<\tau<\tau_{M}\) then any solution initiating from \(((1\theta)V_{L}, y_{0}^{+})\) with \(y_{0}^{+}>0\) will experience finite statedependent feedback control actions and enter into \(\operatorname{Int}\Omega_{h}\) eventually. If so, for model (2.2) no limit cycle or periodic solution exists for this case. In the following we prove that if \(\tau\geq\tau_{M}\), then for model (2.2) no limit cycle with order no less than 3 exists for case (SC_{2}).
Therefore, for the series \(y_{k}^{+}\) we have either all \(y_{k}^{+}\geq b/p\) (i.e. \(y_{k}^{+}\geq Y_{\max}^{h}\)) for \(k\in{\mathcal{K}}\) or \({\mathcal{Y}}=\overline{{\mathcal{Y}}}\cup\underline{{\mathcal{Y}}}\). If the former case occurs, then we have all \(y_{k}^{+}> \frac{b}{p}\). It follows from Lemma 10.2 that model (2.2) does not have any limit cycle with order no less than 2. If the latter case occurs, then without loss of generality we assume \(y_{0}^{+}< Y_{\min}^{h}< b/p\) and claim that there must exist the smallest positive integer j such that \(y_{j}^{+}\leq Y_{\min}^{h}\) and \(y_{j+1}^{h}\geq Y_{\max}^{h}\). Otherwise, we have \(y_{k}^{+}\leq Y_{\min}^{h}< b/p\) for \(k\in{\mathcal{K}}\) and this is impossible based on discussions above. Therefore, \(y_{j+1}^{+}\geq Y_{\max}^{h}>b/p\) must hold true. Based on Lemma 10.4 we conclude the \(y_{k}^{+}\geq Y_{\max}^{h}\geq b/p\) for \(k\geq j+1\) and once again according to Lemma 10.2 for model (2.2) a limit cycle with order no less than 3 does not exist.
For case (SC_{11}), we can employ the same methods as those for case (SC_{2}) to prove that for model (2.2) a limit cycle with order no less than 3 does not exist. So we omit the details here.
For case (SC_{12}), it follows from Theorem 9.3 that if \(\tau_{3}^{h_{1}}\leq\tau\leq\tau_{2}^{h_{1}}\), then any solution initiating from \(((1\theta)V_{L}, y_{0}^{+})\) with \(y_{0}^{+}>0\) will experience finite statedependent feedback control actions and enter into \(\operatorname{Int}\Omega_{h_{1}} \cup \partial\Omega_{h_{1}}\) eventually. Thus, for model (2.2) no limit cycle or periodic solution exists if \(\tau_{3}^{h_{1}}\leq \tau\leq\tau_{2}^{h_{1}}\).
Therefore, the sequence \(y_{k}^{+}\) for any solution initiating from \(((1\theta)V_{L}, y_{0}^{+})\) with \(y_{0}^{+}\in[y_{m}^{+}, y_{M}^{+}]\), which experiences infinite pulse effects, satisfies \(y_{k}^{+}>Y_{\max}^{h_{1}}\) for \(k\geq1\). For the solution with \(y_{0}^{+}\notin[y_{m}^{+}, y_{M}^{+}]\), there must exist a positive integer j such that \(y_{j}^{+} \in [y_{m}^{+}, y_{M}^{+}]\) and consequently we have \(y_{k}^{+}>Y_{\max}^{h_{1}}\) for \(k\geq j+1\). According to Lemma 10.2, model (2.2) does not have any limit cycle with order no less than 3.
Thus, according to results for cases (a) and (b) that if \(\tau>\tau_{2}^{h_{1}}\) then model (2.2) does not have any orderk (\(k>2\)) limit cycle. In conclusion, we have proved that model (2.2) does not have a limit cycle with order no less than 3 for all cases, and consequently the result shown in Theorem 10.1 is true. □
Corollary 10.1
For (SC_{12}), if \(0<\tau<\tau_{3}^{h_{1}}\), then for model (2.2) there exists a unique order1 limit cycle which is globally stable with respect to the phase set \({\mathcal{N}}_{2}^{h_{1}}\).
