We present a design strategy in terms of ordinary differential equations which creates chaotic attractors with an increasing number of positive Lyapunov exponents as the (finite) dimension of the system is increased. First, we introduce the most simple abstract equation containing only one nonlinearity. Second, we suggest a piecewise linear version of the abstract equation. Third, we propose a set of chemical reactions and demonstrate that the corresponding rate equations produce hyperchaotic behavior equivalent to the abstract system.

Design of hyperchaotic flows.

Original Articles: Predictive control and synchronization of chaotic and hyperchaotic systems based on a T-S fuzzy model

We propose a criterion for the destruction of a two-dimensional torus through the formation of an infinite set of cusp points on the closed invariant curves defining the resonance torus. This mechanism is specific to noninvertible maps. The cusp points arise when the tangent to the torus at the point of intersection with the critical curve ${L}_{0}$ coincides with the eigendirection corresponding to vanishing eigenvalue for the noninvertible map. Further parameter changes lead typically to the generation of loops (self-intersections of the invariant manifolds) followed by the transformation of the torus into a complex chaotic set.

/pdf/torus-breakdown-in-noninvertible-maps-1qqlob9t2l.pdf

Torus breakdown in noninvertible maps.

The paper investigates the special clustering phenomena that one can observe in systems of nonlinear oscillators that are coupled via a shared flow of primary resources (or a common power supply). This type of coupling, which appears to be quite frequent in nature, implies that one can no longer separate the inherent dynamics of the individual oscillator from the properties of the coupling network. Illustrated by examples from microbiological population dynamics, renal physiology, and electronic oscillator theory, we show how competition for primary resources in a resource distribution chain leads to a number of new generic phenomena, including partial synchronization, sliding of the synchronization region with the resource supply, and coupling-induced inhomogeneity.

http://chaos.sgu.ru/science_work/publications/live_systems/chaos15-1_2005.pdf

Oscillator clustering in a resource distribution chain.

Chemical reaction chains with feedback of one of the products on the source of the chain are considered. A strategy is presented in terms of ordinary differential equations which creates one, two, and three positive Lyapunov exponents as the finite dimension of the system is increased. In particular, a nonlinear inhibition loop in a chemical reaction sequence controls the type of chaos. The bifurcation scenarios are studied and chaos and hyperchaos are found for broad regions of bifurcation parameter. Some implications for the occurrence of higher chaos in real systems are discussed.

Hyperchaos and chaotic hierarchy in low‐dimensional chemical systems

Chaotic hierarchy in a model of competing populations

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Chaotic hierarchy in a model of competing populations