Abstract
This paper deals with the dynamic behavior of the chaotic nonlinear time delay systems of general form dot{x}(t)=g(x(t),x(t-tau )). We carry out stability analysis to identify the parameter zone for which the system shows a stable equilibrium response. Through the bifurcation analysis, we establish that the system shows a stable limit cycle through supercritical Hopf bifurcation beyond certain values of delay and parameters. Next, a numerical simulation of the prototype system is used to show that the system has different behaviors: stability, periodicity and chaos with the variation of delay and other parameters, which demonstrates the validity of our method. We give the single- and two-parameter bifurcation diagrams which are employed to explore the dynamics of the system over the whole parameter space.
Highlights
For the last decades, time-delayed dynamical systems have been attracting the attention of researchers of various fields, including mathematics, biology, economics, physics, engineering, etc. [1,2,3,4]
We will carry out bifurcation analysis to study the rich dynamical behavior of the system in theory, and we will use a numerical simulation of the system to show that with the variation of delay and other system parameters, the system shows stability, periodicity and chaos
We study the dynamic behavior of system (2) by varying the time delay τ and taking γ = 1 and β = 1
Summary
Time-delayed dynamical systems have been attracting the attention of researchers of various fields, including mathematics, biology, economics, physics, engineering, etc. [1,2,3,4]. We will carry out bifurcation analysis to study the rich dynamical behavior of the system in theory, and we will use a numerical simulation of the system to show that with the variation of delay and other system parameters, the system shows stability, periodicity and chaos. Theorem 3 The equilibrium points E2,3 = 0 are asymptotically stable for τ ∈ [0, τ0) and unstable for τ > τ0, and equation (17) undergoes a Hopf bifurcation at E2,3 when τ = τk for = 0, 1, 2,. Proof This theorem can be proof by Lemma 3. When τ ≥ 0.26, the fixed point loses its stability through Hopf
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