Stability of cycling behaviour near a heteroclinic network model of Rock–Paper–Scissors–Lizard–Spock

Postlethwaite, Claire M.; Rucklidge, Alastair M.

doi:10.1088/1361-6544/ac3560

Cited by 18 publications

(11 citation statements)

References 65 publications

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“…After substituting the equation for x c into those for x e and y e , and those in turn into an expression for T, we derive, after substituting in (27) and again using the simplification that T 1 and λ − t < 0,…”

Section: Complex Expanding Eigenvalues In the σ Cyclementioning

confidence: 99%

Travelling waves and heteroclinic networks in models of spatially-extended cyclic competition

Groothuizen Dijkema,

Postlethwaite

2023

Nonlinearity

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Dynamical systems containing heteroclinic cycles and networks can be invoked as models of intransitive competition between three or more species. When populations are assumed to be well-mixed, a system of ordinary differential equations (ODEs) describes the interaction model. Spatially extending these equations with diffusion terms creates a system of partial differential equations which captures both the spatial distribution and mobility of species. In one spatial dimension, travelling wave solutions can be observed, which correspond to periodic orbits in ODEs that describe the system in a steady-state travelling frame of reference. These new ODEs also contain a heteroclinic structure. For three species in cyclic competition, the topology of the heteroclinic cycle in the well-mixed model is preserved in the steady-state travelling frame of reference. We demonstrate that with four species, the heteroclinic cycle which exists in the well-mixed system becomes a heteroclinic network in the travelling frame of reference, with additional heteroclinic orbits connecting equilibria not connected in the original cycle. We find new types of travelling waves which are created in symmetry-breaking bifurcations and destroyed in an orbit flip bifurcation with a cycle between only two species. These new cycles explain the existence of ‘defensive alliances’ observed in previous numerical experiments. We further describe the structure of the heteroclinic network for any number of species, and we conjecture how these results may generalise to systems of any arbitrary number of species in cyclic competition.

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Section: Complex Expanding Eigenvalues In the σ Cyclementioning

confidence: 99%

Travelling waves and heteroclinic networks in models of spatially-extended cyclic competition

Groothuizen Dijkema,

Postlethwaite

2023

Nonlinearity

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“…which has eigenvalues δ, ±i, but the eigenvector for the eigenvalue δ has a zero in the second component, and so this cycle can also never be fragmentarily asymptotically stable. There are other routes trajectories can take whilst still approaching the network: in fact, this network is equivalent to the Rock-Paper-Scissors-Lizard-Spock network investigated by Postlethwaite and Rucklidge (for ODEs) 16 , which has some very complicated dynamics: see figure 12 for a typical time series. 1) , .…”

Section: Analysis Of Ring Graph With M-nearest Neighbour Couplingmentioning

confidence: 99%

Stability of heteroclinic cycles in rings of coupled oscillators

Postlethwaite¹,

Sturman²

2022

Preprint

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Networks of interacting nodes connected by edges arise in almost every branch of scientific enquiry. 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 networks with nearest-neighbour or nearest-mneighbour 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 which can be asymptotically stable, and thus the attracting dynamics of the network are expected to be non-ergodic. We conjecture that much of this behaviour persists in less structured networks and as such, non-ergodic behaviour is somehow typical.

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“…For the cycle ξ 1 → ξ 4 → ξ 2 → ξ 5 → ξ 3 , the transition matrix is   δ 1 0 −1 0 1 δ 0 0   , which has eigenvalues δ, ±i, but the eigenvector for the eigenvalue δ has a zero in the second component and so this cycle can also never be fragmentarily asymptotically stable. There are other routes trajectories can take while still approaching the network: in fact, this network is equivalent to the Rock-Paper-Scissors-Lizard-Spock network investigated by Postlethwaite and Rucklidge (for ODEs), 16 which has some very complicated dynamics: see Fig. 12 for a typical time series.…”

Section: Analysis Of Ring Graph With M-nearest-neighbor Couplingmentioning

confidence: 99%

Stability of heteroclinic cycles in ring graphs

Postlethwaite

Sturman

2022

Chaos: An Interdisciplinary Journal of Nonlinear Science

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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-[Formula: see text]-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.

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Stability of cycling behaviour near a heteroclinic network model of Rock–Paper–Scissors–Lizard–Spock

Cited by 18 publications

References 65 publications

Travelling waves and heteroclinic networks in models of spatially-extended cyclic competition

Travelling waves and heteroclinic networks in models of spatially-extended cyclic competition

Stability of heteroclinic cycles in rings of coupled oscillators

Stability of heteroclinic cycles in ring graphs

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