Corollary 10.2
For model (2.3) a limit cycle with order no less than 3 does not exist.
It follows from [1] that the result shown in Corollary 10.2 for model (2.3) has also been addressed and the proof provided only for \(\tau\geq b/p\), and a conjecture for \(\tau< b/p\) has been proposed. Thus, in this paper we have solved this problem completely.
11 Multiple attractors and their basins of attraction, interior structure
Based on the key parameters θ, \(V_{L}\), and τ, we can investigate the dynamics of model (2.2) and model (2.3) in terms of different parameter spaces (i.e. (SC_{123}), (SC_{1}) and (SC_{2})) and the critical values of τ. So far, the dynamics for (SC_{1}) and (SC_{2}) have not been solved completely. For example: the global existence of order2 limit cycles and their stabilities have not been solved yet. Moreover, as mentioned in Section 9, for certain intervals of τ model (2.2) there exist multiple attractors including an order1 limit cycle and invariant set \(\operatorname{Int}\Omega_{h}\) or \(\operatorname{Int}\Omega_{h_{1}} \cup\partial\Omega_{h_{1}}\), and the question is how to determine the basins of attraction once multiple attractors exist in model (2.2). Note that for some special cases this question for model (2.3) has been discussed in [1]. Thus, we will focus on those points in this section, aiming to find all the types of multiple attractors for system (2.2) and their regions of attraction.
11.1 Multiple attractors and their basins of attraction for (SC_{2})
To address the existence of multiple attractors of model (2.2) for (SC_{2}), it follows from Table 3 that the parameter τ can be divided into two parts: (a) \(\tau\in I_{\tau}^{1}=(0, \tau_{M})\) and (b) \(\tau\in I_{\tau}^{2}=[\tau_{M}, \tau_{2}]\cup (\tau_{2}, +\infty)\). If \(\tau\in I_{\tau}^{1}\), then according to Theorem 9.1 and Corollary 9.1 the set \(\operatorname{Int}\Omega_{h}\) is a unique global attractor of model (2.2) under case (SC_{2}). Thus, we assume \(\tau\in I_{\tau}^{2}\) in this subsection. That is, we have \(\tau_{M}\leq\tau\) and in the following we consider two cases: (a) \(\tau_{M}\leq\tau< b/p\) and (b) \(\max\{\tau_{M}, b/p\}\leq\tau\).
Lemma 11.1
Proof
Denote the region \(\Omega_{Q_{1}}\) bounded by the trajectory \(\Gamma_{Q_{1}}\), two line segments \(\overline{P_{1}^{+}P_{4}^{+}}\) and \(\overline{Q_{0}Q_{1}}\) and a piece of closed trajectory \(\Gamma_{h}\), i.e. arc \(\widehat{Q_{0}P_{1}^{+}}\). Then we have the following results.
Theorem 11.1
Proof
Remark 11.1
In Theorem 6.1 of [1], only the special case (i.e. \((1\theta)V_{L}=x_{2}^{*}\)) for model (2.3) has been proven. However, in Theorem 11.1 we have proved that the results for model (2.2) hold true for all \((1\theta)V_{L}>x_{4}^{*}\) and of course hold true for model (2.3) under case (SC_{2}) and \(\tau_{M}\leq\tau< b/p\).
Lemma 11.2
For (SC_{2}), if \(\tau\in[\tau_{M}, b/p)\), then there exists a \(\tau_{2}^{*}\in(\tau_{M}, b/p)\) such that model (2.2) has an order2 limit cycle provided \(y_{3}(b/p)> Y_{\max}^{h}\frac{b}{p}=\tau_{M}\).
By using methods similar to those used in Theorem 11.1, we have the following results.
