Ultimate boundary estimations and topological horseshoe analysis of a new 4 D hyper-chaotic system ∗

Abstract. In this paper, we first estimate the boundedness of a new proposed 4-dimensional (4D) hyper-chaotic system with complex dynamical behaviors. For this system, the ultimate bound set Ω1 and globally exponentially attractive set Ω2 are derived based on the optimization method, Lyapunov stability theory, and comparison principle. Numerical simulations are presented to show the effectiveness of the method and the boundary regions. Then, to prove the existence of hyper-chaos, the hyper-chaotic dynamics of the 4D nonlinear system is investigated by means of topological horseshoe theory and numerical computation. Based on the algorithm for finding horseshoes in three-dimensional hyper-chaotic maps, we finally find a horseshoe with two-directional expansions in the 4D hyper-chaotic system, which can rigorously prove the existence of the hyper-chaos in theory.


Introduction
A 4D hyper-chaotic system is usually considered as a chaotic system with two positive Lyapunov exponents, which can enhance the randomness and unpredictability of the nonlinear system.So constructing a new hyper-chaotic system with complex dynamical behaviors may be more useful in some research fields, such as communication [1,37], encryption [5,20], and synchronization [3,22,31,32].
It is well known that the dissipative chaotic system often has a typical strange attractor in its phase space and the strange attractor is usually bounded.Therefore, an interesting research topic is how to estimate the bound of the dissipative chaotic or hyper-chaotic contracted closely to a certain surface.Based on this feature, we can use a remarkable algorithm proposed by Li and Tang [16] to detect a horseshoe with two-directional expansions effectively by deducting the dimension along the direction of contraction.
Motivated by the above discussions, a new 4D smooth quadratic autonomous hyperchaotic system is constructed directly, and the complex dynamical behaviors are studied.We first investigate the boundedness of the new hyper-chaotic system.The ultimate bound and the positively invariant set Ω 1 are obtained, which is based on the optimization method and the comparison principle.Then we discuss the relationship between ultimate bound and the two positive Lyapunov exponents of the system.Besides, we find the globally exponentially attractive set Ω 2 of the system and conclude that the trajectories of the system go from the exterior of Ω 2 to the interior of Ω 2 at exponential rate.Here we use a combination of Lyapunov stability theory with the comparison principle method.Secondly, since the values of the two positive Lyapunov exponents are not large enough to tolerate the numerical errors during the computation, we give the computer-assisted verification of hyper-chaos.Based on the algorithm for finding horseshoes in three-dimensional hyper-chaotic maps, we find a horseshoe with two-directional expansions in the 4D hyperchaotic system, which can rigorously prove the existence of the hyper-chaos in theory.
The organization of this paper is as follows.In Section 2, we will introduce mathematical model and study complex dynamical behaviors of the new proposed 4D hyperchaotic system.In Section 3, we will estimate the boundedness of the new 4D hyperchaotic system including the ultimate bound and the positively invariant set Ω 1 and the globally exponentially attractive set Ω 2 .In Section 4, we will give the computer-assisted verification of hyper-chaos by virtue of the algorithm for finding horseshoes in threedimensional hyper-chaotic maps.Finally, conclusions will be drawn in Section 5.

Mathematical model and complex dynamical behaviors
The new proposed 4D hyper-chaotic system is given as follows: where a, b, c, d, e, g, k are positive constant parameters determining the chaotic and hyper-chaotic behaviors of system (1).

Symmetry and invariant set
Since system (1) is invariant under the coordinate transformation (x, y, z, w) → (−x, −y, z, −w), it is symmetrical about z-axis.
z-axis is a positively invariant set of system (1), and we can get the equation ż = −cz when the system is restricted to the z-axis.Since the solution of the equation is z(t) = z(t 0 )e −c(t−t0) , the trajectories from arbitrary point on the z-axis will tend to (0, 0, 0, 0) when t → ∞.

Dissipation
Denote V (t) as the volume element and f as the vector field with components ( ẋ, ẏ, ż, ẇ).According to the divergence theorem, we can get the following equation: So system (1) is dissipative.That is to say, for the initial volume element V (0), V (t) = V (0)e −(a+c+d+1)t indicates that each volume element containing the trajectory of the system shrinks to zero at an exponential rate, and its dynamics behavior will be fixed on an attractor [28,29].Thus, system (1) is globally bounded.

The stability of equilibria
Before the detailed analysis, the study of equilibria is necessary.S 0 = (0, 0, 0, 0) is always a equilibrium point of system (1).Other equilibrium points can be shifted to S 0 by means of coordinate translation, so we just consider the stability of S 0 .For the zero equilibrium point S 0 , the characteristic equation is When a = 18, b = 60, c = 8, d = 0.2, e = 0.1, g = 0.1, k = 10, a + d + k − ab + ad < 0, the characteristic equation at least has one positive real eigenvalue.Therefore, the equilibrium point S 0 is unstable, which indicates that the system will generate chaos and hyper-chaos.The stability of the other equilibrium points of the system can be verified by the same method.

