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

Degenerate Fold-Hopf Bifurcations in a Rössler-Type System

We study the Hopf and the fold--Hopf bifurcations of the R\"ossler--type differential system * =-y-z, =x ay, =-cz byz, * with b 0. We show that the classical Hopf bifurcation cannot be applied to this system for detecting the fold--Hopf bifurcation, which here is studied using the averaging theory. Our results show that a Hopf bifurcation takes place at the equilibrium (-ac/b,c/b,-c/b) when c=a<0. This Hopf bifurcation becomes a fold--Hopf bifurcation when c=a=0

A Weak Signal Detection Application Based on Hyperchaotic Lorenz System

Due to accurate capability to detect weak signal, chaotic oscillators have become an interesting topic for many scientific researches. In this paper, two hyperchaotic Lorenz systems are presented to detect weak signal. These systems are chosen because of their parametric variety and high applicability. Dynamic behaviors of two hyperchaotic systems are analysed in detail. For this purpose, the Lyapunov exponent values and bifurcation behaviours of two hyperchaotic systems are analysed for weak signal detection applications. The relationship between the system state and the amplitude of the weak signal is defined by examining the Lyapunov exponents of the system. So, dynamic characteristics of two chaotic oscillators are observed by this way. The critical chaotic parametric threshold value of a chaotic system is easily determined by the bifurcation analysis. The bifurcation threshold value named as tangent bifurcation point is the most suitable one to detect weak signal. For this purpose, the tangent bifurcation points of these systems are determined via bifurcation analysis. Additionally, weak signal detection applications of two hyperchaotic systems are also studied. The applicability of the proposed systems is shown by these applications. These systems also detect the weak signal with low signal to noise ratio (SNR). Simulation results obtained from Matlab-Simulink® program verify the studied method

Dynamics at infinity, degenerate Hopf and zero-Hopf bifurcation for Kingni–Jafari system with hidden attractors

To understand the complex dynamics of Kingni-Jafari system with hidden attractors, the first objective of this paper was to study the global dynamics, and give a complete description of the dynamics of Kingni-Jafari system at infinity by using the Poincar´e compactification of a polynomial vector field in R3. The second objective of this paper was to proof the existence of periodic solutions in the Kingni-Jafari system by classic Hopf bifurcation and degenerate Hopf bifurcation. Moreover, based on averaging theory, a small amplitude periodic solution that bifurcate from a zero-Hopf equilibrium was derived in the Kingni-Jafari system. The theoretical analysis and simulations demonstrate the rich dynamics of the syste