Analysis on bifurcations of multiple limit cycles for a parametrically and externally excited mechanical system

Research on the bifurcations of the multiple limit cycles for a parametrically and externally excited mechanical system is presented in this paper. The original mechanical system is first transformed to the averaged equation in the Cartesian form, which is in the form of a Z₂-symmetric perturbed polynomial Hamiltonian system of degree 5. Then, using the bifurcation theory of planar dynamical system and the method of detection function, the bifurcations of the multiple limit cycles of the system are investigated and the configurations of compound eyes are also obtained.     

Synchronization of chaos in non-identical parametrically excited systems

In this paper, we investigate the synchronization of chaotic systems consisting of non-identical parametrically excited oscillators. The active control technique is employed to design control functions based on Lyapunov stability theory and Routh–Hurwitz criteria so as to achieve global chaos synchronization between a parametrically excited gyroscope and each of the parametrically excited pendulum and Duffing oscillator. Numerical simulations are implemented to verify the results.     

On cascades of bifurcations leading to chaos in several nonlinear dissipative systems of ODEs

This paper considers several nonlinear dissipative systems of ordinary differential equations. The studied systems undergo a full analysis of corresponding singular points on a whole set of parameters’ values variation. Specifically, types of singular points, boarders of stability regions, as well as presented local bifurcations, are determined. By using numerical methods a consideration of scenarios of transition to chaos in these systems with one bifurcation parameter variation is held. The aim of this research is a confirmation of a Feigenbaum–Sharkovskii–Magnitskii mechanism of transition to chaos unique for all dissipative systems of ODEs. As the result of analysis of one of the systems the lack of any chaotic behavior is shown with the help of Poincare sections.     

On the influence of a constant force on the appearance of period-doubling bifurcations and chaos in a harmonically excited pure cubic oscillator

A pure cubic oscillator with a constant and a harmonic force acting on it, which represents a nonlinear asymmetric system, is considered. Building on previous studies on the matter, analytical and numerical approaches are used to examine and illustrate its dynamics related to the phenomenon of period-doubling bifurcations and their development into chaos for different values of the constant force. The region of control parameters in which this scenario is possible is determined and discussed with a view to revisiting literature results and to giving novel and deeper insights into the phenomenon related to the influence of the magnitude of the constant force and certain resonances.     

Global chaos synchronization of new chaotic systems via nonlinear control

Nonlinear control is an effective method for making two identical chaotic systems or two different chaotic systems be synchronized. However, this method assumes that the Lyapunov function of error dynamic (e) of synchronization is always formed as V (e) = 1/2e^Te. In this paper, modification based on Lyapunov stability theory to design a controller is proposed in order to overcome this limitation. The method has been applied successfully to make two identical new systems and two different chaotic systems (new system and Lorenz system) globally asymptotically synchronized. Since the Lyapunov exponents are not required for the calculation, this method is effective and convenient to synchronize two identical systems and two different chaotic systems. Numerical simulations are also given to validate the proposed synchronization approach.     

Blowout bifurcations in a model for chaos in spin-wave dynamics

Blowout bifurcations are investigated numerically in a model for chaos in the coincidence regime of high-power ferromagnetic resonance, based on interactions between the uniform mode and two pairs of parametric spin waves. This model possess two orthogonal invariant manifolds corresponding to the excitation of only one spin–wave pair above the first-order Suhl instability threshold. Marginal synchronization of the amplitudes of spin–wave pairs, the exchange of stability of the invariant manifolds, as well as both supercritical and subcritical blowout bifurcations are observed as the system parameters are varied, with the accompanying on–off intermittency, attractor bubbling and intermingled basins of attraction.     

Resonant vibrations in harmonically excited weakly coupled mechanical systems with cyclic symmetry

The resonant vibrations in weakly coupled nonlinear cyclic symmetric structures are studied. These structures consist of weakly coupled identical nonlinear oscillators. A careful bifurcation analysis of the amplitude equations is performed in the fundamental resonance case for an illustrative example consisting of a three particle system. In case of a uniformly distributed excitation, a localized response is identified in which one of the particles exhibits large amplitude motions compared to those of the other particles. In case of single-particle excitation, it is found that for very small coupling strength and large external mistuning, a large stable localized periodic response coexists with an extended small response. With an increase in the coupling strength, multiple extended solutions arise near the exact external resonance via saddle-node bifurcations. Further increase in coupling strength and a decrease in damping results in isolated asymmetric solution branches, which bifurcate from the symmetric solutions via symmetry-breaking bifurcations. The role of coupling strength in creating/destroying localized solutions is discussed.     

