A fractional-order form of a system with stable equilibria and its synchronization

Abstract There has been an increasing interest in studying fractional-order chaotic systems and their synchronization. In this paper, the fractional-order form of a system with stable equilibrium is introduced. It is interesting that such a three-dimensional fractional system can exhibit chaotic attractors. Full-state hybrid projective synchronization scheme and inverse full-state hybrid projective synchronization scheme have been designed to synchronize the three-dimensional fractional system with different four-dimensional fractional systems. Numerical examples have verified the proposed synchronization schemes

A new megastable chaotic oscillator with singularity

While multistability is known as a hot topic in nonlinear dynamics, two exceptional cases of multistable systems have been investigated less: extreme multistable systems and megastable systems are two newer categories of multistable dynamical systems. In this paper, for the first time, a chaotic megastable oscillator is introduced which has a singularity in its equations. The effect of the amplitude and frequency of forcing term on the dynamical behavior of the designed system is investigated. With the help of the bifurcation diagram and the Lyapunov exponents’ diagram, it is shown that the proposed oscillator can show a variety of dynamical behaviors, including limit cycle, torus, and strange attractor

A novel chaotic system in the spherical coordinates

Investigating new chaotic flows has been a hot topic for many years. Studying the chaotic attractors of systems with various properties illuminates a lamp to reveal the vague of the generation of chaotic attractors. A new chaotic system in the spherical coordinates is proposed in this paper. The system’s solution is inside a predefined sphere, and its attractor cannot cross the sphere. Investigation of equilibrium points of the system shows that the system has eight equilibria, and all of them are saddle. Bifurcation analysis of the system depicts the period-doubling route to chaos with changing the bifurcation parameter. Also, Lyapunov exponents in the studied interval of the bifurcation parameter are discussed. The basin of attraction of the system is investigated to show the sensitivity of the system to initial conditions

A new memristive chaotic flow with a line of equilibria

A new 4-D memristive chaotic flow is proposed in this paper. Dynamical investigation of the proposed system shows some specific properties. The system has a line of equilibria which are unstable in some limited intervals and stable at other intervals. Investigating bifurcation diagram of the system shows an inverse period-doubling route to chaos. Also, different initiation in plotting bifurcation diagram shows its multistability. In some intervals of the parameter, two Lyapunov exponents of the system are positive, and the attractor is hyper-chaotic. However, in some other ranges of parameters, only one Lyapunov exponent is positive, and the attractor is chaotic. Basin of attraction of the system is studied which shows a vast region of attraction for the chaotic attractor. Entropy analysis of the system provides a viewpoint into the unpredictability of the system

Extreme multi-stability analysis of a novel 5D chaotic system with hidden attractors, line equilibrium, permutation entropy and its secure communication scheme

In this paper a new 5D chaotic system with line equilibrium is designed and described to reveal its extreme multi-stability. Hence, all of the resulting attractors are hidden. The suggested system owns many complex dynamic behaviors in comparison with other chaotic systems. System initial state-associated complex dynamical behaviors are considered and we discover that it possesses an immeasurable number of coexisting attractors, which expresses the occurrence of extreme multi-stability. Besides, we also demonstrate the line equilibrium stability in detail, bifurcation diagrams, Lyapunov exponents, and basins of attraction. Also, in order to analyze the new 5D chaotic system we have considered the permutation entropy technique. Finally, the application of the novel 5D chaotic system with line equilibrium to the problem of chaos synchronization and secure communication through observer design is presented