Stable Heteroclinic Cycles Research Articles

Role of adaptive intraspecific competition on collective behavior in the rock–paper–scissors game

Density-dependent selection is a universal feature in the evolution of populations, and such an adaptive behavior can change the survival strategy of species during evolution. In this paper, we investigate the role of adaptive behavior in biodiversity in the system of cyclic competition. By incorporating a density-dependent mechanism into intraspecific competition, which is well-known as a key mechanism leading to biodiversity, we studied how such adaptive intraspecific competition can affect biodiversity and collective behavior during the evolution of cyclically competing species. Microscopically, we found that species can coexist strongly and are spirally entangled or collectively united by presenting two distinct pattern formations on spatially extended systems. While the adaptive mechanism can always promote species coexistence in a mean-field manner for particular sensitivity to the group scale, corresponding spatial dynamics exhibit nonmonotonic features for the robustness of extinction at moderately high mobility regime over the critical mobility when the associated mean-field system exhibits asymptotically stable heteroclinic cycles. The findings can shed light on a new aspect of the collective behavior of coexisting populations which may indicate the possibility of changing the survival strategies of each group to maintain the coexistence of cyclically competing populations.

Replicator dynamics of public goods games with global exclusion

Studies to date on the role of social exclusion in public cooperation have mainly focused on the peer or pool sanctioning types of excluding free-riders from the share of common goods. However, the exclusive behaviors are not necessarily performed by individuals or local organizations but may rather be implemented by a centralized enforcement institution at a global scale. Besides, previous modeling methods of either peer or pool exclusion often presuppose some particular forms of feedback between the individual or collective efforts and the efficiency of social exclusion and, therefore, cannot comprehensively evaluate their effects on the evolution of cooperation in the social dilemma situations. Here, we construct a general model of global exclusion by considering the successful construction of the centralized exclusive institution as an arbitrary non-decreasing and smooth function of the collective efforts made by the global excluders and then theoretically analyze its potential impacts in the replicator dynamics of the public goods game. Interestingly, we have shown that, despite the presence of both the first- and second-order free-riding problems, global exclusion can indeed lead to the emergence or even stabilization of public cooperation without the support of any other evolutionary mechanism. In addition, we have also observed rich dynamical behaviors, such as the occurrence of a global or local family of neutrally stable closed orbits revolving around a nonlinear center or the existence of stable heteroclinic cycles between defectors, cooperators as well as global excluders, which give rise to a classification of up to 21 different phases.

Stability of heteroclinic cycles in ring graphs.

Networks of interacting nodes connected by edges arise in almost every branch of scientific inquiry. The connectivity structure of the network can force the existence of invariant subspaces, which would not arise in generic dynamical systems. These invariant subspaces can result in the appearance of robust heteroclinic cycles, which would otherwise be structurally unstable. Typically, the dynamics near a stable heteroclinic cycle is non-ergodic: mean residence times near the fixed points in the cycle are undefined, and there is a persistent slowing down. In this paper, we examine ring graphs with nearest-neighbor or nearest- m-neighbor coupling and show that there exist classes of heteroclinic cycles in the phase space of the dynamics. We show that there is always at least one heteroclinic cycle that can be asymptotically stable, and, thus, the attracting dynamics of the network are expected to be non-ergodic. We conjecture that much of this behavior persists in less structured networks and as such, non-ergodic behavior is somehow typical.

