Universal theory of dynamical chaos in nonlinear dissipative systems of differential equations

A new universal theory of dynamical chaos in nonlinear dissipative systems of differential equations including ordinary and partial, autonomous and non-autonomous differential equations and differential equations with delay arguments is presented in this paper. Four corner-stones lie in the foundation of this theory: the Feigenbaum’s theory of period doubling bifurcations in one-dimensional mappings, the Sharkovskii’s theory of bifurcations of cycles of an arbitrary period up to the cycle of period three in one-dimensional mappings, the Magnitskii’s theory of rotor type singular points of two-dimensional non-autonomous systems of differential equations as a bridge between one-dimensional mappings and differential equations and the theory of homoclinic cascade of bifurcations of stable cycles in nonlinear differential equations. All propositions of the theory are strictly proved and illustrated by numerous analytical and computing examples.     

Universal theory of dynamical chaos in nonlinear dissipative systems of differential equations

The article presents a new universal theory of dynamical chaos in nonlinear dissipative systems of differential equations, including autonomous and nonautonomous ordinary differential equations (ODE), partial differential equations, and delay differential equations. The theory relies on four remarkable results: Feigenbaum’s period doubling theory for cycles of one-dimensional unimodal maps, Sharkovskii’s theory of birth of cycles of arbitrary period up to cycle of period three in one-dimensional unimodal maps, Magnitskii’s theory of rotor singular point in two-dimensional nonautonomous ODE systems, acting as a bridge between one-dimensional maps and differential equations, and Magnitskii’s theory of homoclinic bifurcation cascade that follows the Sharkovskii cascade. All the theoretical propositions are rigorously proved and illustrated with numerous analytical examples and numerical computations, which are presented for all classical chaotic nonlinear dissipative systems of differential equations.     

On the Cauchy problem for nonlinear dissipative systems

The large-time asymptotic behavior of solutions of the Cauchy problem for a system of nonlinear evolutionary equations with dissipation is studied. The approach used in the case of small initial data is based on the construction of solutions by the method of contracting mappings. In the case of large initial data, we will obtain the large-time asymptotics of solutions with a certain symmmetry of a nonlinear term taken into account. In the critical case, it is proved that if the initial data has a nonzero total mass, then the principal term of the large-time asymptotics of a solution is given by the self-similar solution uniquely determined by the total mass of the initial data. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 29, Voronezh Conference-1, 2005.     

Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems

This paper first summarizes the theory of quasi-periodic bifurcations for dissipative dynamical systems. Then it presents algorithms for the computation and continuation of invariant circles and of their bifurcations. Finally several applications are given for quasiperiodic bifurcations of Hopf, saddle-node and period-doubling type.     

Global bifurcations and chaos in externally excited cyclic systems

The global bifurcations in mode interaction of a nonlinear cyclic system subjected to a harmonic excitation are investigated with the case of the primary resonance, the averaged equations representing the evolution of the amplitudes and phases of the interacting normal modes exhibit complex dynamics. The energy-phase method proposed by Haller and Wiggins is employed to analyze the global bifurcations for the cyclic system. The results obtained here indicate that there exist the Silnikov-type multi-pulse orbits homoclinic to certain invariant sets for the resonant case in both Hamiltonian and dissipative perturbations, which imply that chaotic motions occur for this class of systems. Homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are found and the visualizations of these complicated structures are presented.     

On global bifurcations in three-dimensional diffeomorphisms leading to wild Lorenz-like attractors

We study dynamics and bifurcations of three-dimensional diffeomorphisms with nontransverse heteroclinic cycles. We show that bifurcations under consideration lead to the birth of wild-hyperbolic Lorenz attractors. These attractors can be viewed as periodically perturbed classical Lorenz attractors, however, they allow for the existence of homoclinic tangencies and, hence, wild hyperbolic sets.      

Stability for uncertain nonlinear dissipative systems

This paper presents a technique for constructing a storage function for a broad class of nonlinear passive systems with mismatched uncertainties. And the problem for H_∞ disturbance attenuation with internal stability is studied by using robust adaptive output feedback controller which need not construct any state observer. The paper also shows how to explicitly design output feedback control that attenuates the disturbances effect on the output to an arbitrary degree of accuracy. Further, the results derived in this paper complement the previous work in the literature.     

