Hidden chaotic attractors in a class of two-dimensional maps.

This paper studies the hidden dynamics of a class of two-dimensional maps inspired by the Hénon map. A special consideration is made to the existence of fixed points and their stabilities in these maps. Our concern focuses on three typical scenarios which may generate hidden dynamics, i.e., no fixed point, single fixed point, and two fixed points. A computer search program is employed to explore the strange hidden attractors in the map. Our findings show that the basins of some hidden attractors are tiny, so the standard computational procedure for localization is unavailable. The schematic exploring method proposed in this paper could be generalized for investigating hidden dynamics of high-dimensional maps

Hidden attractors in fundamental problems and engineering models

Recently a concept of self-excited and hidden attractors was suggested: an attractor is called a self-excited attractor if its basin of attraction overlaps with neighborhood of an equilibrium, otherwise it is called a hidden attractor. For example, hidden attractors are attractors in systems with no equilibria or with only one stable equilibrium (a special case of multistability and coexistence of attractors). While coexisting self-excited attractors can be found using the standard computational procedure, there is no standard way of predicting the existence or coexistence of hidden attractors in a system. In this plenary survey lecture the concept of self-excited and hidden attractors is discussed, and various corresponding examples of self-excited and hidden attractors are considered

Chaotic behavior and modified function projective synchronization of a simple system with one stable equilibrium

summary:By introducing a feedback control to a proposed Sprott E system, an extremely complex chaotic attractor with only one stable equilibrium is derived. The system evolves into periodic and chaotic behaviors by detailed numerical as well as theoretical analysis. Analysis results show that chaos also can be generated via a period-doubling bifurcation when the system has one and only one stable equilibrium. Based on Lyapunov stability theory, the adaptive control law and the parameter update law are derived to achieve modified function projective synchronized between the extended Sprott E system and original Sprott E system. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive controllers

Synchronization of Coupled Nonidentical Fractional-Order Hyperchaotic Systems

Synchronization of coupled nonidentical fractional-order hyperchaotic systems is addressed by the active sliding mode method. By designing an active sliding mode controller and choosing proper control parameters, the master and slave systems are synchronized. Furthermore, synchronizing fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system is performed to show the effectiveness of the proposed controller

Periodic solutions and circuit design of chaos in a unified stretch-twist-fold flow

Based on the original stretch-twist-fold flow, we propose a unified stretch–twist–fold (USTF) flow. We explore the conditions that zero-Hopf bifurcation occurs at the origin. Using the first-order averaging theorem, a periodic solution produced from the zero-Hopf equilibrium is derived. In addition, we obtain the conclusion that for parameter $$\alpha$$ large enough, the periodic orbit of USTF flow exists as well unstable. Finally, circuit design has been built for implementing the new system, showing a good agreement between computer simulations and experimental observations

Comparison theorems of tempered fractional differential equations

In this paper, we study the first comparison theorem and the second comparison theorem of Caputo (and Riemann–Liouville) tempered fractional differential equations with order $$\alpha \in (0, 1)$$. The detailed proof process is given. At the same time, continuous dependence of the solutions of the equation on the parameter is analyzed, which is used in the previous process of proof. In addition, we give two examples to support the theoretical analysis

Zero-Hopf bifurcation analysis in an inertial two-neural system with delayed Crespi function

In this paper, we study a four-dimensional inertial two-nervous system with delay. By analyzing the distribution of eigenvalues, the critical value of zero-Hopf bifurcation is obtained. Complex dynamic behaviors are considered when two parameters change simultaneously. Pitchfork and Hopf bifurcation critical lines at near the zero-Hopf point are obtained by using the central manifold reduction and the normal form theory. The bifurcation diagram is given, and the results of period-doubling bifurcation into chaotic region in the inertial two-neural system with delayed Crespi function are shown

Zero-Hopf bifurcation and Hopf bifurcation for smooth Chua’s system

Abstract Based on the fact that Chua’s system is a classic model system of electronic circuits, we first present modified Chua’s system with a smooth nonlinearity, described by a cubic polynomial in this paper. Then, we explore the distribution of the equilibrium points of the modified Chua circuit system. By using the averaging theory, we consider zero-Hopf bifurcation of the modified Chua system. Moreover, the existence of periodic solutions in the modified Chua system is derived from the classical Hopf bifurcation theorem

Melnikov-type method for a class of planar hybrid piecewise-smooth systems with impulsive effect and noise excitation: heteroclinic orbits

The classical Melnikov method for heteroclinic orbits is extended theoretically to a class of hybrid piecewisesmooth systems with impulsive effect and noise excitation. We assume that the unperturbed system is a piecewise Hamiltonian system with a pair of heteroclinic orbits. The heteroclinic orbit transversally jumps across the first switching manifold by impulsive effect, and crosses the second switching manifold continuously. In effect, the trajectory of the corresponding perturbed system crosses the second switching manifold by applying the reset map describing the impact rule instantaneously. The random Melnikov process of such systems is then derived by measuring the distance of the perturbed stable and unstable manifolds, the criteria for the onset of chaos with or without noise excitation is established. In this derivation process, we overcome the difficulty that the derivation method of the corresponding homoclinic case cannot be directly used due to the difference between the symmetry of the homoclinic orbit and the asymmetry of the heteroclinic orbit. Finally, we investigate the complicated dynamics of a particular piecewise-smooth system with and without noise excitation under the reset maps, impulsive effect, non-autonomous periodic and damping perturbations by this new extended method and numerical simulations. The numerical results verify the correctness of the theoretical results, and demonstrate that this extended method is simple and effective for studying the dynamics of such systems