Saddle Cycle Research Articles

Intermittent bursts in a directly modulated laser diode

Based on a rate equation description of a directly modulated laser diode, we investigate the origin of intermittent bursts that appear in the output power of the device. Sudden drops in the maximum amplitude of the laser's output characterize this behavior, found in the period doubling sequence that appears when the modulation index varies. From the investigation of this behavior the paper reveals the cause for the extreme sensitivity to noise of modulated laser diodes. Using bifurcation theory techniques, we explain how the unstable periodic solutions and the intrinsic noise of the device play a key role in the mechanism giving rise to these bursts. The paper shows that the mechanism is composed of two fundamental stages, which have been referred to as capture and reinjection. As we clarify the origin of the two stages, the different time scales involved in each one are also identified. From this analysis, we can conclude that the intermittent bursts are a characteristic signature of the process by which the laser stochastic dynamics progressively embeds one of the saddle cycles as the modulation index is varied.

Underlying dynamics of the pedestal components in a modulated laser diode with noise

We demonstrate that the pedestal components observed in the power spectra of a directly modulated laser diode, which were interpreted as a sign of instability of the periodic regime, are an indication of the coexistence of a chaotic regime with the periodic one. We present the underlying dynamics behind the rise of these pedestals, showing two different situations in which the pedestals appear. In both, a periodic regime coexists with another attractor, a saddle cycle in one case and a chaotic attractor in the other. The random fluctuations included in the laser diode model allow the coexisting attractors to merge in the observed behavior of the laser.

Multiple Attractors, Saddles, and Population Dynamics in Periodic Habitats

Mathematical models predict that a population which oscillates in the absence of time-dependent factors can develop multiple attracting final states in the advent of periodic forcing. A periodically-forced, stage-structured mathematical model predicted the transient and asymptotic behaviors of Tribolium(flour beetle) populations cultured in periodic habitats of fluctuating flour volume. Predictions included multiple (2-cycle) attractors, resonance and attenuation phenomena, and saddle influences. Stochasticity, combined with the deterministic effects of an unstable ‘saddle cycle’ separating the two stable cycles, is used to explain the observed transients and final states of the experimental cultures. In experimental regimes containing multiple attractors, the presence of unstable invariant sets, as well as stochasticity and the nature, location, and size of basins of attraction, are all central to the interpretation of data.

Homoclinic and heteroclinic orbits to a cycle in a tri-trophic food chain

The asymptotic behavior of a tri-trophic food chain model is studied. The analysis is carried out numerically, by finding both local and global bifurcations of equilibria and limit cycles. The existence of transversal homoclinic orbits to a limit cycle is shown. The appearance of homoclinic orbits, by moving through a homoclinic bifurcation point, is associated with the sudden disappearance of a chaotic attractor. A homoclinic bifurcation curve, which bounds a region of extinction, is continued through a two-dimensional parameter space. Heteroclinic orbits from an equilibrium to a limit cycle are computed. The existence of these heteroclinic orbits has important consequences on the domains of attraction. Continuation of non-transversal heteroclinic orbits through parameter space shows the existence of two codimension-two bifurcations points, where the saddle cycle is non-hyperbolic. The results are summarized by dividing the parameter space in subregions with different asymptotic behavior.

Fluctuation-induced escape from the basin of attraction of a quasiattractor

Noise-induced escape from the basin of attraction of a strange attractor (SA) in a periodically excited nonlinear oscillator is investigated. It is shown by numerical simulation methods that escape occurs in two steps: transfer of the system from the SA to a close-lying saddle cycle along several optimal trajectories, and a subsequent fluctuation-induced transfer from the basin of attraction of the SA along a single optimal trajectory. The possibility of using the results of this work to solve problems of the optimal control of switchings from an attractor and for constructing theoretical estimates of the escape probability is discussed.

Loss of chaos synchronization in coupled Rossler systems

We study process of loss of complete chaos synchronization in interacting oscillators which demonstrates cascade of period—doubling bifurcations. It is shown that bifurcations of main family of saddle cycles embedded in chaotic attractor lead to loss of the synchronous regime robustness and appearance of limit sets located in vicinity of symmetrical subspace. Bifurcations of nonsymmetrical limit sets lead to formation of complex structure of attractor basins. The transition to hyperchaos regime which appears as a result of merge of some chaotic sets completes the loss of chaos synchronization.

