Hopf bifurcation and chaos analysis of Chen’s system with distributed delays

doi:10.1016/j.chaos.2004.11.007

Chaos, Solitons & Fractals

Volume 25, Issue 1, July 2005, Pages 197-220

https://doi.org/10.1016/j.chaos.2004.11.007 Get rights and content

Abstract

In this paper, a general model of nonlinear systems with distributed delays is studied. Chen’s system can be derived from this model with the weak kernel. After the local stability is analyzed by using the Routh–Hurwitz criterion, Hopf bifurcation is studied, where the direction and the stability of the bifurcating periodic solutions are determined by using the normal form theory and the center manifold theorem. Some numerical simulations for justifying the theoretical analysis are also presented. Chaotic behavior of Chen’s system with the strong kernel is also found through numerical simulation, in which some waveform diagrams, phase portraits, and bifurcation plots are presented and analyzed.

Introduction

In 1963, Lorenz found the first chaotic attractor in a three-dimensional (3D) autonomous system when he studied atmospheric convection [16]. As the first chaotic model, the Lorenz system has become a paradigm of chaos research. Notably, over the last two decades, chaos in engineering systems, such as nonlinear circuits, has gradually moved from simply being a scientific curiosity to a promising subject with practical significance and applications. It has been noticed that purposefully creating chaos can be a key issue in many technological applications. In this pursuit, Chen constructed a 3D autonomous chaotic system from an engineering feedback control approach [4], [17], [21], followed by a closely related Lü system [13], and a unified system that combines the three systems as its special cases [14]. According to the canonical-form classification of Vanecek and Celikovský [18], in the linear part A = [a_ij] of 3D autonomous systems with quadratic nonlinearities, the Lorenz system satisfies a₁₂a₂₁ > 0, the Chen system satisfies a₁₂a₂₁ < 0, while the Lü system satisfies a₁₂a₂₁ = 0, and they are not topologically equivalent. In this sense, they together constitute a complete family of generalized Lorenz dynamical systems.

Over the last three years, there have been some detailed investigations on Chen’s system [1], [2], [3], [9], [15], [17], [19], [20], [21]. In particular, Hopf bifurcation analysis of Chen’s system has been carried out [7], [9], [15], [17]. On the other hand, dynamical systems with distributed delays have been studied for population dynamics and neural networks [6], [8], [10], [11], [12]. In this paper, a general model of nonlinear systems with distributed delays is further studied. Chen’s system can be derived from this model with a weak kernel. After the linear stability is analyzed by using the Routh–Hurwitz criterion, Hopf bifurcation is studied, where the direction and the stability of the bifurcating periodic solutions are determined by using the normal form theory and the center manifold theorem. Some numerical simulations for justifying the theoretical analysis are also presented. Chaotic behavior of Chen’s system with a strong kernel is also simulated, where some waveform diagrams, phase portraits, and bifurcation plots are presented and analyzed.

More precisely, consider the following system: $\{\begin{matrix} {\dot{x}}_{1} (t) = (c - a) \int_{- \infty}^{0} x_{1} (τ) k (t - τ) d τ - x_{2} (t) \int_{- \infty}^{0} x_{1} (τ) k (t - τ) d τ + {cx}_{1} (t), \\ {\dot{x}}_{2} (t) = x_{1} (t) \int_{- \infty}^{0} x_{1} (τ) k (t - τ) d τ - {bx}_{2} (t) \end{matrix}$ in which a, b and c are positive constants and the delay kernel k is assumed to satisfy the following properties: $k : [0, \infty) \to [0, \infty), k is piecewise continuous$ and $\int_{0}^{\infty} k (s) d s = 1; \int_{0}^{\infty} sk (s) d s < \infty .$ It is also assumed that system (1) is supplemented with initial conditions of the form $\begin{matrix} x_{1} (s) = ϕ_{1} (s), \\ x_{2} (s) = ϕ_{2} (s), \end{matrix} s \in (- \infty, 0], ϕ_{1}, ϕ_{2} are bounded continuous on [0, \infty) .$ In system (1), if the weak kernel k of the form $k (s) = a e^{- as}$ is used, where a is a parameter varying in (0, ∞), which denotes the decay rate of the effect of the past memories, then system (1) becomes the following Chen’s system: $\{\begin{matrix} {\dot{x}}_{1} (t) = (c - a) x_{3} (t) - x_{2} (t) x_{3} (t) + {cx}_{1} (t), \\ {\dot{x}}_{2} (t) = x_{1} (t) x_{3} (t) - {bx}_{2} (t), \\ {\dot{x}}_{3} (t) = a (x_{1} (t) - x_{2} (t)) . \end{matrix}$ If, instead, the strong kernel of the form $k (s) = a^{2} s e^{- as}$ is used, then system (1) becomes $\{\begin{matrix} {\dot{x}}_{1} (t) = (c - a) x_{4} (t) - x_{2} (t) x_{4} (t) + {cx}_{1} (t), \\ {\dot{x}}_{2} (t) = x_{1} (t) x_{4} (t) - {bx}_{2} (t), \\ {\dot{x}}_{3} (t) = a (x_{1} (t) - x_{3} (t)), \\ {\dot{x}}_{4} (t) = a (x_{3} (t) - x_{4} (t)) . \end{matrix}$

