A novel entanglement functions-based 4D fractional-order chaotic system and its bifurcation analysis

Abstract

A novel 4D fractional-order chaotic entanglement system based on sinusoidal functions is established in this paper. We aim to reveal the relationship between the dynamical behavior of the new system and its entanglement coefficients. It is found that the equilibrium point of the system varies regularly with the successive change of the entanglement coefficient. The supercritical pitchfork bifurcation phenomenon of the new system is discussed based on the fractional-order stability theory. Furthermore, sufficient conditions and threshold for supercritical Hopf bifurcation caused by the entanglement coefficient are provided. Finally, the route to chaos of the new system is explored utilizing multiple numerical indicators, such as spectral entropy complexity, bifurcation diagrams, Lyapunov exponential spectrum, phase portraits, and 0-1 test curves. The results indicate that in addition to various chaotic attractors, there are phenomena such as period-doubling bifurcations, period windows, and coexisting symmetric attractors (periodic or chaotic).

A secure communication scheme based on generalized modified projective synchronization of a new 4-D fractional-order hyperchaotic system

“…4. Finally, the normalized spectral entropy is obtained by computing Shannon entropy using the following formula [68] Here, we set q ä [0.743, 1] with a step size of 0.001 and a ä [5,20] with a step size of 0.01. In practice, the chaos diagram can help us select the suitable parameters to obtain a highly complex system.…”

“…Fractional calculus is an advanced field in mathematics that studies real-world phenomena that are modeled by non-integer-order derivatives. Since fractional calculus provides more accurate models than integer order calculus, it has garnered significant attention in the last few years for both its derivative and integration calculus [20,21]. Nowadays, there are many types of fractional derivatives, with or without singular kernals [22].…”