Rank One Strange Attractors in Periodically Kicked Lorenz System with Time-Delay

Abstract

Rank one strange attractor in periodically kicked Lorenz system with time-delay is investigated. Our discussion is based on the theory of rank one maps formulated by Wang and Young. First, we develop the rank one chaotic theory to delayed systems. It is shown that strange attractors occur when periodically kicked delayed system undergoes a generic Hopf bifurcation. Then we use the theory to the periodically kicked Lorenz system with delay, and derivation of conditions for Hopf bifurcation and rank one chaos along with the results of numerical simulations are presented.

1. Introduction

In the 1970s, chaos characterized by SRB measure was first introduced by Sinai, Ruelle, and Bowen [1–3]. Since then, many results on SRB measure, strange attractor, Lyapunov exponent, and their applications were obtained [4–6]. In 2001, Wang and Young gave simple conditions that guarantee, for strongly dissipative maps, the existence of strange attractors with a single direction of instability and certain controlled behaviors [7]. In 2008, Wang and Young accomplished a more comprehensive understanding of the complicated geometric and dynamical structures of a specific class of nonuniformly 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 [8]. In 2009, Chen and Han studied the existence of rank one chaos in the neighborhood of a periodically kicked stable limit cycle close to a heteroclinic cycle of a planar equation [9]. In 2012, Fang studied the synchronization between rank one chaotic systems without and with delay using linear delayed feedback control method [10]. In 2013, Wang and Young gave a dynamical study of certain classes of rank one attractors [11].

The system we consider in this paper is the well-known Lorenz system with time-delay. Since Lorenz presented the Lorenz system as (1) in [12], people had considerable interest in the three-dimensional autonomous chaotic system (see, e.g., [13, 14]):

The following Lorenz system with time-delay has been researched [15]: The authors found that a simple time-delay transforms the Lorenz system to the generalized Chen system or the generalized L system without any coefficient changes.

In this paper, we consider the derivative of with respect to and the derivative of with respect to a time-periodic forcing term . Then we consider the following system: where , , and are real parameters. The stability of equilibrium, the bifurcating of periodic solutions, and the rank one chaos of the periodic kicked delayed system are investigated.

This paper is organized as follows. In Section 2, we give preliminaries about the rank one theory. In Section 3, we derived the rank one theory for delayed systems. In Section 4, we consider rank one chaos for the periodically kicked Lorenz system, and at last, numerical simulations are presented in Section 5.

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 according to [16].

Let , let be the phase variable, and let be the time. Consider the following system of equations: where is a real by matrix and is a vector valued real analytic function in defined on a given neighborhood of such that , . Both and are smooth dependents of parameter around . is a parameter that controls the magnitude of the forcing. is a real vector valued function of that represents the shape of the forcing and . When , the undisturbed system is

Firstly, we give several definitions.

Definition 1 (see [17]). Let be a diffeomorphism of a compact Riemannian manifold onto itself. One says is an Anosov diffeomorphism if the tangent space at every is split into where and are -invariant subspaces, is uniformly expanding, and is uniformly contracting. A compact -invariant set is called an attractor if there is a neighborhood of called its basin such that for every . is called an Axiom A attractor if the tangent bundle over is split into as above.

Definition 2 (see [17]). Let be a diffeomorphism with an Axiom attractor . Then there is a unique -invariant Borel probability measure on that is characterized by each of the following (equivalent) conditions:(i) has absolutely continuous conditional measures on unstable manifolds;(ii)consider  where is the metric entropy of ;(iii)there is a set having full Lebesgue measure such that, for every continuous observable , we have, for every , (iv) is the zero-noise limit of small random perturbations of .

Then the invariant measure is called the Sinai-Ruelle-Bowen measure or SRB measure of .

Definition 3. Given a map , the Lyapunov exponent is defined at as follows: where .

For such that where is open, let be the attractor. By strange attractor, here we refer to attractors with the following three properties, see [18].(SA1)Positive Lyapunov exponents: For Lebesgue, almost every ; the orbit of has a positive Lyapunov exponent.(SA2)Existence of SRB measure and basin property:(a) admits at least one and at most finitely many ergodic SRB measures all of which have no zero Lyapunov exponents, denoted by .(b)For Lebesgue, almost every ; such that, for every continuous function , (SA3)Statistical properties of dynamical observations:(a)For every ergodic SRB measure and every continuous function , the sequence obeys a central limit theorem; that is, if , then converges in distribution to the normal distribution, and the variance is strictly positive unless for some .(b)Suppose that, for some , has a SRB measure that is mixing. Then given exponent , such that, for all with exponent , such that, for all ,

Now, we consider . Assume the following.