The present interest in particular is to consider Hopf bifurcation to periodicity as a is varied in system (1). The local nature of bifurcation and the orbital asymptotic stability of the bifurcating periodic solutions etc. are of concern. Although there are several investigations dealing with local Hopf bifurcation by using b as the bifurcation parameter in Chen’s system (4), the stability of the local bifurcating solutions has been rarely discussed [7], [9], [15], [17].

The rest of this paper is organized as follows. In Section 2, local stability property of model (1) is discussed and some sufficient conditions for the stability are derived based on the Routh–Hurwitz criterion. In Section 3, model (1) with the general kernel is further studied, where both the direction and stability of Hopf bifurcation are analyzed by the normal form theory and the center manifold theorem [5], with some criteria for the stability derived. In Section 4, some specific results for the case of a weak kernel are obtained by applying the results of Section 3. Numerical simulations will be shown, justifying the theoretical results reported and discovering some specific results for the case with a strong kernel. By varying the bifurcation parameter, a new chaotic system is found and reported in Section 5. Finally, conclusions are drawn with further research directions given in Section 6.

Section snippets

Local stability analysis

In this section, consider the local stability of the equilibrium solutions of system (1).

From the special nature of the delay kernel (2) embedded in system (1), it is founded that an equilibrium solution ( $x_{1}^{*}, x_{2}^{*}$ ) of (1) is given by the solution of $\{\begin{matrix} (c - a) x_{1}^{*} - x_{2}^{*} x_{1}^{*} + {cx}_{1}^{*} = 0, \\ (x_{1}^{*})^{2} - {bx}_{2}^{*} = 0 . \end{matrix}$

The following observations regard the basic dynamical behaviors of system (1):

Lemma 1

[4], [15]

If a ⩾ 2c, then system (1) has only one equilibrium point, S₀ = (0, 0); if a < 2c, then system (1) has three equilibrium points: $S_{0} = (0, 0), S_{+} = (\sqrt{b (}$

Stability of bifurcating periodic solutions: the general kernel case

In this section, the stability of the bifurcating periodic solutions system (1) with the kernel satisfying (2) is studied.

For convenience in the study of the Hopf bifurcation problem, first transform system (9) into an operator equation of the form $\frac{d y_{t}}{d t} = A (μ) y_{t} + {Ry}_{t},$ where y = col(y₁, y₂), y_t = y(t + θ), θ ∈ (−∞, 0), μ = a − a₀, and operators A and R are defined as $A (μ) ϕ (θ) = \{\begin{matrix} \frac{d ϕ (θ)}{d θ}, & - \infty < θ < 0, \\ L ϕ (θ) + \int_{- \infty}^{0} F (s) ϕ (s) d s, & θ = 0, \end{matrix}$ $R ϕ (θ) = \{\begin{matrix} (\begin{matrix} 0 \\ 0 \end{matrix}), & - \infty < θ < 0, \\ (\begin{matrix} f_{1} \\ f_{2} \end{matrix}), & θ = 0, \end{matrix}$ where $\begin{matrix} f_{1} = - ϕ_{2} (0) \int_{- \infty}^{0} k (- s) ϕ_{1} (s) d s, \\ f_{2} = ϕ_{1} (0) \int_{- \infty}^{0} k (- s) ϕ_{1} (s) d s, \end{matrix}$ with L and F being