On the occurrence of chaos via different routes to chaos: period doubling and border-collision bifurcations

This paper introduces a new 2D piecewise smooth discrete-time chaotic mapping with rarely observed phenomenon – the occurrence of the same chaotic attractor via different and distinguishable routes to chaos: period doubling and border-collision bifurcations as typical futures. This phenomenon is justified by the location of system equilibria of the proposed mapping, and the possible bifurcation types in smooth dissipative systems. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 61, Optimal Control, 2008.     

Study of intermittent bifurcations and chaos in boost PFC converters by nonlinear discrete models

This paper mainly deals with nonlinear phenomena like intermittent bifurcations and chaos in boost PFC converters under peak-current control mode. Two nonlinear models in the form of discrete maps are derived to describe precisely the nonlinear dynamics of boost PFC converters from two points of view, i.e., low- and high-frequency regimes. Based on the presented discrete models, both the evolution of intermittent behavior and the periodicity of intermittency are investigated in detail from the fast and slow-scale aspects, respectively. Numerical results show that the occurrence of intermittent bifurcations and chaos with half one line period is one of the most distinguished dynamical characteristics. Finally, we make some instructive conclusions, which prove to be helpful in improving the performances of practical circuits.     

Nature of chaos in conservative and dissipative systems of the Duffing-Holmes oscillator

We show that the chaotic dynamics of the conservative Duffing-Holmes oscillator obeys the universal Feigenbaum-Sharkovskii-Magnitskii theory of passage to chaos in dynamical systems of ordinary differential equations. Moreover, the cascades of bifurcations of the conservative and dissipative oscillators are continuously related to each other. Our study uses the stable control method, which permits rapidly stabilizing nearly any periodic solution and dynamically changing system parameters without moving far away from that periodic solution.     

Chaotic attractors in nonlinear dissipative systems of ordinary differential equations

A mechanism is proposed describing the formation of irregular attractors in a wide class of three-dimensional nonlinear autonomous dissipative systems of ordinary differential equations with singular cycles. The attractors of such systems, called singular attractors, lie on two-dimensional surfaces in the phase space and have no positive Lyapunov exponents. In all systems of this class the onset of chaos follows the same universal mechanism: a cascade of Feigenbaum’s period doubling bifurcations, a subharmonic cascade of Sharkovskii’s bifurcations, and eventually a homoclinic cascade. All classical chaotic systems, including Lorenz, Rössler, and Chua systems, satisfy these conditions.     

On Lagrangians and Hamiltonians of some fourth-order nonlinear Kudryashov ODEs

We derive the Lagrangians of the reduced fourth-order ordinary differential equations studied by Kudryashov, when they satisfy the conditions stated by Fels [Fels ME, The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations. Trans Am Math Soc 1996;348:5007–29] using Jacobi’s last multiplier technique. In addition the Hamiltonians of these equations are derived via Jacobi–Ostrogradski’s theory. In particular, we compute the Lagrangians and Hamiltonians of fourth-order Kudryashov equations which pass the Painlevé test.     

Transition to chaos in nonlinear dynamical systems described by ordinary differential equations

We investigate scenarios that create chaotic attractors in systems of ordinary differential equations (Vallis, Rikitaki, Rossler, etc.). We show that the creation of chaotic attractors is governed by the same mechanisms. The Feigenbaum bifurcation cascade is shown to be universal, while subharmonic and homoclinic cascades may be complete, incomplete, or not exist at all depending on system parameters. The existence of a saddle-focus equilibrium plays an important and possibly decisive role in the creation of chaotic attractors in dissipative nonlinear systems described by ordinary differential equations. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 3, pp. 73–98, 2003.     

Robust reliable dissipative control of nonlinear networked control systems

This article focuses on the robust reliable dissipative control issue for a class of switched discrete‐time nonlinear networked control systems with external energy bounded disturbances. In particular, nonlinearities are modeled in a probabilistic way according to Bernoulli distributed white sequence with known conditional probability. A Lyapunov–Krasovskii functional is proposed based on which sufficient conditions for the existence of the reliable dissipative controller are derived in terms of linear matrix inequalities (LMIs) which ensures exponentially stability as well as dissipative performance of the resulting closed‐loop system. The explicit expression of the desired controller gains can be obtained by solving the established LMIs. Finally, a numerical example is presented to demonstrate the effectiveness and applicability of the proposed design strategy. © 2016 Wiley Periodicals, Inc. Complexity 21: 427–437, 2016