Dynamical Analysis and Periodic Solution of a Chaotic System with Coexisting Attractors

Chaotic attractors with no equilibria, with an unstable node, and with stable node-focus are presented in this paper. The conservative solutions are investigated by the semianalytical and seminumerical method. Furthermore, multiple coexisting attractors are investigated, and circuit is carried out. To study the system’s global structure, dynamics at infinity for this new chaotic system are studied using Poincaré compactification of polynomial vector fields in R 3 . Meanwhile, the dynamics near the infinity of the singularities are obtained by reducing the system’s dimensions on a Poincaré ball. The averaging theory analyzes the periodic solution’s stability or instability that bifurcates from Hopf-zero bifurcation.

Multistability in a Fractional-Order Centrifugal Flywheel Governor System and Its Adaptive Control

In this paper, a 4D fractional-order centrifugal flywheel governor system is proposed. Dynamics including the multistability of the system with the variation of system parameters and the derivative order are investigated by Lyapunov exponents (LEs), bifurcation diagram, phase portrait, entropy measure, and basins of attraction, numerically. It shows that the minimum order for chaos of the fractional-order centrifugal flywheel governor system is q = 0.97, and the system has rich dynamics and produces multiple coexisting attractors. Moreover, the system is controlled by introducing the adaptive controller which is proved by the Lyapunov stability theory. Numerical analysis results verify the effectiveness of the proposed method.

Generating coexisting attractors from a new four-dimensional chaotic system

This paper introduces a new four-dimensional chaotic system with a unique unstable equilibrium and multiple coexisting attractors. The dynamic evolution analysis shows that the system concurrently generates two symmetric chaotic attractors for fixed parameter values. Based on this system, an effective method is established to construct an infinite number of coexisting chaotic attractors. It shows that the introduction of some non-linear functions with multiple zeros can increase the equilibria and inspire the generation of coexisting attractor of the system. Numerical simulations verify the availability of the method.

Network Dynamics of a Fractional-Order Phase-Locked Loop with Infinite Coexisting Attractors

We have investigated a fractional-order phase-locked loop characterised by a third-order differential equation. The integer-order mathematical model of the phase-locked loop (PLL) is first converted to fractional order using the Caputo-Fabrizio method. The stability of the equilibrium points is discussed in detail in both parameter and fractional-order domain. The proposed fractional-order phase-locked loop (FOPLL) model shows multiple coexisting attractors which was not discussed in the earlier literature of PLL. The significance of these infinite coexisting attractors is that they exist in the operation region of the PLL between [−π,π] which increases the complexity of operation of the PLLs. Mainly when such FOPLLs are used in large-scale networks, the synchronisation of the FOPLLs becomes complicated and will result in unstable control conditions. Hence, studying the network dynamics of such FOPLLs is significant which motivates us to investigate the synchronisation phenomenon of the FOPLLs constructed in a square network. We could show that, because of the multiple coexisting attractors, the FOPLLs show various synchronisation phenomena, and more importantly in the chaotic region for lower fractional-order values, we could show that the FOPLLs are synchronised and this finding is very useful to completely analyse the FOPLL networks in high-frequency operations.

Dynamic Analysis, Circuit Design, and Synchronization of a Novel 6D Memristive Four-Wing Hyperchaotic System with Multiple Coexisting Attractors

In this work, a novel 6D four-wing hyperchaotic system with a line equilibrium based on a flux-controlled memristor model is proposed. The novel system is inspired from an existing 5D four-wing hyperchaotic system introduced by Zarei (2015). Fundamental properties of the novel system are discussed, and its complex behaviors are characterized using phase portraits, Lyapunov exponential spectrum, bifurcation diagram, and spectral entropy. When a suitable set of parameters are chosen, the system exhibits a rich repertoire of dynamic behaviors including double-period bifurcation of the quasiperiod, a single two-wing, and four-wing chaotic attractors. Further analysis of the novel system shows that the multiple coexisting attractors can be observed with different system parameter values and initial values. Moreover, the feasibility of the proposed mathematical model is also presented by using Multisim simulations based on an electronic analog of the model. Finally, the active control method is used to design the appropriate controller to realize the synchronization between the proposed 6D memristive hyperchaotic system and the 6D hyperchaotic Yang system with different structures. The Routh–Hurwitz criterion is used to prove the rationality of the controller, and the feasibility and effectiveness of the proposed synchronization method are proved by numerical simulations.

Fast–Slow Dynamics and Bifurcation Mechanism in a Novel Chaotic System

This paper proposes a novel three-dimensional chaotic system with multiple coexisting attractors, where different values of a constant control parameter may drive the chaotic behaviors to evolve from single-scroll to double-scroll attractors. When the controlling term is replaced by a periodic harmonic excitation where the exciting frequency is far less than the natural frequency, chaotic movement may disappear, while periodic bursting oscillations will take place. Based on the fact that during a period defined by the natural frequency, the exciting term keeps almost a constant, the whole exciting term can be regarded as a slow-varying parameter resulting in a generalized autonomous system, its equilibrium branches as well as the related bifurcations occurring with the variation of the slow-varying parameter are derived. With the increase of the exciting amplitude, asymmetric and symmetric bursting attractors can be observed, for which the mechanism can be analyzed by the overlap of the equilibrium branches and the transformed phase portraits. With different values of the exciting amplitude corresponding to the change region of the slow-varying parameter, different bifurcations such as fold and Hopf bifurcations may involve the bursting structures, leading to different types of bursting oscillations. Furthermore, the phase space can be divided into two regions by a line boundary because of the symmetry of the vector field. When the trajectory from one region returning to the region arrives at the boundary, two asymmetric bursting attractors located in different regions coexist, which are symmetric to each other. However, when the trajectory passes across the boundary, an enlarged symmetric bursting attractor can be observed, whose trajectory connects the two original asymmetric attractors. Furthermore, it is found that when the trajectory runs along a stable equilibrium branch to the bifurcation point, it may move almost strictly along an unstable equilibrium branch of the fast subsystem because of the delay influence of the bifurcation.

Coexistence of Two-Dimensional Attractors in Border Collision Normal Form

The two-dimensional border collision normal form is considered. It is known that multiple attractors can exist in this piecewise smooth system. We show that in appropriate parameter regions there can be a robust transition from a stable fixed point to multiple coexisting attractors with toological dimensions equal to two.