Theorem 11.2
Therefore, to address the existence of the attractor for cases (SC_{2}) with \(\tau\in[\tau_{M}, b/p)\) (i.e. case (a)) completely, we need to discuss the following three subcases: (a_{1}) \(y_{3}(b/p)> Y_{\max}^{h}\frac{b}{p}=\tau_{M}\) and \(\tau\in[\tau_{M}, \tau_{2}^{*})\); (a_{2}) \(y_{3}(b/p)> Y_{\max}^{h}\frac{b}{p}=\tau_{M}\) and \(\tau\in [\tau_{2}^{*}, b/p)\); (a_{3}) \(y_{3}(b/p)< Y_{\max}^{h}\frac{b}{p}=\tau_{M}\). By using the same methods as those in Theorem 11.1 the three subcases can be studied and the attractors and their regions of attraction can be obtained similarly, so we omit them here.
Case (b): \(\max\{\tau_{M}, b/p\}\leq\tau\).
To discuss the existence of the attractors and their regions of attraction, we consider the following two subcases: (b_{1}) \(\max\{\tau_{M}, b/p, Y_{\max}^{h}\}\leq\tau\); (b_{2}) \(\max\{\tau_{M}, b/p\}\leq\tau< Y_{\max}^{h}\).
Therefore, for subcase (b_{1}), we can measure off \(PP_{2}^{+}\) on \(L_{5}\) equal to τ. There exists a trajectory through \(P_{2}^{+}=((1\theta)V_{L}, \tau)\) that intersects with the line \(L_{4}\) at point \(Q_{2}=(V_{L},y_{2})\).
Note that any trajectory initiating from \({\mathcal{N}}_{2}^{h}\) either stays in the positive invariant \(\Omega_{b_{1}}\) or jumps into it after a single impulsive effect. This implies that the horseshoelike positive invariant set \(\Omega_{b_{1}}\) is an attractor whose region of attraction is \({\mathcal{N}}_{2}^{h}\), as shown in Figure 18(A). Therefore, we have the following results for subcase (b_{1}).
Theorem 11.3
For (SC_{2}), if \(\max\{\tau_{M}, b/p, Y_{\max }^{h}\}\leq\tau\), then the horseshoelike positive invariant set \(\Omega_{b_{1}}\) is an attractor whose region of attraction is the set \({\mathcal{N}}_{2}^{h}\).
The interesting question arising here is what the interior structure of the horseshoelike positive invariant set \(\Omega_{b_{1}}\) is, and we have the following main results.
Theorem 11.4
 (i)
\(\Pi_{z_{0}^{+}}\) is an order1 limit cycle;
 (ii)
\(\Pi_{z_{0}^{+}}\) is an order2 limit cycle;
 (iii)
\(\lim_{t\rightarrow \infty}\rho(\Pi_{z_{0}^{+}}(t)O_{1})=0\);
 (iv)
\(\lim_{t\rightarrow \infty}\rho(\Pi_{z_{0}^{+}}(t)O_{2})=0\),
Proof
Therefore, according to the uniqueness of \(y^{*}\) we have either \(y_{1}^{*}=y^{*}=y_{2}^{*}\) or \(y_{1}^{*}>y^{*}>y_{2}^{*}\). If the former case occurs, then the trajectory \(\Pi_{z_{0}^{+}}(t)\) tends to the order1 limit cycle; if the later case occurs, then the trajectory \(\Pi_{z_{0}^{+}}(t)\) tends to an order2 limit cycle. □
It follows from the relations discussed in Section 5.1 and the necessary conditions of the existence of an order2 limit cycle discussed in Section 8.2 that we have the following results.
Corollary 11.1
For (SC_{2}), if \(\max\{\tau_{M}, b/p, Y_{\max}^{h}\}\leq \tau\leq\tau_{2}\), then the unique order2 limit cycle of model (2.2) is globally stable with respect to the phase set \({\mathcal{N}}_{2}^{h}\).
Proof
It follows from Figure 6 and the relations discussed in Section 5.1 that if the conditions of Corollary 11.1 hold, then for (SC_{2}) the order1 limit cycle is unstable. Thus, if the order2 limit cycle is unique, then it follows from the proof of Theorem 11.4 that the results of Corollary 11.1 are true. □
Corollary 11.2
For (SC_{2}), if \(\max\{\tau_{M}, b/p, Y_{\max}^{h}\}\leq \tau\) and model (2.2) does not have any order2 periodic solution (i.e. \(y_{1}^{*}=y_{2}^{*}\)), then the order1 limit cycle is globally stable with respect to the phase set \({\mathcal{N}}_{2}^{h}\).
