A new three-dimensional chaotic system without equilibrium points, its dynamical analyses and electronic circuit application

U radu se predstavlja i analizira novi trodimenzionalni kaotični sustav bez točaka ekvilibrija. Osnovna dinamička analiza tog novog kaotičnog sustava bez točaka ekvilibrija izvodi se pomoću sustava ekvilibrija, faznih slika (portreta), osjetljivosti na početne uvjete, fraktalne dimenzije i kaotičnog ponašanja. Uz to je izvedena analiza spektra Lyapunovljevih eksponenata i bifurkacijska analiza predloženog kaotičnog sustava primjenom izabranih parametara. Kaotični sustav bez točaka ekvilibrija dobiven je detaljnom teorijskom analizom kao i simulacijama s dizajniranim elektroničkim krugom. Sustav kaosa bez točaka ekvilibrija također je poznat kao sustav kaosa sa skrivenim atraktorom i o tome postoji mali broj istraživanja. Budući da ne postoje homokliničke i heterokliničke orbite, Schilnikova metoda se ne može primijeniti kako bi se ustanovilo je li sustav kaotičan ili nije kaotičan. Stoga kaotični sustav bez točaka ekvilibrija može biti od koristi u mnogim tehničkim primjenama, naročito u kriptologiji i kodiranju zasnovanom na kaosu. Nadalje, ovaj predloženi kaotični sustav bez točaka ekvilibrija može se ponašati na mnogo dinamički nepoznatih načina. Takve vrste ponašanja nepoznatih kaotičnih atraktora zahtijevaju dodatna istraživanja.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

Synchronization in Coupled Multistable Systems with Hidden Attractors

In this paper, we study the results of coupling multistable systems which have hidden attractors with each other. Three modified Sprott systems were coupled and their synchronization was observed. The final state of the synchronized system changes with the change in the coupling strength. This was seen for two different types of coupling, one with a single variable and the other with two system variables

Singular Orbits and Dynamics at Infinity of a Conjugate Lorenz-Like System

A conjugate Lorenz-like system which includes only two quadratic nonlinearities is proposed in this paper. Some basic properties of this system, such as the distribution of its equilibria and their stabilities, the Lyapunov exponents, the bifurcations are investigated by some numerical and theoretical analysis. The forming mechanisms of compound structures of its new chaotic attractors obtained by merging together two simple attractors after performing one mirror operation are also presented. Furthermore, some of its other complex dynamical behaviours, which include the existence of singularly degenerate heteroclinic cycles, the existence of homoclinic and heteroclinic orbits and the dynamics at infinity, etc, are formulated in detail. In the meantime, some problems deserving further investigations are presented

Research of Chaotic Dynamics of 3D Autonomous Quadratic Systems by Their Reduction to Special 2D Quadratic Systems

New results about the existence of chaotic dynamics in the quadratic 3D systems are derived. These results are based on the method allowing studying dynamics of 3D system of autonomous quadratic differential equations with the help of reduction of this system to the special 2D quadratic system of differential equations