Rank one strange attractors in periodically kicked Chua’s system with time delay
- Wenjie Yang^{1},
- Yiping Lin^{1}Email author,
- Yunxian Dai^{1} and
- Yusheng Jia^{2}
https://doi.org/10.1186/s13662-015-0397-7
© Yang et al.; licensee Springer. 2015
Received: 10 October 2014
Accepted: 30 January 2015
Published: 4 March 2015
Abstract
In this paper, rank one strange attractor in a periodically kicked time-delayed system is investigated. It is shown that rank one strange attractors occur when the delayed system under a periodic forcing undergoes Hopf bifurcation. Our discussion is based on the theory of rank one maps formulated by Wang and Young. As an example, periodically kicked Chua’s system with time-delay is considered, conditions for rank one chaos along with the results of numerical simulations are presented.
Keywords
1 Introduction
Recently, a chaos theory on rank one maps has been developed by Wang and Young. In 2001, Wang and Oksasoglu [1] gave simple conditions that guarantee the existence of strange attractors with a single direction of instability and certain controlled behaviors. In 2008, Wang and Young accomplished a more comprehensive understanding of the complicated geometric and dynamical structures of a specific class of non-uniformly hyperbolic homoclinic tangles. For certain differential equations, through their well-defined computational process, the existence of the indicated phenomenon of rank one chaos was verified [2]. In 2009, Chen and Han studied the existence of rank one chaos in a periodically kicked planar equation with heteroclinic cycle [3]. In 2012, Fang studied the synchronization between rank-one chaotic systems without and with delay using linear delayed feedback control method [4].
This paper is organized as follows. In Section 2, we give preliminaries about the rank one chaotic theory. In Section 3, we derive the rank one chaotic theory for a time-delayed system. In Section 4, we take Chua’s system as an example. In Section 5, numerical simulations are presented. Conclusions are given in Section 6.
2 Preliminaries
To properly motivate the studies presented in this paper, we first give a brief overview on the studies of rank one strange attractors, which can be constructed in the following way.
Firstly, we give several definitions.
Definition 1
[8]
Let \(f:M\rightarrow M\) be a diffeomorphism of a compact Riemannian manifold onto itself. We say that f is an Anosov diffeomorphism if the tangent space at every \(x \in M\) is split into \(E^{u}(x) \oplus E^{s}(x)\), where \(E^{u}\) and \(E^{s}\) are Df-invariant subspaces, \(Df|_{E^{u}}\) is uniformly expanding and \(Df|_{E^{s}}\) is uniformly contracting. A compact f-invariant set \(\Lambda\subset M\) is called an attractor if there is a neighborhood U of Λ called its basin such that \(f^{n} x \rightarrow \Lambda\) for every \(x \in U\). Λ is called an Axiom A attractor if the tangent bundle over Λ is split into \(E^{u} \oplus E^{s}\) as above.
Definition 2
[8]
- (i)
μ has absolutely continuous conditional measures on unstable manifolds;
- (ii)where \(h_{\mu}(f)\) is the metric entropy of f;$$h_{\mu}(f)=\int\bigl|\operatorname{det}(Df|_{E^{u}})\bigr|\,d\mu, $$
- (iii)there is a set \(V \subset U\) having full Lebesgue measure such that for every continuous observable \(\varphi: U \rightarrow R\), we have, for every \(x \in V\),$$\frac{1}{n}\sum^{n-1}_{i=0}\varphi \bigl(f^{i}x\bigr)\rightarrow\int\varphi \,d\mu; $$
- (iv)
μ is the zero-noise limit of small random perturbations of f.
Then the invariant measure μ is called the Sinai-Ruelle-Bowen measure, or SRB measure, of f.
Definition 3
Now, we consider (5)_{ ε }. Assume the following:
(A2) \(E(0)>0\).
- (a)
(A1)-(A2) hold for Eq. (5)_{ ε };
- (b)
\(\varphi(\theta)\) in Eq. (10) is a Morse function; and
- (c)
μ, ε are such that \(0 <\mu\ll 1\), \(0 < \varepsilon \ll 1\).
The following theorem is obtained by Wang and Oksasoglu [9].
Theorem 1
3 Rank one strange attractors of a delayed system
(B2) \(E(0)>0\).
Then we know that system (11)_{0} has a supercritical Hopf bifurcation near the equilibrium.
- (a)
(B1)-(B2) hold for Eq. (11)_{ ε };
- (b)
\(\phi(\theta)\) in Eq. (23) is a Morse function; and
- (c)
μ, ε are such that \(0 < \mu\ll 1\), \(0 < \varepsilon\ll 1\).
Then we obtain the following.
Theorem 2
4 Analysis of rank one strange attractors in delayed Chua’s system
We consider the following cases.
(1) \(E^{*} = E^{*}_{1}\)
Since \(G(0)=-b^{2}\beta_{1}^{2}<0\), \(G(+\infty)=+\infty\), we immediately obtain the following.
Lemma 1
Eq. (29) has at least one positive root since \(e_{3}<0\).
We assume that Eq. (29) has three positive roots, \(z_{1}\), \(z_{2}\), \(z_{3}\), and \(\omega_{1}=\sqrt{z_{1}}\), \(\omega_{2}=\sqrt{z_{2}}\), \(\omega_{3}=\sqrt{z_{3}}\).
In order to investigate the distribution of the roots of Eq. (25), we need to introduce the following lemma [11].
Lemma 2
Lemma 3
Suppose that \(z_{k} = \omega_{k}^{2} \) and \(G'(z_{k} ) \ne0\), then \(\frac{\operatorname{Re} \lambda (\tau_{k}^{(j)} )}{{d\tau}} \) has the same sign as \(G'(z_{k} ) \).
Proof
Now we apply the Hopf bifurcation theorem for functional differential equations [12] and obtain the following results.
Theorem 3
- (i)
when \(\tau\in[0,\tau_{0})\), the equilibrium \(E_{1}^{*}\) of system (4)_{0} is locally asymptotically stable,
- (ii)
when \(\tau>\tau_{0}\), the equilibrium \(E_{1}^{*}\) of system (4)_{0} is unstable,
- (iii)
when \(\tau=\tau_{0}\), system (4)_{0} undergoes Hopf bifurcation at \(E_{1}^{*}\).
(2) \(E^{*} = E^{*}_{2},\mbox{or }E^{*}_{3}\)
When \(E^{*} = E^{*}_{2}, \mbox{or }E^{*}_{3}\), the coefficients of Eq. (25) are \(\alpha _{1}=1-2ac\), \(\alpha_{2}=-2ac-a\), \(\beta_{1}=-2ac\). Under condition (H_{1}), Eq. (25) has at least one positive real root, so \(E^{*}_{2}\) and \(E^{*}_{3}\) are unstable equilibria.
We can get \(\langle q^{*} (s),\bar{q}(\theta) \rangle = 0\).
5 Numerical simulations
6 Conclusions
We have developed rank one theory from an ordinary differential equation to a time-delayed system and considered the existence of rank one chaos in time-delayed Chua’s system. It is shown that rank one strange attractors occur when the delayed system under a periodic kick undergoes supercritical Hopf bifurcation. We also show some results of numerical simulations to support the rank one theory.
Declarations
Acknowledgements
This research is supported by the National Natural Science Foundation of China (Nos. 11061016, 11461036).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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