35 research outputs found
Resonance bifurcations of robust heteroclinic networks
Robust heteroclinic cycles are known to change stability in resonance
bifurcations, which occur when an algebraic condition on the eigenvalues of the
system is satisfied and which typically result in the creation or destruction
of a long-period periodic orbit. Resonance bifurcations for heteroclinic
networks are more complicated because different subcycles in the network can
undergo resonance at different parameter values, but have, until now, not been
systematically studied. In this article we present the first investigation of
resonance bifurcations in heteroclinic networks. Specifically, we study two
heteroclinic networks in and consider the dynamics that occurs as
various subcycles in each network change stability. The two cases are
distinguished by whether or not one of the equilibria in the network has real
or complex contracting eigenvalues. We construct two-dimensional Poincare
return maps and use these to investigate the dynamics of trajectories near the
network. At least one equilibrium solution in each network has a
two-dimensional unstable manifold, and we use the technique developed in [18]
to keep track of all trajectories within these manifolds. In the case with real
eigenvalues, we show that the asymptotically stable network loses stability
first when one of two distinguished cycles in the network goes through
resonance and two or six periodic orbits appear. In the complex case, we show
that an infinite number of stable and unstable periodic orbits are created at
resonance, and these may coexist with a chaotic attractor. There is a further
resonance, for which the eigenvalue combination is a property of the entire
network, after which the periodic orbits which originated from the individual
resonances may interact. We illustrate some of our results with a numerical
example.Comment: 46 pages, 20 figures. Supplementary material (two animated gifs) can
be found on
http://www.maths.leeds.ac.uk/~alastair/papers/KPR_res_net_abs.htm
Stability of cycling behaviour near a heteroclinic network model of Rock-Paper-Scissors-Lizard-Spock
The well-known game of Rock--Paper--Scissors can be used as a simple model of
competition between three species. When modelled in continuous time using
differential equations, the resulting system contains a heteroclinic cycle
between the three equilibrium solutions representing the existence of only a
single species. The game can be extended in a symmetric fashion by the addition
of two further strategies (`Lizard' and `Spock'): now each strategy is dominant
over two of the remaining four strategies, and is dominated by the remaining
two. The differential equation model contains a set of coupled heteroclinic
cycles forming a heteroclinic network. In this paper we carefully consider the
dynamics near this heteroclinic network. We are able to identify regions of
parameter space in which arbitrarily long periodic sequences of visits are made
to the neighbourhoods of the equilibria, which form a complicated pattern in
parameter space.Comment: Submitted to Nonlinearit
Two-state intermittency near a symmetric interaction of saddle-node and Hopf bifurcations: a case study from dynamo theory
Copyright © 2004 Elsevier. NOTICE: This is the author’s version of a work accepted for publication by Elsevier. Changes resulting from the publishing process, including peer review, editing, corrections, structural formatting and other quality control mechanisms, may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Physica D, Vol 194, Issues 1-2, 2004, DOI:10.1016/j.physd.2004.02.002We consider a model of a Hopf bifurcation interacting as a codimension 2 bifurcation with a saddle-node on a limit cycle, motivated by a low-order model for magnetic activity in a stellar
dynamo. This model consists of coupled interactions between a saddle-node and two Hopf bifurcations,
where the saddle-node bifurcation is assumed to have global reinjection of trajectories.
The model can produce chaotic behaviour within each of a pair of invariant subspaces, and also
it can show attractors that are stuck-on to both of the invariant subspaces. We investigate the
detailed intermittent dynamics for such an attractor, investigating the effect of breaking the
symmetry between the two Hopf bifurcations, and observing that it can appear via blowout
bifurcations from the invariant subspaces.
We give a simple Markov chain model for the two-state intermittent dynamics that reproduces
the time spent close to the invariant subspaces and the switching between the different
possible invariant subspaces; this clarifes the observation that the proportion of time spent near
the different subspaces depends on the average residence time and also on the probabilities of
switching between the possible subspaces
Rectangle--triangle soft-matter quasicrystals with hexagonal symmetry
Aperiodic (quasicrystalline) tilings, such as Penrose's tiling, can be built
up from e.g. kites and darts, squares and equilateral triangles, rhombi or
shield shaped tiles and can have a variety of different symmetries. However,
almost all quasicrystals occurring in soft-matter are of the dodecagonal type.
