Showing papers on "Infinite-period bifurcation published in 1985"

Super persistent chaotic transients

Abstract: The unstable-unstable pair bifurcation is a bifurcation in which two unstable fixed points or periodic orbits of the same period coalesce and disappear as a system paremeter is raised. For parameter values just above that at which unstable orbits are destroyed there can be chaotic transients. Then, as the bifurcation is approached from above, the average length of a chaotic transient diverges, and, below the bifurcation point, the chaotic transient may be regarded as having been converted into a chaotic attractor. It is argued that unstable-unstable pair bifurcations should be expected to occur commonly in dynamical systems. This bifurcation is an example of the crisis route to chaos. The most striking fact about unstable-unstable pair bifurcation crises is that long chaotic transients persist even for parameter values relatively far from the bifurcation point. These long-lived chaotic transients may prevent the time asymptotic state from being reached during experiments. An expression giving a lower bound for the average lifetime of a chaotic transient is derived and shown to agree well with numerical experiments. In particular, this bound on the average lifetime, (T), satisfies

The Bifurcation and Secondary Bifurcation of Capillary-Gravity Waves

Abstract: The bifurcation and secondary bifurcation of capillary-gravity waves is analysed when the surface tension is close to or equal to a value where the eigenspace of the critical phase speed has multiplicity two. The existence and multiplicity of solutions is seen, via the implicit function theorem, to be a special case of the secondary bifurcation phenomena, which occur when a double eigenvalue splits, under perturbation, into two simple eigenvalues in the presence of a symmetry in the problem.

A codimension three bifurcation for the laser with saturable absorber

Abstract: We study the five mode equations which are used to model the dynamical behavior of a laser with saturable absorber in the mean field limit and exact resonance. We show that in this system a codimension three bifurcation exists where a tricritical point of the stationary solution encounters a double zero eigenvalue. A center manifold reduction is performed to fix the three-dimensional submanifold in parameter space where this degeneracy occurs. The associated Takens-normal form is given. By unfolding the normal form we obtain all structurally stable phase portraits near this bifurcation point and display them in the form of bifurcation diagrams with the laser pumping rate as a distinguished bifurcation parameter. These diagrams allow a unifying analytical and geometrical description of many different numerical solutions of the equations describing a laser with absorber. In particular, they yield the connection of the small amplitude periodic solutions with passiveQ-switching and suggest new bifurcation processes, which one can expect to occur for physical parameters near the critical submanifold. The existence of a codimension four bifurcation is indicated.

Dynamics of a system exhibiting the global bifurcation of a limit cycle at infinity

Abstract: This paper concerns the dynamics of a class of non-linear oscillators of the form: x″ + x − ϵx′(1−ax 2 −bx′ 2 ) = 0 . The non-linear term contains two parameters a and b which may be varied to give the Rayleigh and Van der Pol differential equations as special cases. The existence and approximation of limit cycles in this system are investigated using the Poincare-Bendixson theorem and the Lindstedt perturbation method. Analysis of the system at infinity is used to study the global bifurcation through which the limit cycle is created from four saddle-saddle connections between equilibrium points at infinity. Center manifold theory is used to determine the stability of the equilibrium points at infinity. Numerical integration is used to verify the analytical results. It is shown that an arbitrarily small perturbation to the damping term of the Rayleigh equation results in points close to the stable limit cycle escaping to infinity.

Profuse multiplicity of solutions in mixed convection flow in ducts

Abstract: In this investigation the question of the existence of multiple steady-state solutions for the mixed convection flow problem in horizontal rectangular ducts is examined. The numerical study employs the theoretical results of Benjamin on the bifurcation theory for a bounded incompressible fluid as a guide in systematically exploring the bifurcation phenomena for this problem. The observed bifurcations of solutions are discussed in terms of the dynamic processes involved. Each cellular flow may be represented by a solution surface in the parametric space of Grashof number and aspect ratio. These are delimited by loci of bifurcation points. The projection of these surfaces on the Gr-..gamma.. plane overlaps. The primary modes exchange roles via the formation of a tilted cusp. Other salient features, such as primary mode hysteresis, quasi-critical range of cellular development, and transcritical bifurcations are present. One-sided bifurcation is the most common type occurring away from the cusp region. The bifurcation curves represent the limits of stability for a particular cellular flow beyond which it will mutate into another cellular flow via a certain dynamic process. It is believed that these curves are compositional of which each segment represents a certain mode of bifurcation and that the curves located inmore » this study are the ones which make the solution surface minimum.« less

Arnold tongues and subharmonics in the forced oscillations of a mechanical clock

Abstract: We consider a simple model of a mechanical clock and its response to periodic disturbances. The model consists of a damped linear oscillator subjected to impulses whenever the system achieves a prescribed state (position and velocity). The unforced system possesses a limit cycle which, due to the piecewise linear nature of the model, is known in an explicit form. The response of this system to periodic excitation depends in a delicate manner on the ratio of the limit cycle period to the forcing period. Arnold tongues emerge, as the driving amplitude is increased form zero, from points where this ratio is a rational number and correspond to the appearance of subharmonic motions. We present explicit formulae for some of these bifurcation curves and for some secondary bifurcation curves where the subharmonics lose their stability, these bifurcations are either period doubling or Hopf bifurcations.

Period-Doublings And Chaos In Electro-Optic Bistable Device With Short Delay Times In The Feedback

Abstract: The bifurcation structure of the transmitted light power for an electro-optic bistable device with delayed feedback has been investigated. We report a bifurcation scheme with hysteresis in the bifurcation points, so that we obtain different values of the bifurcation points when the bifurcation parameter is increas-ing and decreasing. This hysteresis phenomenon has been investigated experimentally and numerically, and we find that the time dependence of the bifurcation parameter is crucial to the bifurcation mechanism and to the critical values for which the transitions to period -2, period-4, etc. occur.