Infinitely Many Hidden Attractors in a Simple Six-Term Chaotic 2-torus Snap System with a Line of Equilibria

Hidden and self-excited attractors in a heterogeneous Cournot oligopoly model

Hidden attractors, singularly degenerate heteroclinic orbits, multistability and physical realization of a new 6D hyperchaotic system

In the present decade, chaotic systems are used and appeared in many fields like in information security, communication systems, economics, bioengineering, mathematics, etc. Thus, developing of chaotic dynamical systems is most interesting and desirable in comparison with dynamical systems with regular behaviour. The chaotic systems are categorised into two groups. These are (i) system with self-excited attractors and (ii) systems with hidden attractors. A self-excited attractor is generated depending on the location of its unstable equilibrium point and in such case, the basin of attraction touches the equilibria. But, in the case of hidden attractors, the basin of attraction does not touch the equilibria and also finding of such attractors is a difficult task. The systems with (i) no equilibrium point and (ii) stable equilibrium points belong to the category of hidden attractors. Recently chaotic systems with infinitely many equilibria/a line of equilibria are also considered under the cattegory of hidden attractors. Higher dimensional chaotic systems have more complexity and disorders compared with lower dimensional chaotic systems. Recently, more attention is given to the development of higher dimensional chaotic systems with hidden attractors. But, the development of higher dimensional chaotic systems having both hidden attractors and self-excited attractors is more demanding. This chapter reports three hyperchaotic and two chaotic, 5-D new systems having the nature of both the self-excited and hidden attractors. The systems have non-hyperbolic equilibria, hence, belong to the category of self-excited attractors. Also, the systems have many equilibria, and hence, may be considered under the category of a chaotic system with hidden attractors. A systematic procedure is used to develop the new systems from the well-known 3-D Lorenz chaotic system. All the five systems exhibit multistability with the change of initial conditions. Various theoretical and numerical tools like phase portrait, Lyapunov spectrum, bifurcation diagram, Poincare map, and frequency spectrum are used to confirm the chaotic nature of the new systems. The MATLAB simulation results of the new systems are validated by designing their circuits and realising the same.

Extreme multi-stability is a newly introduced property observed in nonlinear dynamical systems. Such systems have very rich dynamical solutions depending on both parameters and initial conditions. On the other hand, designing dynamical systems with special features related to their equilibria is of great interest. In this paper, a novel chaotic system with extreme multi-stability and a line of equilibrium is presented. Such systems are so infrequent. It also should be noted that this designed chaotic system belongs to the category of dynamical systems with hidden attractors. Complete dynamical properties of this new system are investigated. Also, by the assistance of FPGA and electronic circuit implementation, this system is implemented.

In 1695, G. Leibniz laid the foundations of fractional calculus, but mathematicians revived it only 300 years later. In 1971, L.O. Chua postulated the existence of a fourth circuit element, called memristor, but Williams’s group of HP Labs realized it only 37 years later. In recent years, few unusual dynamical systems, such as those with a line of equilibriums, with stable equilibria or without equilibrium, which belong to chaotic systems with hidden attractors, have been reported. By looking at these interdisciplinary and promising research areas, in this chapter, a fractional-order 4-D memristive system with a line of equilibria is introduced. In particular, a hyperchaotic behavior in a simple fractional-order memristor-based system is presented. Systematic studies of the hyperchaotic behavior in the integer and fractional-order form of the system are performed using phase portraits, Poincare maps, bifurcation diagrams and Lyapunov exponents. Simulation results show that both integer-order and fractional-order system exhibit hyperchaotic behavior over a wide range of control parameter. Finally, the electronic circuits for the evaluation of the theoretical model of the proposed integer and fractional-order systems are presented.

Extreme multistable systems can show vibrant dynamical properties and infinitely many coexisting attractors generated by changing the initial conditions while the system and its parameters remain unchanged. On the other hand, the frequency of extreme events in society is increasing which could have a catastrophic influence on human life worldwide. Thus, complex systems that can model such behaviors are very significant in order to avoid or control various extreme events. Also, hidden attractors are a crucial issue in nonlinear dynamics since they cannot be located and recognized with conventional methods. Hence, finding such systems is a vital task. This paper proposes a novel five-dimensional autonomous chaotic system with a line of equilibria, which generates hidden attractors. Furthermore, this system can exhibit extreme multistability and extreme events simultaneously. The fascinating features of this system are examined by dynamical analysis tools such as Poincaré sections, connecting curves, bifurcation diagrams, Lyapunov exponents spectra, and attraction basins. Moreover, the reliability of the introduced system is confirmed through analog electrical circuit design so that this chaotic circuit can be employed in many engineering fields.

The Chameleon hidden chaotic system is a chaotic system with exciting and particular properties. One of its specific features is that by changing its constant parameters, the flow exhibits three classes of hidden attractors containing one stable equilibrium, line of equilibria, or no equilibria, and self-excited attractors. In this paper, a new Chameleon hidden hyperchaotic flow is proposed and the corresponding hidden attractors and self-excited attractor are evaluated. Then, a new super-twisting fast terminal adaptive sliding mode control technique is suggested for finite-time stability of Chameleon hidden hyperchaotic flows. Dynamics of this chaotic system with different values of constant parameters has been studied using phase portraits, stability analysis and bifurcation diagrams. Descriptive simulations on the Chameleon hidden hyperchaotic system with external disturbances and parametric uncertainties are presented to approve the effectiveness of the proposed method.

This paper introduces a newly designed four-dimensional memristive chaotic system. The novel oscillator is chaotic regarding the findings that the system's dynamic has one positive Lyapunov exponent (LE). Also, due to the results of the equilibrium points analysis, it is shown that the oscillator has a line of equilibria, so the attractors of this system are hidden. Moreover, the study of energy dissipation of this system, power spectrum, and Poincaré sections are conducted to investigate the system's dynamics. The complex features of this system are investigated with the aid of bifurcation diagrams, LEs spectra, approximate entropy, and basin of attraction.

Chapter 5 - A New Five Dimensional Multistable Chaotic System With Hidden Attractors

In this paper, a new commensurate fractional-order chaotic oscillator is presented. The mathematical model with a weak feedback term, which is named hypogenetic flow, is proposed based on the Liu system. And with changing the parameters of the system, the hidden attractor can have no equilibrium points or line equilibrium. What is more interesting is that under the occasion that no equilibrium point can be obtained, the phase trajectory can converge to a minimal field under the lead of some initial conditions, similar to the fixed point. We call it the virtual equilibrium point. On the other hand, when the value of parameters can produce an infinite number of equilibrium points, the line equilibrium points are nonhyperbolic. Moreover than that, there are coexistence attractors, which can present hyperchaos, chaos, period, and virtual equilibrium point. The dynamic characteristics of the system are analyzed, and the parameter estimation is also studied. Then, an electronic circuit implementation of the system is built, which shows the feasibility of the system. At last, for the fractional system with hidden attractors, the finite-time synchronization control of the system is carried out based on the finite-time stability theory of the fractional system. And the effectiveness of the controller is verified by numerical simulation.