The interplay of rock-paper-scissors competition and environments mediates species coexistence and intriguing dynamics

Asymmetrical rock-paper-scissors (RPS) competition has been perceived as a crucial factor in shaping species biodiversity, and understanding this ecological issue in a multi-species paradigm is rather difficult because community dynamics usually depend on distinct factors such as abiotic environments, biotic interactions and symmetry-breaking phenomenon. To address this problem, we employ a Lotka-Volterra competitive system consisting of both symmetrical, asymmetrical interactions and abiotic environment components. We discover that that asymmetrical RPS competition in heterogeneous environments can yield much richer dynamical behaviors, compared to the symmetrical and asymmetrical competition in homogeneous environments. While it is observed that species coexistence outcomes and/or oscillatory solutions are maintained as in the case of homogeneous environments, the nonuniformity in the environmental carrying capacities may lead to extra dynamics with regards to the appearance of survival states; for instance, coexistence of any two-species and single-species persistence states, which are not evident in the previous modelling studies. By means of bifurcation analysis, various salient features of the dynamical systems, including the emergence of certain attractors (e.g., different steady states, stable limit cycles and heteroclinic cycles) and co-dimension one bifurcations (e.g., transcritical and supercritical Hopf bifurcations) are realized in this ecological model. Overall, this modelling work provides a novel attempt to simultaneously encompass not only symmetry-breaking phenomenon through RPS competition, but also heterogeneity in the environments. This framework can provide additional insights to better understand various mechanisms underlying the effects of distinct ecological processes on multi-species communities.

Asymptotically stable heteroclinic cycles in discrete-time -equivariant cubic dynamical systems

We consider a discrete-time -equivariant cubic dynamical system depending upon a real parameter. The system under consideration is a particular case of a discrete analogue to the principal -equivariant differential equations. We rigorously prove that the system has an asymptotically stable heteroclinic cycle, relatively to an open subset of a compact subspace of the plane, for values of the parameter in the interval . We explore properties of the omega-limit sets for the points attracted by the heteroclinic cycle.

Multistability in the cyclic competition system.

Cyclically competition models have been successful to gain an insight of biodiversity mechanism in ecosystems. There are, however, still limitations to elucidate complex phenomena arising in real competition. In this paper, we report that a multistability occurs in a simple rock-paper-scissor cyclically competition model by assuming that intraspecific competition depends on the logistic growth of each species density. This complex stability is absent in any cyclically competition model, and we investigate how the proposed intraspecific competition affects biodiversity in the existing society of three species through macroscopic and microscopic approaches. When the system is multistable, we show basins of the asymptotically stable heteroclinic cycle and stable attractors to demonstrate how the survival state is determined by initial densities of three species. Also, we find that the multistability is associated with a subcritical Hopf bifurcation. This surprising finding will give an opportunity to interpret rich dynamical phenomena in ecosystems which may occur in cyclic competition systems with different types of interactions.

Pseudo-simple heteroclinic cycles in [formula omitted

We study pseudo-simple heteroclinic cycles for a Γ-equivariant system in R4 with finite Γ⊂O(4), and their nearby dynamics. In particular, in a first step towards a full classification – analogous to that which exists already for the class of simple cycles – we identify all finite subgroups of O(4) admitting pseudo-simple cycles. To this end we introduce a constructive method to build equivariant dynamical systems possessing a robust heteroclinic cycle. Extending a previous study we also investigate the existence of periodic orbits close to a pseudo-simple cycle, which depends on the symmetry groups of equilibria in the cycle. Moreover, we identify subgroups Γ⊂O(4), Γ⊄SO(4), admitting fragmentarily asymptotically stable pseudo-simple heteroclinic cycles. (It has been previously shown that for Γ⊂SO(4) pseudo-simple cycles generically are completely unstable.) Finally, we study a generalized heteroclinic cycle, which involves a pseudo-simple cycle as a subset.

On Takens’ last problem: tangencies and time averages near heteroclinic networks

We obtain a structurally stable family of smooth ordinary differential equations exhibiting heteroclinic tangencies for a dense subset of parameters. We use this to find vector fields C2-close to an element of the family exhibiting a tangency, for which the set of solutions with historic behaviour contains an open set. This provides an affirmative answer to Takens’ last problem (Takens 2008 Nonlinearity 21 T33–6). A limited solution with historic behaviour is one for which the time averages do not converge as time goes to infinity. Takens’ problem asks for dynamical systems where historic behaviour occurs persistently for initial conditions in a set with positive Lebesgue measure.The family appears in the unfolding of a degenerate differential equation whose flow has an asymptotically stable heteroclinic cycle involving two-dimensional connections of non-trivial periodic solutions. We show that the degenerate problem also has historic behaviour, since for an open set of initial conditions starting near the cycle, the time averages approach the boundary of a polygon whose vertices depend on the centres of gravity of the periodic solutions and their Floquet multipliers.We illustrate our results with an explicit example where historic behaviour arises C2-close of a -equivariant vector field.

