A chaotic system with only one stable equilibrium

If you are given a simple three-dimensional autonomous quadratic system that has only one stable equilibrium, what would you predict its dynamics to be, stable or periodic? Will it be surprising if you are shown that such a system is actually chaotic? Although chaos theory for three-dimensional autonomous systems has been intensively and extensively studied since the time of Lorenz in the 1960s, and the theory has become quite mature today, it seems that no one would anticipate a possibility of finding a three-dimensional autonomous quadratic chaotic system with only one stable equilibrium. The discovery of the new system, to be reported in this Letter, is indeed striking because for a three-dimensional autonomous quadratic system with a single stable node-focus equilibrium, one typically would anticipate non-chaotic and even asymptotically converging behaviors. Although the new system is not of saddle-focus type, therefore the familiar \v{S}i'lnikov homoclinic criterion is not applicable, it is demonstrated to be chaotic in the sense of having a positive largest Lyapunov exponent, a fractional dimension, a continuous broad frequency spectrum, and a period-doubling route to chaos.Comment: 13 pages, 10 figure

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

Entropy in Dynamic Systems

In order to measure and quantify the complex behavior of real-world systems, either novel mathematical approaches or modifications of classical ones are required to precisely predict, monitor, and control complicated chaotic and stochastic processes. Though the term of entropy comes from Greek and emphasizes its analogy to energy, today, it has wandered to different branches of pure and applied sciences and is understood in a rather rough way, with emphasis placed on the transition from regular to chaotic states, stochastic and deterministic disorder, and uniform and non-uniform distribution or decay of diversity. This collection of papers addresses the notion of entropy in a very broad sense. The presented manuscripts follow from different branches of mathematical/physical sciences, natural/social sciences, and engineering-oriented sciences with emphasis placed on the complexity of dynamical systems. Topics like timing chaos and spatiotemporal chaos, bifurcation, synchronization and anti-synchronization, stability, lumped mass and continuous mechanical systems modeling, novel nonlinear phenomena, and resonances are discussed