Stable Heteroclinic Cycles Research Articles

Heteroclinic cycles in the chemostat models and the winnerless competition principle

Winnerless competition principle (WLC) is a type of competition that does not have a winner; all species take turns (or switch) to win. In the phase space, it appears as a stable heteroclinic contour connecting single-species equilibria. In ecology, May and Leonard [R.M. May, W.J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math. 29 (1975) 243–253] were the first to discover the behavior in their famous paper that the competition of three species experiences a special type of WLC competition, the rock-paper-scissors competition. Recently, WLC concepts are used for the design in neural network dynamics. In this manuscript, it is shown that WLC can also appear in the chemostat model. We consider a chemostat model of n species of microorganisms competing for k essential and growth-limiting nutrients. Sufficient conditions for a stable heteroclinic cycle connecting single-species equilibria in the limit sets are given. The heteroclinic cycle can be constructed so that the equilibria are connected in the following order: E 1 → E 2 → E 3 → ⋯ → E n → E 1 in which E i 's are the ith species equilibria. This heteroclinic cycle describes the rock-paper-scissors winnerless competition; all of the n species take turns to win, there is no final winner.

The effect of symmetry breaking on the dynamics near a structurally stable heteroclinic cycle between equilibria and a periodic orbit

The effect of small forced symmetry breaking on the dynamics near a structurally stable heteroclinic cycle connecting two equilibria and a periodic orbit is investigated. This type of system is known to exhibit complicated, possibly chaotic dynamics including irregular switching of sign of various phase space variables, but details of the mechanisms underlying the complicated dynamics have not previously been investigated. We identify global bifurcations that induce the onset of chaotic dynamics and switching near a heteroclinic cycle of this type, and by construction and analysis of approximate return maps, locate the global bifurcations in parameter space. We find there is a threshold in the size of certain symmetry-breaking terms, below which there can be no persistent switching. Our results are illustrated by a numerical example.

A NEW MECHANISM FOR INTERMITTENCY IN RINGS OF CELLS

We discuss a general mechanism that induces periodic switching between different dynamic phenomena in rings of cells. Such intermittent behavior is normally modeled by asymptotically stable heteroclinic cycles. Our mechanism does not require a heteroclinic cycle in the ring of cells, instead a Van der Pol oscillator forces our system so that it periodically changes state. We investigate this mechanism numerically in a number of different examples. These simulations demonstrate a variety of interesting forms of intermittency.

Continuous model for the rock–scissors–paper game between bacteriocin producing bacteria

In this work, important aspects of bacteriocin producing bacteria and their interplay are elucidated. Various attempts to model the resistant, producer and sensitive Escherichia coli strains in the so-called rock-scissors-paper (RSP) game had been made in the literature. The question arose whether there is a continuous model with a cyclic structure and admitting an oscillatory dynamics as observed in various experiments. The May-Leonard system admits a Hopf bifurcation, which is, however, degenerate and hence inadequate. The traditional differential equation model of the RSP-game cannot be applied either to the bacteriocin system because it involves positive interaction terms. In this paper, a plausible competitive Lotka-Volterra system model of the RSP game is presented and the dynamics generated by that model is analyzed. For the first time, a continuous, spatially homogeneous model that describes the competitive interaction between bacteriocin-producing, resistant and sensitive bacteria is established. The interaction terms have negative coefficients. In some experiments, for example, in mice cultures, migration seemed to be essential for the reinfection in the RSP cycle. Often statistical and spatial effects such as migration and mutation are regarded to be essential for periodicity. Our model gives rise to oscillatory dynamics in the RSP game without such effects. Here, a normal form description of the limit cycle and conditions for its stability are derived. The toxicity of the bacteriocin is used as a bifurcation parameter. Exact parameter ranges are obtained for which a stable (robust) limit cycle and a stable heteroclinic cycle exist in the three-species game. These parameters are in good accordance with the observed relations for the E. coli strains. The roles of growth rate and growth yield of the three strains are discussed. Numerical calculations show that the sensitive, which might be regarded as the weakest, can have the longest sojourn times.

Robust heteroclinic cycles in the one-dimensional complex Ginzburg–Landau equation

Numerical evidence is presented for the existence of stable heteroclinic cycles in large parameter regions of the one-dimensional complex Ginzburg–Landau equation (CGL) on the unit, spatially periodic domain. These cycles connect different spatially and temporally inhomogeneous time-periodic solutions as t → ± ∞ . A careful analysis of the connections is made using a projection onto five complex Fourier modes. It is shown first that the time-periodic solutions can be treated as (relative) equilibria after consideration of the symmetries of the CGL. Second, the cycles are shown to be robust since the individual heteroclinic connections exist in invariant subspaces. Thirdly, after constructing appropriate Poincaré maps around the cycle, a criteria for temporal stability is established, which is shown numerically to hold in specific parameter regions where the cycles are found to be of Shil’nikov type. This criterion is also applied to a much higher-mode Fourier truncation where similar results are found. In regions where instability of the cycles occurs, either Shil’nikov–Hopf or blow-out bifurcations are observed, with numerical evidence of competing attractors. Implications for observed spatio-temporal intermittency in situations modelled by the CGL are discussed.