(A1) Let be the eigenvalues of . There is a conjugated pair such that , , , and there exists such that , .

Then the flow on the central manifold of system can be explicitly written by using a complex variable in the following normal form:

According to the standard normal form theory, there exists a change of variables near identity, which we write as which transfers (10) to Using the formula given in [19], we have Assume the following.

(A2) . Suppose that a real linear coordinate change transfers into where , are scalars. is an -vector, is an by matrix of stable eigenvalues, and

Let be such that , and define

Further let be the unit circle in -plane in -space, and define

The time- map of is denoted by , where is the bifurcation parameter of the unperturbed and , are the parameters of forcing. Assume that(a)(A1)-(A2) hold for ;(b) in (13) is a Morse function;(c), are such that and .

The following Theorem is obtained by Wang and Oksasoglu [16].

Theorem 4. Let the values of and be fixed and assume that (a)–(c) hold. Regard period of the forcing as a parameter and denote . Then there exists a constant , determined exclusively by , such that if then there exists a positive measure set for , so that, for , has a strange attractor admitting no periodic sinks. That is to say, there exists an open neighborhood of in such that has a positive Lyapunov exponent for Lebesgue almost every point in . Furthermore, admits an ergodic SRB measure, with respect to which almost every point of is generic.

3. Rank One Strange Attractors of Delayed System

In this section we consider a nonlinear delayed differential equation: where , for , is a linear operator, and is a nonlinear term satisfying , . and depend on analytically for is sufficiently small, and . When , the undisturbed system is

For linear system , there is an matrix , such that for any

Let the spectral set of satisfy the following.

(B1) There is a simple conjugated pair such that , , , and there exists such that, for any , ; , .

Then the flow on the central manifold of system can be written by using Hassard’s method [20] in the following form: at . Let

Suppose the following.

(B2) . Then we know that system has a supercritical Hopf bifurcation near the equilibrium.

We define for

Since , becomes For (25) is .

Following Hassard’s method, we know that satisfies equation where is a eigenvector corresponding to eigenvalue of which is the adjoint operator of .

When , let and we define

Then from and the definitions of , , we obtain At , We can calculate Then (25) can be written as

Let , where is a Banach space, and let in (31). Define

We further let , , and in (32); then is the unit circle in ()-plane in -space. Define

The time- map of is denoted by where is the bifurcation parameter of the unperturbed and , are the parameters of forcing. Assume that(a)(B1)-(B2) hold for ;(b) in (33) is a Morse function;(c) are such that , .

Then we obtain the following.

Theorem 5. Let the values of and be fixed and assume that (a)–(c) hold. Regard period of the forcing as a parameter and denote . Then there exists a constant , determined exclusively by , such that if then there exists a positive measure set for , so that, for , has a strange attractor admitting no periodic sinks. That is to say, there exists an open neighborhood of such that has a positive Lyapunov exponent for Lebesgue almost every point in . Furthermore, admits an ergodic SRB measure, with respect to which almost every point of is generic.

Proof. We can easy see that (B1) and (B2) in Theorem 5 are corresponding to (A1) and (A2) in Theorem 4. After transformations, (33) in -plane is corresponding to (17) in -plane, and they are on the central manifolds. Obviously the conditions in Theorem 4 are satisfied.

4. Analysis of Rank One Strange Attractors in Delayed Lorenz System

We consider the corresponding undisturbed system of (3) as follows: Obviously, system (35) always has equilibria , , where

Let be the arbitrary equilibrium, and denote , where , , and still denote , , and by , , and , respectively; then the linearized system of the corresponding equation at is as follows: where , , , , , , , , , , and all the others of and are 0.

The characteristic equation of system (37) is where , , , , and . We consider the following cases.

(1) One has . The coefficients of (38) are , , , , and .

When , (38) becomes

Routh-Hurwitz criterion implies that if then (38) has at least one positive real root, so when holds, is unstable equilibrium.

(2) One has , or . Due to the symmetry of and , here we only consider . When , the coefficients of system (38) are , , , , and . According to Routh-Hurwitz criterion, if holds, then all roots of (38) have negative real parts.

Suppose that is a root of (38) in the imaginary axis. Substituting it in (38) and separating the real and imaginary parts, we have which is equivalent to

Let ; then (41) becomes where , , and .

Lemma 6. For (42), one has the following.(i)If , then (42) has at least one positive root.(ii)If , then (42) has at least one positive root if and only if there exists such that and .

We assume that (42) has three positive roots , , and and , , and .

According to (40), we have where and ; then is a pair of imaginary roots of (38) with .

Define

In order to investigate the distribution of the roots of (38), we need to introduce the following lemmas [21].

