Narrow Results

Chaotic dynamical systems without fixed points are promising in the generation of pseudo-random sequences because orbits will not converge to a fixed point. In this class of systems there are no final fixed points, so the basin of attraction of a fixed point does not exist. The absence of fixed points makes it difficult to analyze the dynamics of the systems because a fixed point gives us a lot of information about the dynamics of the system. In addition, if there is an amplitude control parameter for the generated chaotic signals, the tolerance and adaptability are higher and better in the application process. In view of the aforementioned, in this paper, a novel class of discrete maps is presented and described by a kind of piecewise linear (PWL) maps. Necessary and sufficient conditions are given to guarantee that this class of discrete maps does not have any fixed point. Furthermore, we introduce families of these PWL discrete maps without fixed points that present positive Lyapunov exponents and have chaotic dynamics with amplitude control. From these families, we select one particular map, which is analyzed theoretically and proved to be chaotic in the sense of Devaney.

We study the complex dynamics of a discrete analogue of the classical flow dynamical system — Rössler oscillator. Minimal ensembles of two and three coupled discrete oscillators with different topologies are considered. As the main research tool we used the method of Lyapunov exponents charts. For coupled systems, the possibility of two-, three- and four-frequency quasi-periodicity is revealed. Illustrations in the form of Fourier spectra are presented. Doublings of invariant curves, two- and three-dimensional tori are found. The transition from two-dimensional tori to three-dimensional ones occurs through a quasi-periodic saddle-node bifurcation of invariant tori or through a quasi-periodic Hopf bifurcation. A discrete version of the hyperchaotic Rössler oscillator is also discussed. It exhibits dynamical behavior close to a flow system in some measure.

In this study, we investigate the occurrence of a three-frequency quasi-periodic torus in a three-dimensional Lotka–Volterra map. Our analysis extends to the observation of a doubling bifurcation of a closed invariant curve, leading to a subsequent transition into a state of hyperchaos. The absorption of various saddle periodic orbits into the hyperchaotic attractor is demonstrated through distance computation, and we explore the dimensionality of both stable and unstable manifolds. Various routes to cyclic and disjoint quasi-periodic structures are presented. Specifically, we showcase the transition from a saddle-node connection to a saddle-focus connection, leading to the formation of quasi-periodic closed cyclic disjoint curves, as revealed by the computation of one-dimensional unstable manifold. Additionally, we show an unusual transition from a period-2 orbit to a period-6 orbit and uncover the mechanism related to two subsequent bifurcations: (a) sub-critical Neimark–Sacker bifurcation, and (b) saddle-node bifurcation. Our approach involves the use of computational methods for constructing one-dimensional manifolds, extending saddle periodic orbits through a one-parameter continuation, and employing a multidimensional Newton–Raphson approach for pinpointing the saddle periodic orbits in the three-dimensional map.