Dynamics in the 1:2 spatial resonance with broken reflection symmetry

The dynamics near the 1:2 spatial resonance in systems with broken O(2) symmetry are described. The system is assumed to remain SO(2) invariant, and attention is focused on the behavior that replaces the structurally stable heteroclinic cycles present in the O(2)-symmetric system. The symmetry-breaking destroys these cycles, and introduces a variety of new bifurcations into the system, some which are associated with complex behavior. The possible scenarios with increasing symmetry-breaking are described and possible applications of the results are discussed.

Shapes and cycles arising at the steady bifurcation with icosahedral symmetry

This paper analyses the steady-state bifurcation with icosahedral symmetry. The Equivariant Branching Lemma is used to predict the generic bifurcating solution branches corresponding to each irreducible representation of the icosahedral group I h . The relevant amplitude equations are deduced from the equivariance condition, and used to investigate the stability of bifurcating solutions. It is found that the bifurcation with icosahedral symmetry can lead to competition between two-fold, three-fold and five-fold symmetric structures, and between solutions with tetrahedral, three-fold and two-fold symmetry. Stable heteroclinic cycles between solutions with D 2 z symmetry are found to exist in one of the irreps. The theoretical scenarios are compared with the observed behaviour of icosahedral viruses and nanoclusters.

Dynamics of polar reversals in spherical dynamos

Structurally stable heteroclinic cycles (SSHCs) are proposed as the mathematical structures that are responsible for the reversals of dipolar magnetic › elds in spherical dynamos. The existence of SSHCs involving dipolar magnetic › elds generated by convection in a spherical shell for a non-rotating sphere is rigorously proven. The possibility of SSHCs in a rotating shell is proposed, and their existence in a lowdimensional model of the magnetohydrodynamic equations is numerically con› rmed. The resulting magnetic time-series shows a dipolar magnetic › eld, aligned with the rotation axis that intermittently becomes unstable, changes the polar axis, starts rotating, disappears completely and eventually re-establishes itself in its original or opposite direction, chosen randomly.

Stable heteroclinic cycles for ensembles of chaotic oscillators.

We study the formation of synchronous clusters in ensembles of globally coupled chaotic oscillators. We reveal that at least three clusters of identical synchronization are formed in such a system for large enough values of coupling strength. Our main result is an unexpected intermittent process of clusterization. This process gives strong indication to the existence of a stable heteroclinic cycle.

New type of complex dynamics in the 1:2 spatial resonance

The amplitude equations describing the 1:2 spatial resonance in O(2) symmetric systems are re-examined. Complex dynamical behavior associated with a new class of global connections is identified and analyzed. These connections form further from the mode interaction point where the structurally stable heteroclinic cycles described by Armbruster et al. [Physica D 29 (1988) 257] no longer exist. The resulting behavior is characteristic of systems with O(2)×Z 2 symmetry in which the symmetry Z 2 is weakly broken.

Theoretical analysis of a dynamic thermoconvective pattern in a circular container

The gravity- and surface-tension-driven thermoconvective instability of a fluid layer contained in a no-slip circular container of aspect ratio $2.5$ is investigated. Linear and nonlinear analyses are developed. From amplitude equations the bifurcation diagram of the system is deduced, and the different stable convective patterns are studied. In particular, it is shown that a stable heteroclinic cycle due to a 1-2 spatial resonance is predicted. This cycle is also shown to be in agreement with recent experimental observations of a dynamic mode switching.

Destabilization by noise of transverse perturbations to heteroclinic cycles: a simple model and an example from dynamo theory

We show that transverse perturbations from structurally stable heteroclinic cycles can be destabilized by surprisingly small amounts of noise, even when each individual fixed point of the cycle is stable to transverse modes. A condition that favours this process is that the linearization of the dynamics in the transverse direction be characterized by a non-normal matrix. The phenomenon is illustrated by a simple two-dimensional switching model and by a simulation of a convectively driven dynamo.

Noise and O(1) amplitude effects on heteroclinic cycles.

The dynamics of structurally stable heteroclinic cycles connecting fixed points with one-dimensional unstable manifolds under the influence of noise is analyzed. Fokker-Planck equations for the evolution of the probability distribution of trajectories near heteroclinic cycles are solved. The influence of the magnitude of the stable and unstable eigenvalues at the fixed points and of the amplitude of the added noise on the location and shape of the probability distribution is determined. As a consequence, the jumping of solution trajectories in and out of invariant subspaces of the deterministic system can be explained. (c) 1999 American Institute of Physics.