Coupled Oscillatory Systems with 𝔻4 Symmetry and Application to van der Pol Oscillators

In this paper, we study the dynamics of autonomous ODE systems with [Formula: see text] symmetry. First, we consider eight weakly-coupled oscillators and establish the condition for the existence of stable heteroclinic cycles in most generic [Formula: see text]-equivariant systems. Then, we analyze the action of [Formula: see text] on [Formula: see text] and study the pattern of periodic solutions arising from Hopf bifurcation. We identify the type of periodic solutions associated with the pairs [Formula: see text] of spatiotemporal or spatial symmetries, and prove their existence by using the [Formula: see text] Theorem due to Hopf bifurcation and the [Formula: see text] symmetry. In particular, we give a rigorous proof for the existence of a fourth branch of periodic solutions in [Formula: see text]-equivariant systems. Further, we apply our theory to study a concrete case: two coupled van der Pol oscillators with [Formula: see text] symmetry. We use normal form theory to analyze the periodic solutions arising from Hopf bifurcation. Among the families of the periodic solutions, we pay particular attention to the phase-locked oscillations, each of them being embedded in one of the invariant manifolds, and identify the in-phase, completely synchronized motions. We derive their explicit expressions and analyze their stability in terms of the parameters.

Sequential dynamics in the motif of excitatory coupled elements

In this article a new model of motif (small ensemble) of neuron-like elements is proposed. It is built with the use of the generalized Lotka–Volterra model with excitatory couplings. The main motivation for this work comes from the problems of neuroscience where excitatory couplings are proved to be the predominant type of interaction between neurons of the brain. In this paper it is shown that there are two modes depending on the type of coupling between the elements: the mode with a stable heteroclinic cycle and the mode with a stable limit cycle. Our second goal is to examine the chaotic dynamics of the generalized three-dimensional Lotka–Volterra model.

Designing responsive pattern generators: stable heteroclinic channel cycles for modeling and control

A striking feature of biological pattern generators is their ability to respond immediately to multisensory perturbations by modulating the dwell time at a particular phase of oscillation, which can vary force output, range of motion, or other characteristics of a physical system. Stable heteroclinic channels (SHCs) are a dynamical architecture that can provide such responsiveness to artificial devices such as robots. SHCs are composed of sequences of saddle equilibrium points, which yields exquisite sensitivity. The strength of the vector fields in the neighborhood of these equilibria determines the responsiveness to perturbations and how long trajectories dwell in the vicinity of a saddle. For SHC cycles, the addition of stochastic noise results in oscillation with a regular mean period. In this paper, we parameterize noise-driven Lotka–Volterra SHC cycles such that each saddle can be independently designed to have a desired mean sub-period. The first step in the design process is an analytic approximation, which results in mean sub-periods that are within 2% of the specified sub-period for a typical parameter set. Further, after measuring the resultant sub-periods over sufficient numbers of cycles, the magnitude of the noise can be adjusted to control the mean period with accuracy close to that of the integration step size. With these relationships, SHCs can be more easily employed in engineering and modeling applications. For applications that require smooth state transitions, this parameterization permits each state’s distribution of periods to be independently specified. Moreover, for modeling context-dependent behaviors, continuously varying inputs in each state dimension can rapidly precipitate transitions to alter frequency and phase.