Quasi-periodic bifurcations in reversible systems

Invariant tori of integrable dynamical systems occur both in the dissipative and in the conservative context, but only in the latter the tori are parameterized by phase space variables. This allows for quasi-periodic bifurcations within a single given system, induced by changes of the normal behavior of the tori. It turns out that in a non-degenerate reversible system all semi-local bifurcations of co-dimension 1 persist, under small non-integrable perturbations, on large Cantor sets.     

Hopf bifurcations in dynamical systems

The onset of instability in autonomous dynamical systems (ADS) of ordinary differential equations is investigated. Binary, ternary and quaternary ADS are taken into account. The stability frontier of the spectrum is analyzed. Conditions necessary and sufficient for the occurring of Hopf, Hopf–Steady, Double-Hopf and unsteady aperiodic bifurcations—in closed form—and conditions guaranteeing the absence of unsteady bifurcations via symmetrizability, are obtained. The continuous triopoly Cournot game of mathematical economy is taken into account and it is shown that the ternary ADS governing the Nash equilibrium stability, is symmetrizable. The onset of Hopf bifurcations in rotatory thermal hydrodynamics is studied and the Hopf bifurcation number (threshold that the Taylor number crosses at the onset of Hopf bifurcations) is obtained.

     

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.     

Global bifurcations in a piecewise-smooth Cournot duopoly game

The object of the work is to perform the global analysis of the Cournot duopoly model with isoelastic demand function and unit costs, presented in Puu [2]. The bifurcation of the unique Cournot fixed point is established, which is a resonant case of the Neimark–Sacker bifurcation. New properties associated with the introduction of horizontal branches are evidenced. These properties differ significantly when the constant value is zero or positive and small. The good behavior of the case with positive constant is proved, leading always to positive trajectories. Also when the Cournot fixed point is unstable, stable cycles of any period may exist.     

Umbilical torus bifurcations in Hamiltonian systems

We consider perturbations of integrable Hamiltonian systems in the neighbourhood of normally umbilic invariant tori. These lower dimensional tori do not satisfy the usual non-degeneracy conditions that would yield persistence by an adaption of KAM theory, and there are indeed regions in parameter space with no surviving torus. We assume appropriate transversality conditions to hold so that the tori in the unperturbed system bifurcate according to a (generalised) umbilical catastrophe. Combining techniques of KAM theory and singularity theory we show that such bifurcation scenarios of invariant tori survive the perturbation on large Cantor sets. Applications to gyrostat dynamics are pointed out.     

Global bifurcations near a degenerate heterodimensional cycle

This article is devoted to investigating the bifurcations of a heterodimensional cycle with orbit flip and inclination flip, which is a highly degenerate singular cycle. We show the persistence of the heterodimensional cycle and the existence of bifurcation surfaces for the homoclinic orbits or periodic orbits. It is worthy to mention that some new features produced by the degeneracies that the coexistence of heterodimensional cycles and multiple periodic orbits are presented as well, which is different from some known results in the literature. Moreover, an example is given to illustrate our results and clear up some doubts about the existence of the system which has a heterodimensional cycle with both orbit flip and inclination flip. Our strategy is based on moving frame, the fundamental solution matrix of linear variational system is chose to be an active local coordinate system along original heterodimensional cycle, which can clearly display the non-generic properties-``orbit flip" and ``inclination flip" for some sufficiently large time.     

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.     

Global bifurcations in a perturbed cubic system withZ 2-symmetry

In this paper we consider global and local bifurcations in disturbed planar Hamiltonian vector fields which are invariant under a rotation over . All calculation formulas of bifurcation curves have been obtained. Various possible distributions and the existence of limit cycles and singular cycles in different parameter regions have been determined. It is shown that for a planar cubic differential system there are infinitely many parameters in the three-parameter space such that Hilbert numberH(3)11.This project is supported by National Natural Science Foundation of China.     

On integrability and chaos in discrete systems

The scalar nonlinear Schrödinger (NLS) equation and a suitable discretization are well known integrable systems which exhibit the phenomena of “effective” chaos. Vector generalizations of both the continuous and discrete system are discussed. Some attention is directed upon the issue of the integrability of a discrete version of the vector NLS equation.     

On flow switching bifurcations in discontinuous dynamical systems

In paper, the sliding dynamics on the separation boundary is discussed based on the set-valued vector field theory. From vector fields in the neighborhood of a specific separation boundary, the passability of the flow from the one domain into another one is further discussed. The switching bifurcation conditions from the passable boundary to the non-passable boundary are developed. The sliding flow fragmentation on the separation boundary surface is also presented. The normal vector product field function is introduced to determine the switching bifurcation and sliding fragmentation.