Sustained oscillation induced by time delay in a commodity market model
- Kejun Zhuang^{1, 2}Email author and
- Gao Jia^{3}
https://doi.org/10.1186/s13662-017-1113-6
© The Author(s) 2017
Received: 7 September 2016
Accepted: 3 February 2017
Published: 21 February 2017
Abstract
In this paper, the existence of local and global Hopf bifurcation for a delay commodity market model is studied in detail. As time delay increases, the commodity price will fluctuate periodically. Furthermore, such fluctuations will occur even if the time delay is sufficiently large.
Keywords
commodity market model stability global Hopf bifurcationMSC
91B55 34K181 Introduction
In most economic and financial processes, mathematical modeling leads to nonlinear delayed dynamical systems, and the interplay of delayed and nonlinear effects is important for many reasons. Capturing the price behavior of commodity and forecasting future developments are essential in commodity management and international policy. Given this, fluctuations in commodity price have long been, and will continue to be, one of the dominant topics in mathematical economics due to its universal existence and importance. Based on some mathematical assumptions, various price adjustment models have been developed to analyze the problems in economics, see [1–4] and the references cited therein.
However, the dynamic behaviors of system (2) still need further investigation. In this paper, we are trying to improve the understanding of the complex dynamics induced by time delay. Motivated by the conjecture on global bifurcation results in [8], we shall focus on the global continuation of a local Hopf bifurcation. In the following sections, the stability of a unique positive equilibrium and a local Hopf bifurcation analysis for system (2) are presented. After that, the global existence of bifurcating periodic solutions is explored with the assistance of global Hopf bifurcation theory developed by Wu [13], and related applications can be found in [14–17]. Finally, some numerical simulations are performed to illustrate the theoretical results.
2 Local Hopf bifurcation analysis
Due to the above inequality, we can deduce that the number of characteristic roots with positive real parts will increase by two when time delay τ passes the critical values \(\tau_{k}\) each time.
Through the above analysis, we can determine the distribution of roots of (4) as follows.
Lemma 1
- (i)
If \(\beta\leq\alpha\), then all roots of (4) have negative real parts for any \(\tau\geq0\).
- (ii)
If \(\beta>\alpha\), then (4) has a pair of imaginary roots \(\pm i\omega_{0}\) when \(\tau=\tau_{k}\) (\(k=0,1,2,\ldots\)).
- (iii)
If \(\beta>\alpha\), then all roots of (4) have negative real parts only when \(\tau\in[0,\tau_{0})\). Equation (4) has \(2(k+1)\) roots with positive real parts when \(\tau\in(\tau_{k},\tau_{k+1}]\).
According to the results regarding the stability of equilibrium in [19], we have the following theorem about the stability of positive equilibrium and the existence of a local Hopf bifurcation.
Theorem 1
- (i)
if \(\beta\leq\alpha\), then the positive equilibrium \(P_{\ast}\) is asymptotically stable;
- (ii)
if \(\beta>\alpha\), then the positive equilibrium \(P_{\ast}\) is stable when \(\tau<\tau_{0}\) and unstable when \(\tau>\tau_{0}\). Moreover, a Hopf bifurcation occurs at the critical value \(\tau_{k}\), and periodic solutions will bifurcate from \(P_{\ast}\).
By using the normal form theory and the center manifold theorem developed by Hassard [20] and the calculation methods in [21], we can determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions. Next, we shall briefly give the formulae for determining the bifurcation properties at the first bifurcation value \(\tau_{0}\).
It is well known that \(\mu_{2}\) determines the direction of the Hopf bifurcation: the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for \(\tau>\tau_{0}\) (\(\tau<\tau _{0}\)) if \(\mu_{2}>0\) (\(\mu_{2}<0\)); \(\beta_{2}\) determines the stability of bifurcating periodic solutions: the bifurcating periodic solutions are orbitally asymptotically stable (unstable) if \(\beta_{2}<0\) (\(\beta_{2}>0\)); \(T_{2}\) determines the period of the bifurcating periodic solutions: the period increases (decreases) if \(T_{2}>0\) (\(T_{2}<0\)).