Here, we investigate a class of aperiodic tilings with hexagonal symmetry that
are based on rectangles and two types of equilateral triangles. We show how to
design soft-matter systems of particles interacting via pair potentials
containing two length-scales that form aperiodic stable states with two
different examples of rectangle--triangle tilings. One of these is the
bronze-mean tiling, while the other is a generalization. Our work points to how
more general (beyond dodecagonal) quasicrystals can be designed in soft-matter.Comment: 15 pages, 13 figures. Submitted to Physical Review E. The data
associated with this paper are openly available from the University of Leeds
Data Repository at https://doi.org/10.5518/118
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
Three-wave interactions and spatio-temporal chaos
Three-wave interactions form the basis of our understanding of many pattern
forming systems because they encapsulate the most basic nonlinear interactions.
In problems with two comparable length scales, it is possible for two waves of
the shorter wavelength to interact with one wave of the longer, as well as for
two waves of the longer wavelength to interact with one wave of the shorter.
Consideration of both types of three-wave interactions can generically explain
the presence of complex patterns and spatio-temporal chaos. Two length scales
arise naturally in the Faraday wave experiment with multi-frequency forcing,
and our results enable some previously unexplained experimental observations of
spatio-temporal chaos to be interpreted in a new light. Our predictions are
illustrated with numerical simulations of a model partial differential
equation.Comment: 4 pages, 3 figures, revised versio
Snaking without subcriticality: grain boundaries as non-topological defects
Non-topological defects such as grain boundaries abound in pattern forming
systems, arising from local variations of pattern properties such as amplitude,
wavelength, orientation, etc. We introduce the idea of treating such
non-topological defects as spatially localised structures that are embedded in
a background pattern, instead of treating them in an amplitude-phase
decomposition. Using the two-dimensional quadratic-cubic Swift--Hohenberg
equation as an example we obtain fully nonlinear equilibria that contain grain
boundaries which are closed curves containing multiple penta-hepta defects
separating regions of hexagons with different orientations. These states arise
from local orientation mismatch between two stable hexagon patterns, one of
which forms the localised grain and the other its background, and do not
require a subcritical bifurcation connecting them. Multiple robust isolas that
span a wide range of parameters are obtained even in the absence of a unique
Maxwell point, underlining the importance of retaining pinning when analysing
patterns with defects, an effect omitted from the amplitude-phase description.Comment: 16 pages, 12 figures and 2 movies in mp4 forma
Coupled environmental and demographic fluctuations shape the evolution of cooperative antimicrobial resistance
There is a pressing need to better understand how microbial populations
respond to antimicrobial drugs, and to find mechanisms to possibly eradicate
antimicrobial-resistant cells. The inactivation of antimicrobials by resistant
microbes can often be viewed as a cooperative behavior leading to the
coexistence of resistant and sensitive cells in large populations and static
environments. This picture is however greatly altered by the fluctuations
arising in volatile environments, in which microbial communities commonly
evolve. Here, we study the eco-evolutionary dynamics of a population consisting
of an antimicrobial resistant strain and microbes sensitive to antimicrobial
drugs in a time-fluctuating environment, modeled by a carrying capacity
randomly switching between states of abundance and scarcity. We assume that
antimicrobial resistance is a shared public good when the number of resistant
cells exceeds a certain threshold. Eco-evolutionary dynamics is thus
characterized by demographic noise (birth and death events) coupled to
environmental fluctuations which can cause population bottlenecks. By combining
analytical and computational means, we determine the environmental conditions
for the long-lived coexistence and fixation of both strains, and characterize a
fluctuation-driven antimicrobial resistance eradication mechanism, where
resistant microbes experience bottlenecks leading to extinction. We also
discuss the possible applications of our findings to laboratory-controlled
experiments.Comment: 19+7 pages, 4+1 figures. Simulation data and codes for all figures
are electronically available from the University of Leeds Data Repository.
DOI: https://doi.org/10.5518/136
Soft-core particles freezing to form a quasicrystal and a crystal-liquid phase
Systems of soft-core particles interacting via a two-scale potential are studied. The potential is responsible for
peaks in the structure factor of the liquid state at two different but comparable length scales and a similar bimodal
structure is evident in the dispersion relation. Dynamical density functional theory in two dimensions is used to
identify two unusual states of this system: a crystal-liquid state, in which the majority of the particles are located
on lattice sites but a minority remains free and so behaves like a liquid, and a 12-fold quasicrystalline state.
Both are present even for deeply quenched liquids and are found in a regime in which the liquid is unstable with
respect to modulations on the smaller scale only. As a result, the system initially evolves towards a small-scale
crystal state; this state is not a minimum of the free energy, however, and so the system subsequently attempts to
reorganize to generate the lower-energy larger-scale crystals. This dynamical process generates a disordered state
with quasicrystalline domains and takes place even when this large scale is linearly stable, i.e., it is a nonlinear
process. With controlled initial conditions, a perfect quasicrystal can form. The results are corroborated using
Brownian dynamics simulations