Evolving Control Strategies for Suppressing Heteroclinic Bursting

We combine a detailed understanding of the dynamics of low-dimensional models of burstingin the turbulent boundary layer with the method of Genetic Programming to obtain appropriate control strategies for the suppressionof such bursting in these models. The study is applicable toO(2) symmetric systems for which structurally stable heteroclinic cycles are the dominant dynamical features. We argue that such a combined approach can prove a useful tool in achieving control in higher-dimensional models where actual analysis is prohibitively complicated. The results of the present study are compared to near-optimal controllers derived in previous studies.

Rotating convection: recent developments

Convection in a rotating layer was a subject close to Professor Chandrasekhar’s interest, and graces the cover of the Dover edition of his famous book on Hydrodynamic and Hydromagnetic Stability. In this article I will briefly review the results described in this book, which concern primarily linear stability theory for an unbounded layer. I will then discuss in detail the differences between the theory for the unbounded layer and that for a finite cylindrical container heated from below and rotating about its axis. These differences can be traced to the differences in the symmetries of the two systems, and lead one to expect time-dependent convection at onset, regardless of the value of the Prandtl number. In addition the boundaries are responsible for the presence of a new nonaxisymmetric mode of instability, called a wall mode. The theory predicts that these wall modes will precess at onset and set in before the body modes characteristic of the unbounded layer. These predictions have been confirmed in elegant experiments by R. Ecke and colleagues. In the nonlinear regime perhaps the most interesting phenomenon is the Küppers–Lortz instability. In this instability a pattern of parallel rolls becomes unstable to another set of rolls oriented at an angle with respect to the first, once the rotation rate exceeds a critical value. This new set is itself unstable in the same fashion etc., resulting in complex dynamics right at onset. I will describe the theory behind this instability and identify similar instabilities for oscillatory convection whether in the form of standing or traveling rolls. Such instabilities are triggered by the formation of heteroclinic orbits connecting two sets of standing or traveling rolls with different orientation. Structurally stable heteroclinic cycles involving four traveling roll states are possible, and result in quite unusual dynamical behavior.

Heteroclinic cycles in lattice dynamical systems

A criterion of spatial chaos occurring in lattice dynamical systems--heteroclinic cycle--is discussed. It is proved that if the system has asymptotically stable heteroclinic cycle, then it has asymptotically stable homoclinic point which implies spatial chaos.

A robust heteroclinic cycle in an steady-state mode interaction

A structurally stable heteroclinic cycle connecting circles of fixed points of opposite parity is identified in a steady-state mode interaction with symmetry. These fixed points correspond to equilibria in the form of zigzag (odd parity) and varicose (even parity) states. Because of the reflection symmetry this 1:1 mode interaction bears certain similarities to earlier work on the 1:2 mode interaction with symmetry. However, in the present case the cycle connects eight equilibria, four of which are selected from the circle of zigzag equilibria and four from the circle of varicose equilibria. Conditions for the existence and asymptotic stability of the cycle are determined and compared with numerical integration of the mode interaction equations.

Collections of heteroclinic cycles in the Kuramoto-Sivashinsky equation

We study the Kuramoto-Sivashinky equation with periodic boundary conditions in the case of low-dimensional behavior. We analyze the bifurcations that occur in a six-dimensional (6D) approximation of its inertial manifold. We mainly focus on the attracting and structurally stable heteroclinic connections that arise for these parameter values. We reanalyze the ones that were previously described via a 4D reduction to the center-unstable manifold (Ambruster et al., 1988, 1989). We also find a parameter region for which a manifold of structurally stable heteroclinic cycles exist. The existence of such a manifold is responsible for an intermittent behavior which has some features of unpredictability.

Suppression of bursting

We investigate the possibility of using a single small amplitude control input and feedback to stabilize equilibrium sets in a class of highly nonlinear O(2) symmetric dynamical systems possessing structurally stable heteroclinic cycles. The leading-order behavior near the equilibria is bilinear and homogeneous in the state variables, while nonlinearities representing the symmetry breaking effect of the controller are crucial. After a series of simplifying transformations, we use ideas from optimal control theory to construct a stabilizing controller. This study is motivated by the desire to stabilize the burst/sweep cycle in low-dimensional models of a turbulent boundary layer. In the last two sections, we apply the techniques to the 10-dimensional system of Aubry et al. (1988). kw]Bilinear systems; dynamical systems; heteroclinic cycles; normal forms; optimal control; symmetry

Heteroclinic cycles in the three-dimensional postbifurcation motion of O(2)-symmetric fluid conveying tubes

The occurrence of heteroclinic cycles is a phenomenon that has been detected in the dynamics of symmetric systems only recently. For the physical problem of an O(2)-symmetric fluid conveying tube the existence of a robust asymptotically stable heteroclinic cycles is shown.