It is not difficult to find that the derivatives of demand and supply functions at the positive equilibrium \(P_{\ast}\) have significant effects on the properties of a local Hopf bifurcation.
3 Global bifurcation analysis
It is known that periodic solutions through Hopf bifurcation are generally local and only exist in a small neighborhood of the critical value. Hence it is interesting and significant to verify the global existence of bifurcating periodic solutions. In this section, we study the global continuation of periodic solutions bifurcating from the positive equilibrium \(P_{\ast}\) of system (2).
Following the work of [13], we need to show the uniform boundedness of the periodic solutions of (2) and the nonexistence of τ-periodic solutions.
Lemma 2
If \(a>bc\), and m and n are even integers, then all periodic solutions of (2) are uniformly bounded.
Proof
Lemma 3
If m and n are even, then system (2) has no τ-periodic solution.
Proof
Because m and n are even integers, system (5) has three steady states \(P(t)=0\), \(P(t)=P_{\ast}\) and \(P(t)=-P_{\ast}\). In system (5), \(\dot{P}(t)<0\) holds when \(P(t)>P_{\ast}\) or \(-P_{\ast}< P(t)<0\), and \(\dot{P}(t)>0\) holds when \(0< P(t)< P_{\ast}\) or \(P(t)<-P_{\ast}\). Therefore, the ordinary differential equation (5) does not have a nonconstant periodic solution. This implies that system (2) has no τ-periodic solution. The proof is complete. □
We then have the following theorem about the global existence of a Hopf bifurcation.
Theorem 2
Suppose that \(a>bc\), \(\beta>\alpha\), and m and n are even. Then, for each \(\tau>\tau_{k}\), \(k=0,1,2,\ldots\) , system (2) still has positive periodic solutions.
Proof
Define that \(p_{0}=2\pi/\omega_{0}\) and \(\varOmega _{\varepsilon,p_{0}}=\{ (0,p):0< u<\varepsilon,|p-p_{0}|<\varepsilon\}\). If \(|\tau-\tau_{k}|\leq\delta\) and \((u,p)\in\partial \varOmega _{\varepsilon}\), then \(\varDelta _{(x_{0},\tau,p)}(u+2\pi i/p)=0\) if and only if \(\tau=\tau_{k}\), \(u=0\), \(p=p_{k}\). Thus, assumption (A_{4}) in [13] holds.
For a contradiction, we suppose that the projection of \(\mathbb {C}(x_{0},\tau_{k},2\pi/\omega_{0})\) onto τ-space is bounded. This means that the projection of \(\mathbb{C}(x_{0},\tau_{k},2\pi/\omega_{0})\) onto τ-space is included in an interval \((0, \tau^{\ast})\).
4 Numerical examples
Thus, our theoretical results and numerical simulations show that the time delay has a substantial effect on the periodic dynamic behaviors in commodity market model (2).
5 Conclusions
This paper presents the results of an investigation into the existence of local and global Hopf bifurcations for a price adjusting model with time delay. It can be concluded that time delay may destabilize the equilibrium of that model and induce periodic oscillations. Moreover, the periodic oscillations will persist even when the delay is sufficiently large, which indicates the global existence of a Hopf bifurcation in the model. Thus, the results obtained here can supplement the previous literature and help people to understand price fluctuation mechanisms.
However, from another perspective, we sometimes need to control commodity price fluctuations, and one effective method is to shorten the time between the initiation of changes in production and the final alteration of supply. More specifically, the finite delay τ is the time between production and price changes. As we know, large delay may induce complex dynamical behaviors, such as drastic periodic fluctuations. Therefore, timely price adjustments are necessary, which can effectively reduce the time delay. Mathematically, we should stabilize the positive equilibrium and control the Hopf bifurcation, and we will consider this in our work in the near future.
Declarations
Acknowledgements
This work is supported by the Key Project for Excellent Young Talents Fund Program of Higher Education Institutions of Anhui Province (gxyqZD2016100) and the Anhui Provincial Natural Science Foundation (1508085MA09 and 1508085QA13).
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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