Phase Portraits of Z7- Equivariant Sextic Hamiltonian System

In this paper, a class of sextic Z7-equivariant Hamiltonian system is considered. Using the methods of qualitative analysis, bifurcations of the above system are analyzed, the phase portraits of the system are classified and representative orbits are shown by Maple software.     

A Perturbation Method for Nonlinear Two-Dimensional Maps

A new perturbation method for a weakly nonlinear two-dimensional discrete-time dynamical system is presented. The proposed technique generalizes the asymptotic perturbation method that is valid for continuous-time systems and detects periodic or almost-periodic orbits and their stability. Two equations for the amplitude and the phase of solutions are derived and their fixed points correspond to limit cycles for the starting nonlinear map. The method is applied to various nonlinear (autonomous or not) two-dimensional maps. For the autonomous maps we derive the conditions for the appearance of a supercritical Hopf bifurcation and predict the characteristics of the corresponding limit cycle. For the nonautonomous maps we show amplitude-response and frequency-response curves. Under appropriate conditions, we demonstrate the occurrence of saddle-node bifurcations of cycles and of jumps and hysteresis effects in the system response (cusp catastrophe). Modulated motion can be observed for very low values of the external excitation and an infinite-period bifurcation occurs if the external excitation increases. Analytic approximate solutions are in good agreement with numerically obtained solutions.     

Continuity of type-I intermittency from a measure-theoretical point of view

We study a simple dynamical system which displays a so-called type-I intermittency bifurcation. We determine the Bowen-Ruelle measure and prove that the expectation (g) of any continuous functiong and the Kolmogoroff-Sinai entropyh() are continuous functions of the bifurcation parameter. Therefore the transition is continuous from a measure-theoretical point of view. Those results could be generalized to any similar dynamical system.     

Intermittency in a cantilever plate in a randomly fluctuating fluid flow

Fluid–elastic systems nearing dynamic instabilities are known to be sensitive to fluctuations in fluid flow. A cantilever plate in axial flow with random temporal fluctuations, is examined numerically for its dynamical behaviour. The numerical model comprises of a nonlinear structural model for the flexible plate, coupled with unsteady lumped vortex model for the fluid forces. As the mean flow velocity is increased, the system transitions to limit cycle oscillations from a state of rest, through a regime of intermittent oscillations. The conditions for onset and disappearance of intermittency are discussed and are interpreted using stochastic bifurcation theories. While the onset of intermittency is found to be unaffected by the time scales of the flow fluctuations, they are observed to affect the length of the intermittency regime. The effect of plate flexibility on intermittency is also discussed.     

Existence of multiple attractors and the nature of bifurcations in a discontinuous logistic map

We present the analytical investigations on a logistic map with a discontinuity at the centre. An explanation for the bifurcation phenomenon in discontinuous systems is presented. We establish that whenever the elements of ann-cycle (n > 1) approach the discontinuities of thenth iterate of the map, a bifurcation other than the usual period-doubling one takes place. The periods of the cycles decrease in an arithmetic progression, as the control parameter is varied. The system also shows the presence of multiple attractors. Our results are verified by numerical experiments as well.     

A polycycle and limit cycles in a non-differentiable predator-prey model

For a non-differentiable predator-prey model, we establish conditions for the existence of a heteroclinic orbit which is part of one contractive polycycle and for some values of the parameters, we prove that the heteroclinic orbit is broken and generates a stable limit cycle. In addition, in the parameter space, we prove that there exists a curve such that the unique singularity in the realistic quadrant of the predator-prey model is a weak focus of order two and by Hopf bifurcations we can have at most two small amplitude limit cycles.     

完全非弹性蹦球的动力学行为

对振动台面上的完全非弹性球的蹦跳行为进行了初步分析.受约化振动加速度的控制，球的运动可以表现出一系列倍周期分岔过程.对几种典型的倍周期运动及分岔情况进行了讨论. 关键词： 蹦球 倍周期分岔 混沌 颗粒物质     

The influence of noise on the logistic model

The behavior of the logistic system which is generated by the functionf(x =ax (1–x) changes in an interesting way if it is perturbed by external noise. It turns out that the chaotic behavior which was predicted by Li and Yorke for orbits of period 3, becomes visible and that a sequence of mergence transitions occurs at the critical parameter. The change of the invariant probability density and the Lyapunov exponents are examined numerically. The power spectrum for the period 3 orbit for different fluctuations is calculated and a recursion formula for the time evolution of the probability density is presented as a discrete-time analog of a Chapman-Kolmogorov equation.     

Generation of a magnetic field by convective flows in a rotating horizontal layer

The generation of a magnetic field by convective flows of a conducting fluid in a rotating plane layer is investigated numerically. The problem is considered in the complete three-dimensional nonlinear formulation. The sequence of temporal regimes that ensue as the Taylor number Ta increases from 0 (no rotation) to 2000 (the fluid motion is suppressed by rapid rotation) when the other parameters are fixed is studied. The Ta intervals on which bifurcations occur are found, and the breakdown and onset of symmetries in the attractors that arise is investigated.     

Scenarios of the onset of unsteady regimes in the problem of plane convective flow through a porous medium

The problem of plane convective flow through a porous medium in a rectangular vessel with a linear temperature profile steadily maintained on the boundary is considered. The onset of unsteady regimes is investigated numerically. It is shown that their onset scenarios depend on the vessel dimensions and the seepage Rayleigh number and may be as follows: the generation of stable and unstable periodic regimes as a result of a one-sided bifurcation, the generation of a stable periodic regime as a result of an Andronov-Hopf cosymmetric bifurcation, the formation of a chaotic attractor, the branching-out of a stable quasi-periodic regime from a point of a single-parameter family of steady-state regimes, and the generation of unstable periodic regimes as a result of disintegration of homoclinic trajectories. The specifics of most of the bifurcations mentioned above are attributable to the cosymmetry of the problem considered.