Abstract
In order to further discover the hidden chaotic attractor and its generating mechanism in the Vijayakumar system, we give a generalized system showing hidden chaotic attractors which are not from homoclinic orbit or heteroclinic orbit and consider Hopf bifurcations (codimension one and two) by first Lyapunov coefficient and second Lyapunov coefficient. The existence of periodic orbits is strictly proved theoretically. We have considered the problem of Hopf bifurcation in the chaotic system with hidden attractors, which will be helpful to reveal the intrinsic relationship between the local stability of equilibria and global complex dynamical behaviors of the chaotic system. Finally, numerical simulations are obtained for showing the correctness of theoretical results.
1. Introduction
The discussion of chaos is very interesting and important for nonlinear theory. The research of related subjects has a long history. Recently, attractors can be classified into two different types: self-excited or hidden [1–8]. Up to now, hidden chaotic attractors for the 3D autonomous chaotic systems can be found with only one stable equilibrium [9, 10], without equilibrium [11, 12], and with infinite equilibria [13–15], which makes the topology of the found chaotic systems different from that of the well-known 3D autonomous chaotic system.
Moreover, multistability allows flexibility of systems without changing parameters’ values and can be used to correct control strategies and parameters to induce switching between different coexistence attractors. The self-excited attractor has an attractive basin associated with unstable equilibrium [1–5]. Especially, in order to study periodic solutions, most researchers are more interested in studying periodic solutions [16–19]. Zoldi and Greenside [20] discovered unstable periodic orbit will be an important factor for chaos and then kicked something in the statistical correspondence between chaos and periodic orbits. Kawahara and Yamada [21] confirmed that the unstable periodic orbit will result in the classical Couette turbulent structure. The value of the connection number between these orbital pairs plays an irreplaceable role in the generation of chaos. Therefore, the existence of periodic orbit is very important for hidden chaos. In recent years, scholars have begun to consider the complex dynamics about the hidden chaos. This represents that multistability is an important feature of many nonlinear problems. However, some deep-seated hidden complex behaviors have not been thoroughly studied in many realistic chaotic systems [8, 16, 22].
As we know, there are still abundant and complex dynamical behaviors, and the topological structure of the hidden chaos should be thoroughly investigated and exploited. In particular, the generation mechanism of hidden chaotic attractors has always been a scientific problem of great concern. We will consider the coexistence of unstable periodic orbits and hidden chaotic attractors through Hopf bifurcation analysis. We study all possible bifurcations (general bifurcations and degenerate bifurcations) by the Lyapunov coefficient of Hopf bifurcation. More precisely, the first Lyapunov coefficient is obtained for the parameter space, which indicates the possibility of giving two branches of codimension, and the second Lyapunov coefficient is calculated. In addition, unstable periodic solutions can be obtained from bifurcation and can help us in better understanding, revealing an intrinsic relationship of the global dynamical behaviors with the stability of the equilibrium point, especially hidden chaotic attractors.
Based on the hidden chaotic attractors [23, 24], we design a new oscillator with coexisting hidden chaos, limit cycles, and point attractors. In Section 2, a generalized system showing hidden chaotic attractors is given. In Section 3, Hopf bifurcation methods about codimensions one and two are given out, in particular, how to obtain the Lyapunov coefficients related to the stability of the equilibrium. In Section 4, the existence of periodic orbits from Hopf bifurcation can be obtained. Finally, in Section 5, we make some concluding remarks and future works.
2. The New Chaotic System with Hidden Chaos
Based on the three-dimensional autonomous system proposed by Vijayakumar et al. [23], we give the generalized system,where are positive constants, and are arbitrary real constant. If , system (1) is the three-dimensional autonomous system proposed by Vijayakumar et.al. [23], which only shows the numerical results. Now, we want to consider why the hidden chaos can be found theoretically. Here, by choosing some parameter values and using certain numerical methods [25, 26], the system has different kinds of chaotic attractors (see Figure 1, Tables 1 and 2).
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The characteristic polynomial of the system (1) at the only one equilibrium is
If the system (1) will have Hopf bifurcation, the parameters should meet . In order to give the generation of hidden chaos, we will consider and as bifurcation parameters, respectively.
Proposition 1. If we choose as bifurcation parameter and in the system (1), characteristic values of have one negative real eigenvalue and a pair of purely imaginary eigenvalues.
Proposition 2. If we choose as bifurcation parameter and in system (1), characteristic values of have one negative real eigenvalue and a pair of purely imaginary eigenvalues.