The rich dynamics evolution
The rich dynamics evolution of system (1) can be analyzed by Lyapunov exponents (LEs) spectrum, bifurcation diagram, and various phase portraits.Lyapunov exponents are the characteristic quantity obtained by the average of system's infinite long orbit and can be used to determine the existence and complexity of chaos and hyper-chaos.Now, the commonly used methods of calculating the LEs are QR decomposition method [2] and Wolf method [30].In our paper, we use the QR decomposition method to calculate the LEs with the variation of parameter c.Bifurcation diagram is a intuitive representation of the system evolutionary process.The key to drawing bifurcation diagram is to find a suitable Poincaré section.The horizontal axis of bifurcation diagram is the system parameter, while the vertical axis is the value of a state variable.In our paper, we give the bifurcation diagram versus parameter c.   in Figs. 3 and 4. Figure 3 shows phase portraits of quasi-periodic and periodic attractors of system (1).Figure 4 shows phase portraits of chaotic and hyper-chaotic attractors of system (1).Especially, the system is in hyper-chaotic state when c = 8, the four LEs are L 1 = 0.7742, L 2 = 0.6539, L 3 = −0.0001,L 4 = −28.6280. https://www.mii.vu.lt/NA 3 Boundedness of the new hyper-chaotic system

Preliminary
In this subsection, we will introduce some basic definitions, which are necessary for proving the proposed theorems in the following two subsections.Consider the following autonomous system: where a compact set, and the distance between X(t, t 0 , X 0 ) and Ω is defined by for all X 0 ∈ R n \ Ω, that is, for any ε > 0, there exists T > t 0 satisfying X(t, t 0 , X 0 ) ∈ Ω ε for all t T .Then the set Ω is called an ultimate bound of system (2).If for any X 0 ∈ Ω and all t t 0 , X(t, t 0 , X 0 ) ∈ Ω, then Ω is called the positively invariant set for system (2).Definition 2. (See [18].)For autonomous system (2), if there exist generalized positive definite and radially unbounded Lyapunov function V (X(t)) and constants L > 0, β > 0, for all X 0 ∈ R n , when V (X(t)) > L, V (X 0 ) > L, t > t 0 , the following inequality is satisfied: is said to be a globally exponentially attractive set of system (2).

The ultimate bound and positively invariant set
In this subsection, we will estimate the ultimate bound and positively invariant set of the system for a > 0, b > 0, c > 0, d > 0, e > 0, g > 0, k > 0. Before the detailed study, let us introduce the following lemma.
Proof.It can be easily proved by the Lagrange multiplier method.
By Lemma 1, we can get the following theorem.
Theorem 1.For a > 0, b > 0, c > 0, d > 0, e > 0, g > 0, k > 0, h > 0, the following set is the ultimate bound and positively invariant set of the system: Here Proof.Define the following generalized positive definite and radially unbounded Lyapunov function: where h = ke/g > 0. Computing the derivative of V 1 along the trajectory of system (1), we can obtain Let V1 = 0, then the following ellipsoidal surface Γ can be obtained: Outside Γ , V1 < 0, while inside Γ , V1 > 0. Thus, the ultimate bound of system (1) can be only reached on Γ .Since the V 1 (X) is a continuous function and Γ is a bounded close set, function (4) can reach its maximum value: max V 1 (X) X∈Γ = R 2 on the surface Γ defined in Eq. ( 5).Obviously, the set defined by {(x, y, z, w) x, y, z, w) ∈ Γ } contains the solutions of system (1).By solving the following conditional extremum problem, we can get the maximum value max V 1 (X): Set x = x, √ hy = ỹ, √ hz = z, e/g w = w as a new variable, and denote (a + hb) 2 c/(4ha) = ã2 , (a + hb) 2 c/(4h) = b2 , (a + hb) 2 /(4h) = c2 , (a + hb) 2 c/ (4hd) = d2 , then the conditional extremum problem (6) can be transformed into the following forms: According to Lemma 1, we can easily get max Therefore, we have obtained the compact set Ω 1 given in Eq. ( 3).According to Definition 1, since Γ ⊂ Ω 1 , next we will use the reduction to absurdity to prove Here X(t) = (x(t), y(t), z(t), w(t)).Suppose Eq. ( 8) does not hold, we can conclude that the orbits of system (1) are outside Ω 1 , thus V1 < 0 and V 1 (X(t)) monotonously decreases. Let t 0 is the initial time.Consequently, we have that δ, Ṽ1 are both positive constants and dV 1 (X(t))/dt −δ.As t → +∞, we can obtain This is inconsistent, thus Eq. ( 8) holds, it is equivalent to say that the set Ω 1 is the ultimate bound set of system (1).
Then we prove that Ω 1 is also the positively invariant set of system (1).Suppose V 1 (X(t)) attains its maximum value on surface Γ at the point P 0 ( x0 , ỹ0 , z0 , w0 ).Since Γ is contained in Ω 1 , for any point X(t) on Γ and X(t) = P 0 , we have V1 < 0. Thus, any orbit X(t) (X(t) = P 0 ) of system (1) will go into the set Ω 1 .When X(t) = P 0 , by the continuation theorem, X(t) will also go into the set Ω 1 .Summarizing the contents above, we conclude that Ω 1 is positively invariant set of system (1).This completes the proof.In Figs.5-7, we give the phase portraits and ultimate bound of system (1) in 3D and 2D planes with corresponding parameter values.By comparison, we can find that with the variation of parameter values, the two positive LEs becomes smaller, and the corresponding Ω 1 also becomes smaller.