For subcase (b_{2}), as shown in Figure 18(B)(D), connect \(Q_{3}\) and \(P_{3}^{+}=((1\theta)V_{L}, \frac{b}{p}+\tau)\), draw the line \(\overline{P_{1}^{+}Q_{2}}\) such that \(\overline{Q_{3}P_{3}^{+}}\parallel \overline{Q_{2}P_{1}^{+}}\). Then we have the following three possibilities:
Note that any trajectory \(\Pi_{z_{0}^{+}}(t)\) with \(z_{0}^{+}\) lying in the line segment \(\overline{P_{3}^{+}P_{4}^{+}}\) will jump into the horseshoelike positive invariant set \(\Omega_{b_{21}}\) after one impulsive effect, and any trajectory \(\Pi_{z_{0}^{+}}(t)\) with initial point above the point \(P_{4}^{+}\) will jump into the interior of closed curve \(\Gamma_{h}\) after one impulsive effect and then be free from impulsive effects.
(b_{22}): \(Q_{1}\) coincides with \(Q_{2}\), as shown in Figure 18(C). By the same method as subcase (b_{21}) we can show that the horseshoelike set \(\Omega_{b_{22}}\) is a positive invariant set whose boundary is an order2 limit cycle or periodic solution. Moreover, no other trajectory enters into the interior of this invariant set from outside.
(b_{23}): \(Q_{1}\) lies below \(Q_{2}\), as shown in Figure 18(D). For this case, we cannot separate the attractors into two subsets as those shown in subcases (b_{21}) and (b_{22}).
The interior structures of the positive invariant sets \(\Omega_{b_{21}}\) and \(\Omega_{b_{22}}\) can be addressed and the results are the same as those shown in Theorem 11.4 can be obtained similarly. For more detailed analyses, please see reference [1].
11.2 Multiple attractors and their basins of attraction for (SC_{11}) and (SC_{12})
Based on the previous investigations, for the existence of multiple attractors of both (SC_{11}) and (SC_{12}) we only need to study cases when \(\tau>\tau_{2}^{h_{1}}\), and then two subcases should be considered, i.e. \(\tau_{2}^{h_{1}}<\tau\leq b/p\) and \(\max \{\tau_{2}^{h_{1}}, b/p \}<\tau\). Moreover, the latter case \(\max \{\tau_{2}^{h_{1}}, b/p \}<\tau\) can be separated into two subcases: (c_{1}) \(\max \{\tau_{2}^{h_{1}}, b/p, Y_{\max}^{h_{1}} \}<\tau\); (c_{2}) \(\max \{\tau_{2}^{h_{1}}, b/p \}< \tau<Y_{\max}^{h_{1}}\). These can be investigated by using the same methods as those in Section 12, and similar results could be obtained, so we omit them here.
12 Discussion
In order to describe the human actions for real word applications such as pest or virus control and disease treatment, impulsive semidynamic systems can be used, which can provide a natural description for threshold control strategies. It is quite challenging to apply the qualitative theorems of impulsive semidynamic systems to investigate real life problems completely, although some special cases of model (2.2) (say \(\omega=0\) and \(q=0\)) have been investigated [1, 4, 58]. In particular, the existence of an order1 limit cycle and its local stability, and the nonexistence of limit cycles with order no less than 3 have been studied. Moreover, the methods developed in [1, 4] have been used to investigate different models arising from several application domains including chemostat cultures [6, 43, 66], epidemiology [19, 30], and IPM strategies [72, 73]. But only very special cases such as any solution that experiences an infinite number of pulse actions have been addressed. That is, few modeling studies have been completed for all possible dynamics of models with statedependent feedback control due to the complexity [1].
Therefore, a commonly used mathematical model with statedependent feedback control has been proposed and analyzed here by employing the definition and properties of impulsive semidynamical systems. The main purpose was to develop novel analytical techniques and to provide comprehensive qualitative analyses for all possible dynamics on the whole parameter space, of particular interest being the effects of the key parameters related to integrated control tactics on the dynamic behavior.