3. Framework of the Hopf Bifurcation Methods
The Hopf bifurcation methods mainly refer to [19, 27–30]. For the system,where and , and is a class of . If we denote (3) has an equilibrium point at , mark the variable also by , and then write
Aswhere and, for ,
Suppose are a pair of complex eigenvalues. Let be vectors such thatwhere represents the transposed matrix . Any vector can be given as , where . The 2D center manifold at the eigenvalues can be parameterized by and . Through the use of a form immersion, , where has a Taylor expansion of the following form:
With and , thenwhere is given by (4). Taking into account the chart for a central manifold, we can obtain
With , the first Lyapunov coefficient can be given aswhere . Denoting as
And , the second Lyapunov coefficient is given by
When the first Lyapunov coefficient , the dynamic behavior of the system (3) is orbitally topologically equivalent to
As , we can find the existence of codimension one Hopf point and stable (unstable) periodic orbits on this manifold. Moreover, when , we can further consider the Hopf point of codimension two. When , the dynamic behavior of the system (3) is orbitally topologically equivalent towhere and are unfolding parameters.
4. Hopf Bifurcation and Hidden Chaos in New System
Using the mark in Section 3, we can write the multilinear symmetric functions (1)
4.1. Hopf Bifurcation about Parameter
Theorem 1. For system (1) with , the first Lyapunov coefficient at the equilibrium is given bywhereIf , the Hopf point at is a weak repelling focus, and an unstable limit cycle can be found near the asymptotically stable equilibrium for each , but close to ; if , the Hopf point at is weak attractive focus, and a stable limit cycle can be found near the unstable equilibrium for each , but close to .
Proof. Considering as the bifurcation parameter, the transversal conditionis met. The first Lyapunov coefficient will show the stability of the equilibrium point and periodic orbits generate from Hopf bifurcation.
In addition, one can also getwhereThen, the following value iswhereTherefore,Moreover, the results of Theorem 1 will be obtained.
Now, we continue to study the influence of the Hopf bifurcation for hidden chaos. We choose parameters from the work in [22]. According to Theorem 1, we have , , and is unstable point. An unstable periodic solution should be obtained near the stable equilibrium point for (see Figure 2). The result will show the generation of hidden chaos with stable equilibrium point (see Table 1).
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Remark 1. Now, we let and obtain coefficient In addition, we can know (i.e., ) from . Therefore, if , the first coefficient are positive, and unstable periodic orbit can be obtained. If , the first coefficient are negative, and stable periodic orbit can be obtained. Now, we will consider the sign of the second Lyapunov coefficient when with .
Theorem 2. For system (1), with, the first Lyapunov coefficient and the second Lyapunov coefficient for is 3.0633. Therefore, system (1) has a transversal Hopf point of codimension two atwhich is unstable.
Proof. If , and , the first Lyapunov coefficient andWe continue to do some detailed calculations and can getBy the above theorem and calculation, one hasTherefore, we can arrive at the above Theorem 2.
4.2. Hopf Bifurcation about Parameter
In this section, we consider Hopf bifurcation about parameter at Taking as the Hopf bifurcation parameter, we consider system (1) with . The transversal condition
is also satisfied.
Theorem 3. For system (1), with, the first Lyapunov coefficient at the equilibriumis given byIf , the Hopf point at is weak repelling focus, and an unstable limit cycle near the asymptotically stable equilibrium can be found for each , but close to ; if , the Hopf point at is weak attractive focus, and a stable limit cycle near the unstable equilibrium can be found for each but close to .
Proof. Here, we haveThe complex vectors and areThe complex coefficient defined in Section 3 isWe then have Theorem 3 from first Lyapunov coefficient . Consider system (1) with . The first Lyapunov coefficient associated with the equilibria is 1.1014. Then, the equilibrium undergoes a transversal Hopf bifurcation when . More specifically, when , but near to , there exists an unstable limit cycle around the asymptotically stable equilibria (see Figures 3(a) and 3(b)). The result will herald the emergence of hidden chaos (see Figure 3(c)).
In order to check Hopf bifurcation and hidden chaos, we choose , and with initial conditions , and bifurcation diagrams and Lyapunov exponents diagram are shown in Figures 4(a) and 4(b), respectively. When , and , multistability can happen from different initial conditions (see Figure 5), initial conditions (see Figure 6), and initial conditions (see Figure 7).
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5. Conclusion
In this paper, Hopf bifurcations in generalized system chaotic systems with hidden chaos are obtained theoretically. Through this analysis, we obtain the parameter conditions for which the system presents Hopf bifurcations. Then, we make an extension of the analysis to the more degenerate cases. The calculation of the first second and second Lyapunov coefficients, which makes possible the determination of the Lyapunov stability at the equilibria, can make the system exhibit Hopf bifurcation in a much larger parameter region. The first and second Lyapunov coefficients are obtained for exhibiting Hopf bifurcation and showing periodic orbits in the parameter region. In addition, numerical simulations of several parameter values are carried out to illustrate and verify some analysis results. The cascade of period-doubling bifurcation and the existence of hidden attractors are related to Hopf bifurcation at the equilibrium point in a sense. This interesting phenomenon is worth further studying, both theoretically and experimentally, to further reveal the intrinsic relationship between the local stability of equilibrium and global complex dynamical behaviors of the chaotic system. In addition, a fractional-order version of the chaotic system can be designed using integrated circuit technology [31–37], as required in wireless systems. It is expected that a more detailed theoretical analysis will be excavated in the forthcoming paper.
Data Availability
All data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.