Stability of bifurcating periodic solutions: the weak kernel case

In the case of the weak kernel, i.e., k(s) = a e^−as, it follows from (33), (37), (38) that $\begin{matrix} B = \frac{\sqrt{b (2 c - a_{0})} (2 a_{0} + i ω_{0})}{(a_{0} + i ω_{0}) (b + i ω_{0})}, \\ C = - \frac{2 c - a_{0}}{b - i ω_{0}}, \\ \bar{E} = \frac{1}{1 + \bar{C} B + (c - \sqrt{b (2 c - a_{0})} \bar{C}) \frac{a_{0}}{(a_{0} + i ω_{0})^{2}}}, \end{matrix}$ where $\begin{matrix} ω_{0} = \sqrt{bc}, \\ a_{0} = \frac{3 c + \sqrt{17 c^{2} - 8 bc}}{4}, \\ or a_{0} = \frac{3 c - \sqrt{17 c^{2} - 8 bc}}{4} . \end{matrix}$ From the above discussion, one obtains $w_{11}^{(1)} (0) = \frac{1}{2 b (2 c - a)} \{- {bc}_{11}^{(1)} - \sqrt{b (2 c - a)} c_{11}^{(2)}\},$ where $\{\begin{matrix} c_{11}^{(1)} = H_{11}^{(1)} (0) + i ω_{0} c + \frac{p_{1}}{a + i ω_{0}} - \frac{p_{2}}{a - i ω_{0}}, \\ c_{11}^{(2)} = H_{11}^{(2)} (0) - i ω_{0} \sqrt{b (2 c - a)} + \frac{p_{1}}{a + i ω_{0}} - \frac{p_{2}}{a - i ω_{0}} \end{matrix}$ and $\begin{matrix} H_{11}^{(1)} (0) = - \frac{a^{2} (B + \bar{B}) + i ω_{0} a (B - \bar{B})}{a^{2} + ω_{0}^{2}}, \\ H_{11}^{(2)} (0) = \frac{2 a}{a^{2} + ω_{0}^{2}}, \\ p_{1} = - \bar{E} [(B - \bar{C}) \frac{a}{i ω_{0} (a - i ω_{0})} + (\bar{B} - \bar{C}) \frac{a}{i ω_{0} (a + i ω_{0})}], q_{1} = p_{1} B, \\ p_{2} = E [(\bar{B} -] \end{matrix}$

Stability of bifurcating periodic solutions: the strong kernel case

We consider system (1) with the strong kernel, i.e., k(s) = a²s e^−as. It follows from Eqs. (33), (37), (38) that $\begin{matrix} B = \frac{\sqrt{b (2 c - a_{0})} [a_{0}^{2} + (a_{0} - i ω_{0})^{2}]}{(a_{0} - i ω_{0})^{2} (b + i ω_{0})}, \\ C = - \frac{2 c - a_{0}}{b - i ω_{0}}, \\ \bar{E} = \frac{1}{1 + \bar{C} B + 2 (c - \sqrt{b (2 c - a_{0})} \bar{C}) \frac{a_{0}^{2}}{(a_{0} + i ω_{0})^{3}}}, \end{matrix}$ where the equilibrium S₊ is considered, and ω₀, a are similar to those in Eq. (72).

It follows from the above discussion that $w_{11}^{(1)} (0) = \frac{1}{2 b (2 c - a)} \{- {bc}_{11}^{(1)} - \sqrt{b (2 c - a)} c_{11}^{(2)}\},$ where $\{\begin{matrix} c_{11}^{(1)} = H_{11}^{(1)} (0) + c \{\frac{p_{1} [(a - i ω_{0})^{2} - a^{2}]}{(a - i ω_{0})^{2}} + \frac{p_{2} [(a + i ω_{0})^{2} - a^{2}]}{(a + i ω_{0})^{2}}\}, \\ c_{11}^{(2)} = H_{11}^{(2)} (0) - \sqrt{b (2 c - a)} \{\frac{p_{1} [(a - i ω_{0})^{2} - a^{2}]}{(a - i ω_{0})^{2}} + p_{2} [(a + i ω_{0})^{2} -\} \end{matrix}$

Conclusions

In this paper, the chaotic Chen’s system has been generalized to a model with distributed delays. With the weak kernel, this new system reduces to the original Chen’s system. By using the average time delay as the bifurcation parameter, it has been shown that Hopf bifurcation occurs when this parameter passes through a critical value. This means that a class of periodic orbits bifurcates from the corresponding equilibrium. Both the stability and the direction of the bifurcating periodic orbits

Acknowledgement

The work described in this paper was supported by grants from the National Natural Science Foundation of China (No. 60271019), the Doctorate Foundation of the Ministry of Education of China (No. 20020611007), the Hong Kong Research Grants Council (CityU 1115/03E) and the Natural Science Foundation of Chongqing.

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View full text

Hopf bifurcation and chaos analysis of Chen’s system with distributed delays

Abstract

Introduction

Section snippets

Local stability analysis

[4], [15]

Stability of bifurcating periodic solutions: the general kernel case

Stability of bifurcating periodic solutions: the weak kernel case

Stability of bifurcating periodic solutions: the strong kernel case

Conclusions

Acknowledgement

Phys Lett A

Chaos, Solitons & Fractals

Chaos, Solitons & Fractals

Chaos, Solitons & Fractals

Physica D

Neural Networks

On a generalized Lorenz canonical form of chaotic systems

Int J Bifurcat Chaos

From chaos to order: methodologies, perspectives and applications

Yet another chaotic attractor

Int J Bifurcat Chaos

Theory and applications of Hopf bifurcation