Numerical simulations
Although Theorem 1 gives the ultimate bound and positively invariant set of system (1), it does not give the estimation of the trajectories rate.In Section 3.3, we will give the globally exponentially attractive set to estimate the trajectories rate.

The globally exponentially attractive set
The objective of this subsection is to derive the globally exponentially attractive set of the hyper-chaotic system (1).
Proof.Define the following generalized positive definite and radially unbounded Lyapunov function: Here h = ke/g > 0.
Computing the derivative of V (X(t)) along the trajectory of system (1) when V (X(t)) > L, V (X 0 ) > L, t t 0 , we have https://www.mii.vu.lt/NA −β This is equivalent to say By the comparison theorem and integrating both sides of formula ( 9) we obtain Thus, if V (X(t)) > L, V (X 0 ) > L, t t 0 , we have the following exponential estimation for system (1): By the definition, taking limit on both sides of the above inequality as t → +∞, we can get lim t→+∞ V X(t) L.
Namely, the set is the globally exponentially attractive set of system (1).This completes the proof.

Numerical simulations
Through the analysis of the above conditions, we can find that the trajectories of system (1) go from the exterior of Ω 2 to the interior of Ω 2 at exponential rate and when t → +∞, the trajectories of system (1) are contained in Ω 2 as shown in Fig. 8.That is to say, the system cannot have the equilibrium points, periodic or quasi-periodic solutions, or other chaotic or hyper-chaotic attractors existing outside the attractive set.

