A New Series of Three-Dimensional Chaotic Systems with Cross-Product Nonlinearities and Their Switching

This paper introduces a new series of three-dimensional chaotic systems with cross-product nonlinearities. Based on some conditions, we analyze the globally exponentially or globally conditional exponentially attractive set and positive invariant set of these chaotic systems. Moreover, we give some known examples to show our results, and the exponential estimation is explicitly derived. Finally, we construct some three-dimensional chaotic systems with cross-product nonlinearities and study the switching system between them.

The Lorenz system plays an important role in the study of nonlinear science and chaotic dynamics [13][14][15][16][17][18].We know that it is extremely difficult to obtain the information of chaotic attractor directly from system.Most of the results in the literature are based on computer simulations.When calculating the Lyapunov exponents of the system, one needs to assume that the system is bounded in order to conclude chaos.Therefore, the study of the globally attractive set of the Lorenz system is not only theoretically significant but also practically important.Moreover, Liao et al. [19,20] gave globally exponentially attractive set and positive invariant set for the classical Lorenz system and the generalized system by constructive proofs.In addition, Yu et al. [21] studied the problem of invariant set of systems, which was considered as a more generalized Lorenz system.
In this paper, we consider the following three-dimensional autonomous systems with cross-product nonlinearities: where   = ( 1 ,  ; with   ,   ,   ∈ , , ,  = 1, 2, 3.This second-order dynamic system may be regarded as the most general Lorenz system.
For such system, we can choose Lyapunov function:

Preliminaries
In this section, we present some basic definitions which are needed for proving all theorems in the next section.
In general, from the definition we see that a globally exponential attractive set is not necessarily a positive invariant set.But our results obtained in the next section indeed show that a globally exponentially attractive set is a positive invariant set.
Note that it is difficult to verify the existence of Ω in Definition 2. Since the Lyapunov direct method is still a powerful tool in the study of asymptotic behaviour of nonlinear dynamical systems, the following definition is more useful in applications.
Definition 4. For three-dimensional autonomous systems with cross-product nonlinearities (1), if there exist a positive definite and radially unbounded Lyapunov function (()) and positive numbers  > 0,  > 0 such that the following inequality is valid for (()) > ( ≥  0 ), then the system (1) is said to be globally exponentially attractive or globally exponentially stable in the sense of Lagrange, and Ω := { | (()) ≤ ,  ≥  0 } is called the globally exponentially attractive set.

Qualitative Analysis
We call the dynamic system (1) the first class three-dimensional chaotic system with cross-product nonlinearities (1), if there are some nonzero numbers { 1 ,  2 ,  3 } so as to satisfy conditions Condition ( 6) is satisfied by some known three-dimensional quadratic autonomous chaotic systems, the wellknown Lorenz system [1-3], the Rössler system [5], the Rucklidge system [6], and the Chen system [7,8].Lorenz systems are widely studied and the references therein [9][10][11][12][19][20][21].For example, consider the classical Lorenz system  Thus it can be seen that condition (6) is very important in qualitative analysis of the exponentially attractive set and positive invariant set of Lorenz systems.
We will research this dynamic system in two cases.First, supposing =  ̸ = , the dynamic system (1) can be rewritten as The construction techniques of this kind of Lorenz systems are to pay attention to satisfing formula where   ,  = 1, 2, 3 are parameters and where   ,  = 1, 2, 3 are undetermined parameters.And we always assume that the supremum (, ) < +∞ in the paper.
Proof.Again applying Lyapunov function given in (19) and evaluating the derivative of  1 / along the trajectory of system ( 16) lead to ( The conclusion of Example 9 is obtained.
Then, the estimate holds and that is the globally exponentially attractive set and positive invariant set of system (7). 2 = 0,  3 =  + 2, system (6), (()), and (, ) can be rewritten as system (8): We have then is the estimation of the globally exponentially attractive and positive invariant sets of system (8).
Theorem 11.Suppose that  0 = ( 0 1 ,  0 2 , . . .,  0  ) is the stable point of the (, ) defined by (33).If the Hesse matrix of the (, ) is a negative definite matrix, the (, ) has maximum and the estimation and the set is the globally exponentially attractive set and positive invariant set of system (32).
Proof.If  0 is the stable point of the (, ), that is, and the Hesse matrix   of the (, ) is a negative definite matrix, namely, The (, ) has the maximum .Differentiating the Lyapunov function (()) in (3) with respect to time  along the trajectory of system (32) yields The proof is complete.

Switched Chaotic Systems
Condition ( 6) has helpfully provided us with instructions on how to find the new chaotic systems.We construct a series  of new chaotic systems that the condition ( 6) is fulfilled and study the switching system between them.

Solution.
Here   The Hesse matrix of the (, ) is a negative definite matrix, max (, ) ≈ 164045.42.The set is the globally exponentially attractive set and positive invariant set of system (40).(c) We call the dynamic system (1) the second class threedimensional chaotic system with cross-product nonlinearities, if it does not satisfy condition (6).For this class of chaotic systems, (, ) is a cubic polynomial and there is not maximum if we choose energy function (3) differentiating this Lyapunov function with respect to  along the trajectory of system (1).It is very useful to research these problems.Example 13.The new chaotic system shown in Figure 2 is Example 14.The chaotic system shown in Figure 3 is (44) Example 15.The chaotic system shown in Figure 4 is Example 16.The chaotic system shown in Figure 5 is

Conclusion
In this paper, the methods in [19][20][21] have been extended to study the globally exponentially or globally conditional   exponentially attractive set and positive invariant set of the three-dimensional chaotic system family with cross-product nonlinearities.We have given two theorems for studying this question and given some examples to show that such system indeed has the globally exponentially or globally conditional exponentially attractive set and positive invariant set, and the exponential estimation is explicitly derived.We have also suggested an idea to construct the chaotic systems, and some new chaotic systems have been illustrated.The simulation results are given for switched system between these new chaotic systems.It is very interesting to further research that the Hesse matrix of the (, ) is not a negative definite matrix, and the dynamic system (1) is a second class threedimensional chaotic system with cross-product nonlinearities.