Lemma 7. Consider the exponential polynomial where , , and (; ) are constants. As vary, the sum of the order of the zeros of on the open right half plane can change if and only if there are zeros crossing the imaginary axis.

Let be the root of (38) near satisfying Then, the following transversal condition holds.

Lemma 8. Suppose that and ; then has the same sign as .

Proof. Substituting into (38) and taking the derivative with respect to , we obtain Therefore, When , , , we have According to (48), (49), and (42), we have where .
Therefore, has the same sign as .
Now we apply the Hopf bifurcation theorem for functional differential equations [22] and get the following results.

Theorem 9. Suppose that and Lemma 6 are satisfied. Then(i)when , equilibrium of system (35) is locally asymptotically stable;(ii)when , equilibrium of system (35) is unstable;(iii)when , then system (35) undergoes Hopf bifurcation at equilibrium .

Now, we will derive the explicit formulae determining the direction and stability of these periodic solutions bifurcating from equilibrium . Let , , and omit “-” above ; we rewrite (3) as where , are as in (37). We write as the form , where where is the Dirac delta function. Then (3) can be written as where

Assume that is the eigenvector of corresponding to ; then . It follows from the definition of that

Thus, we can easily compute , where

Similarly, it can be verified that is the eigenvector of corresponding to , where In order to assure the bilinear inner product , we have

Therefore, we can choose as where , .

We can get .

In what follows, we will obtain the coordinates to describe the center manifold at .

Notice that ; then we have

Thus, from (29) we have

Comparing the coefficients with (31), we can obtain the following formulae:

In order to obtain , we need to compute and . Substituting the corresponding series into (31) and comparing the coefficients, we obtain For , from (31), we know that

Comparing the coefficients with (31), we have From (63), (64), and the definition of , we can easily get Notice that ; hence where is a constant vector. Similarly, we can obtain where is a constant vector.

In what follows, we will seek appropriate and . According to the definition of and (63), we have

From (31), we have Substituting (67) and (71) into (69) and noticing that we have that is,

According to Cramer’s criteria, the solution of (75) is described by where, ,  , and Similarly, substituting (68) and (72) into (70), we have where Thus, we can compute the following values: where , determining the quantities of bifurcating periodic solutions in the center manifold at the critical value ; that is, determines the directions of the Hopf bifurcation: if , (resp., ), then the Hopf bifurcation is supercritical (resp., subcritical) and the periodic solutions exist for . determines the stability of the bifurcation periodic solutions: the bifurcating periodic solutions are stable (unstable) if .

Furthermore, we can get the definite expressions of and . From (32), we have Let and , then where and . Then is easy to verify being a Morse function from (82). According to Theorem 5, if , then there exists such that when , we can get an observable rank one chaos.

5. Numerical Simulations

Now we move to the parameters for which rank one attractors are observable, so we consider the following system: System (83) has a positive equilibrium . It is not difficult to get the following linear system of (83) at without the disturbed term : It follows that the characteristic equation of system (84) is From the discussion of Section 4, we have , , , , , and , so we know that the equilibrium is asymptotically stable when and (see Figure 1). Further, we know that the bifurcation is supercritical and the stable periodic solution emerges from the equilibrium (see Figure 2).

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(b)

(c)

(d)

Figure 1 

The trajectories and the phase graph of system (83) with , , and ; is asymptotically stable.

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(b)

(c)

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Figure 2 

The trajectories and the phase graph of system (83) with , , and ; becomes unstable, and a stable periodic solution bifurcates from equilibrium .

We also get . Choose in interval , which is quite “large” to afford a long relaxation period between consecutive kicks of the external force, and choose in . In Figure 3, we present a rank one strange attractor at . In Figure 4, a rank one strange attractor at is shown. In Figure 5, largest Lyapunov exponent versus for system (83) with and is given. We can see that there exists a positive measure set for , such that, for , the time- map of system (83) has a strange attractor. Largest Lyapunov exponent versus for system (83) with , , and varying from 0 to 120 is given in Figure 6. The results of our numerical simulations are in perfect agreement with the predictions of the rank one theory in Section 3.

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(b)

(c)

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Figure 3 

The trajectories and the phase graph of system (83) with , , and ; is unstable, and a rank one strange attractor occurs.

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(b)

(c)

(d)

Figure 4 

The trajectories and the phase graph of system (83) with , , and ; is unstable, and a rank one strange attractor occurs.

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(b)

Figure 5 

Largest Lyapunov exponent versus for system (83) with (a), (b), , and varying from 0 to 0.4.

Figure 6 

Largest Lyapunov exponent versus for system (83) with , , and varying from 0 to 120.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research is supported by the National Natural Science Foundation of China (nos. 11061016 and 11461036).