To achieve our aims, we employed the definition of the Lambert W function and its properties and the first integral of ODE model (3.1). The exact analytical formula of the Poincaré map determined by the impulsive point series in the phase set and its domain for each case has been provided. The key points are: (i) The impulsive set and phase set have been analyzed and determined firstly on different parameter spaces, please see Table 1 for details; (ii) The effects of key parameters θ, τ and \(V_{L}\) on the signs of \(A_{h}\) and \(A_{h_{1}}\), and consequently on the domains of the Poincaré map have been completely addressed, as shown in Table 2; (iii) The different parameter spaces for the existence, uniqueness and local stability of the order1 limit cycle have been provided completely, as shown in Table 3. We realize that the above three points are the basis for solving all of the dynamic behavior of model (2.2).
Based on different parameter spaces defined in Table 3, the proof of the global stability of the order1 limit cycle with respect to the basic phase set is possible, and our results show that the local stability of an order1 limit cycle indicates the global stability for case (SC_{123}). In particular, the sufficient conditions for the global stability of the boundary order1 limit cycle have been obtained for the first time, which can be used to compare the efficiency of a single control tactic alone with the efficiency of more than one integrated control measure. Further, the existence of an order2 limit cycle can be determined by the flip bifurcation. Although it is hard to find generalized conditions for the existence of an order2 limit cycle, the necessary conditions for the existence of an order2 limit cycle have been investigated in more detail, which can be used to address the nonexistence of an order2 limit cycle. Moreover, the sufficient conditions for any trajectory initiating from the phase set which will be free from impulsive effects after finite statedependent feedback control actions were studied, and the results show that the orderk (\(k\geq3\)) limit cycle does not exist and so one open problem in reference [1] has been solved here. Finally, multiple attractors and their basins of attraction and the interior structure of horseshoelike attractors have been investigated.
Compared with the previous studies mentioned in the introduction, we can see that the innovative analytical techniques developed in this paper are as follows: (i) Exact domains for impulsive and phase sets; (ii) The definition of the Poincaré map in the phase set; (iii) Methods for proving the global stability of the order1 limit cycle including the boundary order1 limit cycle; (iv) The necessary condition for the existence of an order2 limit cycle; (v) Finite statedependent feedback control actions for all cases have been addressed; (vi) The nonexistence of limit cycles with order no less than 3 has been shown. We believe that these methods could easily be employed to study more generalized models with statedependent feedback control.
The models with statedependent feedback control cannot only provide natural descriptions of real life problems, but can also result in the rich dynamics of models. Our results have provided some fundamental theoretical conclusions that could be of applied importance to real life problems. For instance, under some conditions any solution of our main model (2.2) will jump into a positive invariant set and then stabilize with an order 1 or order 2 limit cycle or become free from impulsive effects. At this stage, the system becomes inert with respect to further impulsive effects and so, in theory, the control purposes can be successfully achieved by a sequence of one, two or a few impulsive actions or, alternatively, by periodic interventions. Note that the analytical formula for the period can be calculated based on the initial values by employing the same methods as those used in [2].
Although it is reasonable to assume that the carrying capacity of the pest population could be infinity due to the threshold level considered in the model being quite small compared with the carrying capacity, the disadvantages of this work are: (i) the Lambert W function and its properties are needed for defining the Poincaré map; and (ii) the first integral of the ODE model plays a key role in most of the results. Therefore, if the carrying capacity is a constant rather than +∞, then the first integral of the generalized model does not exist any more, and consequently the Lambert W function cannot be used. Thus, the question is how to extend all analytical techniques developed in this paper to investigate more generalized models with statedependent feedback control. For our near future work, we will focus on model (2.2) with a constant carrying capacity and different releasing strategies and other models arising from different application fields.
Declarations
Acknowledgements
We would like to thank Prof. Chengzhi Li for helping us to prove Theorem 3.1 that greatly improved the presentation of this paper. This work was partially supported by the National Natural Science Foundation of China (NSFCs 1171199, 11471201), and by the Fundamental Research Funds for the Central Universities (GK201003001, GK201401004).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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