Simple heteroclinic cycles in

In generic dynamical systems heteroclinic cycles are invariant sets of codimension at least one, but they can be structurally stable in systems which are equivariant under the action of a symmetry group, due to the existence of flow-invariant subspaces. For dynamical systems in the minimal dimension for which such robust heteroclinic cycles can exist is n = 3. In this case the list of admissible symmetry groups is short and well known. The situation is different and more interesting when n = 4. In this paper, we list all finite groups Γ such that an open set of smooth Γ-equivariant dynamical systems in possesses a simple heteroclinic cycle (a structurally stable heteroclinic cycle satisfying certain additional constraints). This work extends the results which were obtained by Sottocornola in the case when all equilibria in the heteroclinic cycle belong to the same Γ-orbit (in this case one speaks of homoclinic cycles).

Cooperation within triplets in the rock-paper-scissors game

We study a population involved in a cyclic game of three strategies – the rock-paper-scissors game – whose agents interact through groups of three individuals (triplets), considering the possibility that two weak agents cooperate and beat a strong one. In a wide range of parameters the system presents a stable heteroclinic cycle, which implies that in a finite population some of the strategies become extinct and others survive. We find that the cooperation within triplets only benefits the survival of the strategy if the cooperation probability is above a certain threshold. We study the survival probabilities of the different strategies as a function of the cooperation parameters through a analytic approximation and compare with simulations, obtaining a good agreement. Results are generalizable to other systems with heteroclinic cycles.

Dynamics near a periodically-perturbed robust heteroclinic cycle

Robust heteroclinic cycles (RHCs) arise naturally in collections of symmetric differential equations derived as dynamical models in many fields, including fluid mechanics, game theory and population dynamics. In this paper, we present a careful study of the complicated dynamics generated by small amplitude periodic perturbations of a stable robust heteroclinic cycle (RHC). We give a detailed derivation of the Poincaré map for trajectories near the RHC, asymptotically correct in the limit of small amplitude perturbations. This reduces the nonautonomous system in R3 to a 2D map.We identify three distinct dynamical regimes. The distinctions between these regimes depend subtly on different distinguished limits of the two small parameters in the problem. The first regime corresponds to the RHC being only weakly attracting: here we show that the system is equivalent to a damped nonlinear pendulum with a constant torque. In the second regime the periodically-perturbed RHC is more strongly attracting and the system dynamics corresponds to that of a (non-invertible or invertible) circle map. In the third regime, of yet stronger attraction, the dynamics of the return map is chaotic and no longer reducible to a one-dimensional map. This third regime has been noted previously; our analysis in this paper focusses on providing quantitative results in the first two regimes.

Winnerless competition in coupled Lotka-Volterra maps

Winnerless competition is analyzed in coupled maps with discrete temporal evolution of the Lotka-Volterra type of arbitrary dimension. Necessary and sufficient conditions for the appearance of structurally stable heteroclinic cycles as a function of the model parameters are deduced. It is shown that under such conditions winnerless competition dynamics is fully exhibited. Based on these conditions different cases characterizing low, intermediate, and high dimensions are therefore computationally recreated. An analytical expression for the residence times valid in the N-dimensional case is deduced and successfully compared with the simulations.

Classification and stability of simple homoclinic cycles in

Heteroclinic cycles, unions of equilibria and connection trajectories, can be structurally stable in a Γ-equivariant system due to the existence of invariant subspaces. A structurally stable heteroclinic cycle is called simple if all connecting trajectories are one-dimensional. Heteroclinic cycles where equilibria are related by a symmetry γ ∈ Γ are called homoclinic. This paper presents a complete study of simple homoclinic cycles in . We find all symmetry groups Γ such that a Γ-equivariant dynamical system in can possess a simple homoclinic cycle. We introduce a classification of simple homoclinic cycles in based on the action of the system symmetry group. For systems in , we list all classes of simple homoclinic cycles. For each class, we derive necessary and sufficient conditions for asymptotic stability and fragmentary asymptotic stability in terms of eigenvalues of linearization near the steady state involved in the cycle. For any action of the groups Γ which can give rise to a simple homoclinic cycle, we list classes to which the respective homoclinic cycles belong, thus determining the conditions for the asymptotic stability of these cycles.