Computer-assisted verification of hyper-chaos
From the analysis in Sections 2 and 3 we can find that system (1) is hyper-chaotic for different parameters.One important difference among them is the values of two positive Lyapunov exponents.In Section 3, when parameters a = 10, b = 30, c = 3, d = 0.1, e = 0.08, g = 0.06, k = 3.8, the two positive Lyapunov exponents are L 1 = 0.3906, https://www.mii.vu.lt/NAL 2 = 0.3546, respectively.The two positive Lyapunov exponents may be not large enough to tolerate the numerical errors during the computation.So it is significant to find a more reliable method to prove the existence of hyper-chaos.In our cases, we will introduce a rigourous proof of the existence of hyper-chaos in system (1) by directly detecting topological horseshoes with two-directional expansions in the corresponding Poincaré map.
Before the detailed study, some theoretical criteria of topological horseshoes are firstly reviewed, and then we will present our main results.
Let X be a metric space, and D is a compact subset of X.There are m completely disjoint connected compact subsets D 1 , D 2 , . . ., D m of D. For each D i , 1 i m, let D 1 i and D 2 i be its two mutually disjoint connected nonempty compact subsets contained in the boundary of ∂D i .Let the map f be continuous on each D i .Definition 3. (See [9].)Let Γ ⊂ D i be a connected subset.Γ is said to be a separation of D 1 i and [9,34].)Let Γ ⊂ D i be a compact subset.We say that f (Γ ) separates D j with respect to D 1 j and i ), we say that f : D i → D j is a codimension-one crossing with respect to two pairs (D 1 i , D 2 i ) and (D 1 j , D 2 j ).Theorem 3. (See [34].)If the codimension-one crossing relation f : D i → D j , holds for 1 i, j m, then there exists a compact invariant set K ⊂ D such that f |K is semi-conjugate to the m-shift mapping.Then the entropy of f satisfies ent(f ) log m.
Sometimes for verifying existence of chaos, the assumption in Theorem 3 is not easy to be satisfied in practice.So we use the following practical corollary.
Corollary 1. (See [9].)Suppose that the map f : D → X satisfies the following assumptions: (i) There exist two mutually disjoint compact subsets D 1 and D 2 of D, and f m |D 1 and f n |D 2 are homeomorphisms, where m, n are positive integers.
Then there exists a compact invariant set K ⊂ D, such that f m+n |K is semiconjugate to 2-shift dynamics, and the topological entropy of f satisfies ent(f ) log 2/(m + n).
Since f in the above corollary is a homeomorphism, we are going to study a Poincaré map of system (1).By taking the hyperplane Π = {x ∆ = (x, y, z, w) | x = 0, ẋ > 0} as a Poincaré cross-section, the corresponding Poincaré map P : Π → Π can be defined as follows: for each x = (0, y, z, w) ∈ Π, P (x) is taken to be the first return point in Π under the flow of the dynamical system with the initial condition x.
Different from many other studies on topological horseshoes for two-dimensional (2D) chaotic maps [10,15,35], it is too hard to find a horseshoe directly for threedimensional (3D) chaotic maps due to the higher dimensionality.Hence, we utilize the following technique to make it possible.Firstly, we deduct the dimension along the direction of contraction to get a 2D projective map.Then we detect a projective horseshoe with two-directional expansion.At last, we construct a 3D horseshoe back to the original map P .According to the algorithm, we find a horseshoe by three steps.Proof.According to Corollary 1, we only need to show that A, B, and their images under P and P 2 respectively satisfy the following relationships about the codimensionone crossing: The geometrical relations of A, B and P (A) are shown in Fig. 9. Figure 9(a) is a 3D view, which suggests that P (A) expands in two directions and transversely intersects both blocks A and B. Figure 9(b) is a side view, which shows that the intersection happens between their top and bottom surfaces, i.e., (A t , A b ) and (B t , B b ). Figure 9(c) is a top view, which shows that the side surface of A is mapped outside A and B. Therefore, for each separation S of (A t , A b ), f (S) A must be a separation of (A t , A b ), and f (S) B must be a separation of (B t , B b ).Then we have P (A) → A, P (A) → B. Similarly, we can have P 2 (B) → A, P 2 (B) → B from the Fig. 10.
Since system (1) is smooth, i.e., the system has a unique solution from each initial condition, Figs. 9 and 10 also show that P |A and P 2 |B are both continuous, so they must be homeomorphism.Then it follows from Corollary 1 that there exists a compact invariant set Λ ⊂ A B such that P 1+2 |Λ is semi-conjugate to the 2-shift, and the topological entropy of P is not less than log 2/3.Since P |A and P 2 |B both expand in two directions, the expansions along each trajectory in Λ are also in two directions, so there must exist two positive Lyapunov exponents.Therefore, system (1) is hyper-chaotic.

Conclusion
A new 4D hyper-chaotic system with complex dynamical behaviors is presented in this paper.The Lyapunov exponents spectrum and bifurcation diagram are provided and analyzed.The periodic orbit, chaotic, and hyper-chaotic attractors can be found in this nonlinear system.Based on the optimization method, Lyapunov stability theory, and comparison principle, we have obtained the boundedness of the new 4D hyper-chaotic system including the ultimate bound and positively invariant set Ω 1 and the globally exponentially attractive set Ω 2 .The corresponding boundedness has been verified by the numerical simulations, which show the effectiveness of the proposed scheme.Besides, by means of topological horseshoe theory and numerical simulation, a topological horseshoe with two-directional expansions has been also obtained, which can rigorously ensure that system (1) is hyper-chaotic system in theory.

Theorem 4 .
(i) Due to the large negative LE, the dynamics in this direction contract very quickly, and the hyper-chaotic attractor is often contracted closely to a curved surface whose equation w = s(y, z) can be easily fitted in MATLAB.Hence, we deduct the dimension along the direction of contraction to obtain a 2D projective system.(ii) Now, we only need to cast about for a 2D horseshoe of the projective map on the yoz-plane.Through several attempts, we find a horseshoe with two directional expansions by choosing two quadrilaterals in the yoz-plane.The four vertices of the first quadrilateral A yoz in terms of (y, z) are A 1 = (23.107,0.522), A 2 = (21.321,0.125), A 3 = (25.607,−1.125),A 4 = (26.964,−0.639).The four vertices of the second one B yoz are B 1 = (20.178,−0.123), B 2 = (18.821,−0.282), B 3 = (21.107,−1.165), B 4 = (22.428,−1.075).(iii) We construct the 3D horseshoe of the map P utilizing the projective horseshoe by projecting the planar horseshoe back to the 3D space.Finally, we have two blocks A and B in the phase space of the Poincaré map P as shown in Figs.9(a) and 10(a).It is not hard to get the following theorem.For the Poincaré map P : Π → Π, there exists a closed invariant set Λ ⊂ A B on which P 1+2 |Λ is semi-conjugate to the 2-shift, and ent(P ) log 2/3.