Analysis of the Stability and Hopf Bifurcation of a Three-Dimensional System with Delays

We propose a model of three-dimensional autonomous system with delays. We explore the dynamical behavior of the proposed autonomous system by examining bifurcation diagrams, Lyapunov exponents, equilibrium and stability, and the influence of time delay on Hopf bifurcation. A bifurcation theory is used to analyze and detail the problem. In addition, the explicit algorithm that determines the direction of Hopf bifurcation, along with the stability of bifurcating periodic, has been established. Also, there are specific operating conditions that must be met in order to achieve Hopf bifurcation. In the proposed autonomous system, we analyze the procedures for designing chaotic based systems including parameter selection, discretization of the results, as well as exploring the changing regularity of the bifurcation value. A series of numerical simulations is presented to illustrate the analytical results.


Introduction
An autonomous dynamical systems are described by at least three coupled ordinary nonlinear differential equations, that chaos occurs in multiple nonlinear system when numerically computed. As these different values of parameters, variations may occur in the quantitative figure of the solutions for given parameter values. Hopf bifurcation occurs as a entangle conjugate couple of eigenvalues of the linearized system at a certain point becomes purely imaginary. These systems have a unique dynamical appearance, Hopf bifurcation can only occur in systems of dimension two or higher. Lately, researchers started bifurcation analysis and numerical computer simulation for these systems, e.g [1,2], [3,4], [5,6], references therein. In particular, chaos theory has became practicable tools in facilitation secure optical communication devices and secure data encryption.
Over the past one decade, chaos applications have been used in information processing, nonlinear circuits, and etc. Zhang

Hopf Bifurcation Analysis,Kutorzi and Shi
chaos entanglement function. The authors reviewed bifurcations of nonlinear dynamical systems in the new chaotic system [7,8], [9,10], and [11]. The existence of autonomous systems are extensively encountered in many fields such as engineering and chemistry. Delays are inevitable in autonomous activities; everything takes time, with that we proposed a delayed in Hopf bifurcation in three-dimensional based on chaos entanglement function and the results show there are four requirements that are needed to achieve chaos [12],Kutorzi and Yufeng. The stability problem of the dynamic systems with delays is of more intrigue as part of this study.
In this paper, the priority issue is the stability of dynamic systems with delays. We introduced a time delayed in Hopf bifurcation in three-dimensional based on chaos entanglement function. For autonomous models, chaotic behavior has been developed to displays dynamic behaviors. The systems are considered non-linear because of the multiple feedback between the components of the system. In addition we present a specific range of the operating conditions that are needed to achieve Hopf bifurcation.
The outline of the present paper is organized as follows. In section two and three, we foremost described the characteristics of chaotic three dimensional autonomous system. Significantly, the chaotic dynamics are all established when chaotic dynamics are effectively stabilize, explored in details, and the characteristics are obtained by employing varying subsystems. Furthermore, we study the the algorithm for establishing the direction of hopf bifurcation and as well investigated the stability of bifurcating periodic. In section four, detailed numerical simulations are presented. Finally, we present our findings and conclusion in section five.

The Proposed Three-Dimensional Autonomous
Consider 2 linear subsystems: Two-dimensional nonlinear autonomous system and onedimensional.ẋ = ax − bẏ y = cx + ay. (1) According to [2] where he extend the model to explore the dynamical properties.
Where (x, y, z) are steady variable. When a < 0, c < 0 and d < 0 are both the stable subsystems. Now consider time delay into the system (1) & (2) , the proposed three-order time delayed system could be derived thus and so: wherein τ ∈ R + is the integral time delay in system (3).
We obtain system (4) has equilibria at the fixed point E ⋆ = (0, 0, 0). Additionally, linearizing system (4) around E ⋆ = (0, 0, 0) expressed as This leads to the associated characteristic equation The manifold case of (7) with τ = 0, the characteristic equation reduces to The roots of (9) have negative real parts for every nonnegative integer and the positive equilibrium solution with restriction of time delay is asymptotically stable. With the help of Routh-Hurwitz criterion determinant structure all roots of (9) is stable if and only if the following hypothesis holds: (10) is satisfied, then the equilibrium E ⋆ of system (3) implies that appropriate parameters are chosen to admits asymptotic stability of the system for τ = 0. Now we review the effect of the distribution of characteristic roots or delay τ when τ > 0. Hopf bifurcation will occur if at least one eigenvalue has positive real parts, unstable and have negative real parts. Assume that iω(ω > 0) is a root of (7). Then ω must satisfy any nonnegative integer in the following nonlinear equation group By adding both squares, thus have For convenience, let α = ω 2 , then, (14) can be rewritten as where State the following conditions : (H2) X > 0 and Z > 0 holds, then it implies that the positive equilibrium of E ⋆ is usually asymptotically stable for any τ > 0 and that time delayed system (3) has no periodic solution.

Direction and Stability of the Hopf Bifurcation with Delay
In said section, we investigate explicit algorithm for establishing the direction of hopf bifurcation and the stability of bifurcating periodic solution. Based on the discussion in the last section, we derived some conditions under which the system (3) undergoes a Hopf bifurcation. Without the loss of generality, we considered a pair of purely imaginary eigenvalues of system (3); A 0 = {λ 1 , λ 2,3 } same as the delayed system; By applying methods from normal form and centre manifold reduction theory [14] ,we to proof more detailed information of Hopf bifurcation. Denote τ j 0 by τ * and rescaling time τ → t/τ .
According to system (7), equilibria should satisfy ax − by = 0, cx + ay = 0, As has already been mentioned in this section, we are able to apply more detailed conditions under which system undergoes Hopf bifurcation at E ⋆ ; that is the fourparameter family of differential equations (1), (2). We will also consider the direction of Hopf bifurcation at the equilibrium and stability of the bifurcating periodic solutions by using the normal form theory and center manifold for functional differential equations.
Normalizing the delay τ by scaling t → τ and establish the new parameter µ = τ −τ * , then, the new Hopf bifurcation value is µ = 0. The system (3) is transformed intoU

Hopf Bifurcation Analysis,Kutorzi and Shi
Hence µ = (x, y, z) T ∈ R 3 , L µ : C → R 3 is the functional differential equation and f : R → R 3 is the nonlinear part which are given, respectively, as a result of and Then linearized system (25) at origin iṡ Clearly, from the discussion in section 2, the characteristic equation of (7) has a pair of purely imaginary roots Λ 0 = {ω 0 iτ * , −ω 0 iτ * }, we assume that it has the effect of delay τ ; easily cognise that for τ = τ * Let ℑ := C([−1, 0], R 3 ), following this, we consider the following functional differential equation on ℑ: Using Riesz representation theorem, there exists a 3 × 3 matrix function η(θ, τ ) of bounded variation for θ ∈ [−1, 0]. It is obvious L(τ * ) is a continuous linear function In fact, we may take By the Riesz representation theorem,