Global generic dynamics close to symmetry

Our object of study is the dynamics that arises in generic perturbations of an asymptotically stable heteroclinic cycle in S3. The cycle involves two saddle-foci of different type and is structurally stable within the class of (Z2⊕Z2)-symmetric vector fields. The cycle contains a two-dimensional connection that persists as a transverse intersection of invariant surfaces under symmetry-breaking perturbations.Gradually breaking the symmetry in a two-parameter family we get a wide range of dynamical behaviour: an attracting periodic trajectory; other heteroclinic trajectories; homoclinic orbits; n-pulses; suspended horseshoes and cascades of bifurcations of periodic trajectories near an unstable homoclinic cycle of Shilnikov type. We also show that, generically, the coexistence of linked homoclinic orbits at the two saddle-foci has codimension 2 and takes place arbitrarily close to the symmetric cycle.

Stability and bifurcations of heteroclinic cycles of type Z

Dynamical systems that are invariant under the action of a non-trivial symmetry group can possess structurally stable heteroclinic cycles. In this paper, we study stability properties of a class of structurally stable heteroclinic cycles in which we call heteroclinic cycles of type Z. It is well known that a heteroclinic cycle that is not asymptotically stable can nevertheless attract a positive measure set from its neighbourhood. We say that an invariant set X is fragmentarily asymptotically stable, if for any δ > 0 the measure of its local basin of attraction is positive. A local basin of attraction is the set of such points that trajectories starting there remain in the δ-neighbourhood of X for all t > 0, and are attracted by X as t → ∞. Necessary and sufficient conditions for fragmentary asymptotic stability are expressed in terms of eigenvalues and eigenvectors of transition matrices. If all transverse eigenvalues of linearizations near steady states involved in the cycle are negative, then fragmentary asymptotic stability implies asymptotic stability. In the latter case the condition for asymptotic stability is that the transition matrices have an eigenvalue larger than one in absolute value. Finally, we discuss bifurcations occurring when the conditions for asymptotic stability or for fragmentary asymptotic stability are broken.

Phase Resetting in an Asymptotically Phaseless System: On the Phase Response of Limit Cycles Verging on a Heteroclinic Orbit

Rhythmic behaviors in neural systems often combine features of limit-cycle dynamics (stability and periodicity) with features of near heteroclinic or near homoclinic cycle dynamics (extended dwell times in localized regions of phase space). Proximity of a limit cycle to one or more saddle equilibria can have a profound effect on the timing of trajectory components and response to both fast and slow perturbations, providing a possible mechanism for adaptive control of rhythmic motions. Reyn [“Generation of limit cycles from separatrix polygons in the phase plane” in Geometrical Approaches to Differential Equations, Lecture Notes in Math. 810, Springer, New York, 1980, pp. 264–289] showed that for a planar dynamical system with a stable heteroclinic cycle (or separatrix polygon), small perturbations satisfying a net inflow condition will generically give rise to a stable limit cycle (see also [J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 3rd ed...

A new mechanism for stability loss from a heteroclinic cycle

Asymptotically stable robust heteroclinic cycles can lose stability through resonance or transverse bifurcations. In a transverse bifurcation, an equilibrium in the cycle undergoes a local bifurcation, causing a change in stability. A resonance bifurcation is a global phenomenon, determined by an algebraic condition on the eigenvalues, and is generically accompanied by the birth or death of a long-period periodic orbit. In this article we demonstrate a new mechanism causing loss of stability, which is neither resonant nor transverse in the usual sense. The location of the instability is determined by an algebraic condition on the eigenvalues, but the instability occurs in a transverse direction. Furthermore, after the bifurcation, when the cycle is unstable, open sets of trajectories are seen to initially approach the network for an extended period, before moving away in the unstable direction. This should serve as a warning to all those doing numerics near heteroclinic cycles who deduce that the cycle is stable merely because trajectories are observed to initially approach the cycle.