About Two Mechanisms of Reunion of Chaotic Attractors

Considering a family of two-dimensional piecewise linear maps, we discuss two different mechanisms of reunion of two (or more) pieces of cyclic chaotic attractors into a one-piece attracting set, observed in several models. It is shown that, in the case of so-called contact bifurcation of the 2 nd kind, the reunion occurs immediately due to homoclinic bifurcation of some saddle cycle belonging to the basin boundary of the attractor. In the case of so-called contact bifurcation of the 1 st kind, the reunion is a result of a contact of the attractor with its basin boundary which is fractal, including the stable set of a chaotic invariant hyperbolic set appeared after the homoclinic bifurcation of a saddle cycle on the basin boundary

Stochastic potential for a periodically forced nonlinear oscillator

We investigate stationary and nonstationary probability densities for a weakly forced nonlinear physical or chemical system that displays self-oscillations in the absence of forcing. The period and amplitude of forcing are taken as adjustable constraints. We consider a homogeneous reaction system described by a master equation. Our method of solution is based on the Wentzel–Kramers–Brillouin (WKB) expansion of the probability density with the system size as the expansion parameter. The first term in this expansion is the stochastic potential (eikonal). In the absence of forcing, the probability density is logarithmically flat on the limit cycle. With periodic forcing, the phenomenon of phase locking can occur whereby a stable cycle, which is close to the unforced cycle, adopts a constant relative phase to the forcing. A saddle cycle also exists and has a different constant relative phase. For such phase-locked solutions, the distribution over the relative phases is peaked on the stable cycle and exhibits a logarithmically flat region (a plateau) that originates on the saddle cycle. This plateau is due to a nonzero relative phase slippage: large fluctuations from the stable cycle over the saddle cycle are overwhelmingly more probable in a certain relative phase direction, which depends upon the location of the parameters within an entrainment region. This distribution of relative phases is logarithmically equivalent to that of a Brownian particle in a periodic potential with a constant external force in the strong damping and weak noise limits. For parameter values outside of an entrainment region (for which a quasiperiodic solution exists), the distribution in relative phase is logarithmically flat. For this regime, we investigate the evolution of an initially localized density and show that the width grows proportionally with the square root of time. The proportionality factor depends upon both the position (phase) on the cross section of the peak of the density and the distance in parameter space from the boundary of the entrainment region. For parameter values that approach the boundary of an entrainment region, this proportionality factor tends to infinity. We also determine an expression for the first order correction to the stochastic potential for both entrained and quasiperiodic solutions. A thermodynamic interpretation of these results is made possible by the equality of the stochastic potential with an excess work function.

Characterization of homoclinic chaos through double-valued return time maps

For a laser with a saturable absorber (LSA) and for a subnormal glow discharge (GD), both displaying homoclinic chaos, it is shown that double-valued curves in the return time maps, reconstructed from the time evolution of appropriate variables, are related to different time scales associated with the two mechanisms present in the respective chaotic attractors: the escape from an unstable saddle cycle and the reinjection process. For the LSA the investigation is performed numerically on a 3-2 molecular level model and the results are compared with experimental ones obtained from a ${\mathrm{CO}}_{2}\ensuremath{-}{\mathrm{OsO}}_{4}$ LSA system having the laser frequency detuning as the control parameter. The analysis is complemented with experimental results from the GD. For both systems we show how to obtain single-valued multibranched return time maps starting from double-valued return time maps, enabling the characterization of homoclinic chaos.

SOME PROPERTIES OF A TWO-DIMENSIONAL PIECEWISE-LINEAR NONINVERTIBLE MAP

Properties of a piecewise-linear noninvertible map of the plane are studied by using the method of critical curves (two-dimensional extension of the notion of critical point in the one-dimensional case). This map is of (Z0–Z2) type, i.e. the plane consists of a region without preimage, and a region giving rise to two rank one preimages. For the considered parameter values, the map has two saddle fixed points. The characteristic features of the “mixed chaotic area” generated by this map, and its bifurcations (some of them being of homoclinic and heteroclinic type) are examined. Such an area is bounded by the union of critical curves segments and segments of the unstable set of saddle cycles.

