Abstract
Original scientific paper In this paper, a new three-dimensional chaotic system without equilibrium points is introduced and analysed. Basic dynamical analysis of this new chaotic system without equilibrium points is carried out by means of system equilibria, phase portraits, sensitivity to initial conditions, fractal dimension and chaotic behaviours. In addition, in this paper Lyapunov exponents spectrum and bifurcation analysis of the proposed chaotic system have been executed by means of selected parameters. The chaotic system without equilibrium points has been executed by detailed theoretical analysis as well as simulations with designed electronical circuit. A chaotic system without equilibrium points is also known as chaotic system with hidden attractor and there are very few researches in the literature. Since they cannot have homoclinic and heteroclinic orbits, Shilnikov method cannot be applied to find whether the system is chaotic or not. Therefore, it can be useful in many engineering applications, especially in chaos based cryptology and coding information. Furthermore, introduced chaotic system without equilibrium points in this paper can have many unknown dynamical behaviours. These behaviours of the strange chaotic attractors deserve further investigation.
Highlights
Chaotic dynamical equation research is an important problem in nonlinear science
Case 3 covers chaotic attractors with 1 saddle and 2 stable node-foci. Another such 3D chaotic system was introduced by Yang and Chen [11]
The new chaotic system without equilibrium points is described by the following differential equation 1: x = ay − x + zy y = −bxz − cx + yz + d
Summary
Chaotic dynamical equation research is an important problem in nonlinear science. Chaotic systems have been widely studied within scientific and engineering environments in the last years [1÷23]. Case 3 covers chaotic attractors with 1 saddle and 2 stable node-foci. Another such 3D chaotic system was introduced by Yang and Chen [11]. Pehlivan and Uyaroglu [12] and Yang et al [13] recently designed novel chaotic systems with 2 stable node-foci. [24], a systematic search to find 3D chaotic systems with quadratic nonlinearities and no equilibria was performed. Since they cannot have homoclinic and heteroclinic orbits, the Shilnikov method [27] cannot be applied to no equilibria systems.
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