PERIOD-DOUBLING GEOMETRY OF THE BELOUSOV-ZHABOTINSKY REACTION DYNAMICS

The intricate geometry of stable and unstable manifolds of a saddle cycle arising from a super-critical period-doubling bifurcation is explored experimentally for the Belousov-Zhabotinsky (BZ) reaction in a CSTR (Continuous flow Stirred Tank Reactor). We find clear experimental evidence that the stable manifold winds round the stable period-doubled orbit arising at the bifurcation. The stable manifold is probed experimentally by perturbations from the period-doubled limit cycle. The method provides a model-independent experimental test for species essential for the complexity, as well as quantitative information about the geometry of the limit cycles, the associated manifolds, and their embedding in the concentration space. The results are supported by simulations of the experiments with a four-dimensional model of the BZ reaction. Here stable manifold has several branches showing some tendency of curling. However the branches may end on the boundary of the positive orthant of the concentration space.

EXTENSIONS OF THE NOTION OF CHAOTIC AREA IN SECOND-ORDER ENDOMORPHISMS

Properties of chaotic areas (i.e. invariant domains of points positively stable in the Poisson’s sense) of non-invertible maps of the plane are studied by using the method of critical curves (two-dimensional extension of the notion of critical points in the one-dimensional case). The classical situation is that of a chaotic area bounded by a finite number of critical curves segments. This paper considers another class of chaotic areas bounded by the union of critical curves segments and segments of the unstable manifold of a saddle fixed point, or that of saddle cycle (periodic point). Different configurations are examined, as their bifurcations when a map parameter varies.

Bifurcations near 1:2 subharmonic resonance in a structural dynamics model

In this paper we present a numerical bifurcation study of the dynamical behavior of a structural dynamics model of a periodically forced suspended cable. One and two-parameter diagrams of local bifurcation phenomena near the subharmonic 1:2 resonance are computed. Various global bifurcation phenomena near the resonance, including transitions to chaotic oscillations, are exposed by computation of invariant manifolds of saddle cycles.

Global dynamics underlying sharp basin erosion in nonlinear driven oscillators.

For periodically driven damped oscillators with the ability to escape from a potential well, the erosion of the nonescaping basin of attraction begins at the homoclinic tangency of the stable and unstable manifolds of the hilltop saddle cycle. The consequent rate of erosion, however, is intrinsically dependent upon the manifold organization, which to a large extent is determined by the heteroclinic events following the homoclinic tangency. In this paper we outline how, under small parameter changes, there can exist a rapid erosion of the nonescaping basin.

Invariant sets of planar diffeomorphisms in nonlinear vibrations

Using the standard technique of Poincaré sampling, many simple periodically forced nonlinear oscillators can be represented by area-contracting planar diffeomorphisms. A class of such diffeomorphisms, referred to as single-sided systems, are considered. These permit escape of orbits to arbitrarily large displacement. Six closure relations which relate the invariant manifolds of the hilltop saddle cycle to other invariant sets are derived. These relations embody much of the underlying organization that is observed in the dynamics of these systems.

Return maps for intensity and time in a homoclinic-chaos model applied to a laser with a saturable absorber.

Intensity maps from an experiment on a laser with a saturable absorber, in the case of strong dissipation, are compared with theoretical one-dimensional (1D) maps. These 1D maps are derived in the limit of infinite dissipation, starting from 2D maps for a dissipative saddle cycle with stable and unstable manifolds approaching a quadratic tangency and giving rise to a structurally unstable homoclinic orbit. They are multibranched, each branch corresponding to a particular number of turns around the saddle cycle. The good agreement between theory and experiment indicates that the origin of the complicated behavior of a laser with a saturable absorber is a saddle cycle. A multibranched structure is also present in maps of time of flight between planes in phase space. Depending on the choice of these planes, it is possible to obtain time-of-flight maps that give the same symbolic dynamics of the intensity maps.

BASIN ORGANIZATION PRIOR TO A TANGLED SADDLE-NODE BIFURCATION

Heteroclinic and homoclinic connections of saddle cycles play an important role in basin organization. In this study, we outline how these events can lead to an indeterminate jump to resonance from a saddle-node bifurcation. Here, due to the fractal structure of the basins in the vicinity of the saddle-node, we cannot predict to which available attractor the system will jump in the presence of even infinitesimal noise.

Qualitative structures and bifurcations generated by a nonlinear third-order phase synchronization equation: PMM vol. 42, no. 5, 1978, pp 808–819

A qualitative investigation is made of a system of nonlinear third-order differential equations, being a model of a phase synchronization system. The existence is established of bifurcation surfaces separating the parameter space into domains for whose points the system is globally asymptotically stable, contains cycles, has a complex structure (contains a denumerable set of saddle cycles), etc.