401 research outputs found
Recurrent flow analysis in spatiotemporally chaotic 2-dimensional Kolmogorov flow
Motivated by recent success in the dynamical systems approach to transitional
flow, we study the efficiency and effectiveness of extracting simple invariant
sets (recurrent flows) directly from chaotic/turbulent flows and the potential
of these sets for providing predictions of certain statistics of the flow.
Two-dimensional Kolmogorov flow (the 2D Navier-Stokes equations with a
sinusoidal body force) is studied both over a square [0, 2{\pi}]2 torus and a
rectangular torus extended in the forcing direction. In the former case, an
order of magnitude more recurrent flows are found than previously (Chandler &
Kerswell 2013) and shown to give improved predictions for the dissipation and
energy pdfs of the chaos via periodic orbit theory. Over the extended torus at
low forcing amplitudes, some extracted states mimick the statistics of the
spatially-localised chaos present surprisingly well recalling the striking
finding of Kawahara & Kida (2001) in low-Reynolds-number plane Couette flow. At
higher forcing amplitudes, however, success is limited highlighting the
increased dimensionality of the chaos and the need for larger data sets.
Algorithmic developments to improve the extraction procedure are discussed
Applications of dynamical systems with symmetry
This thesis examines the application of symmetric dynamical systems theory to
two areas in applied mathematics: weakly coupled oscillators with symmetry, and
bifurcations in flame front equations.
After a general introduction in the first chapter, chapter 2 develops a theoretical
framework for the study of identical oscillators with arbitrary symmetry group under an
assumption of weak coupling. It focusses on networks with 'all to all' Sn coupling. The
structure imposed by the symmetry on the phase space for weakly coupled oscillators
with Sn, Zn or Dn symmetries is discussed, and the interaction of internal symmetries
and network symmetries is shown to cause decoupling under certain conditions.
Chapter 3 discusses what this implies for generic dynamical behaviour of coupled
oscillator systems, and concentrates on application to small numbers of oscillators (three
or four). We find strong restrictions on bifurcations, and structurally stable heteroclinic
cycles.
Following this, chapter 4 reports on experimental results from electronic oscillator
systems and relates it to results in chapter 3. In a forced oscillator system, breakdown
of regular motion is observed to occur through break up of tori followed by a symmetric
bifurcation of chaotic attractors to fully symmetric chaos.
Chapter 5 discusses reduction of a system of identical coupled oscillators to phase
equations in a weakly coupled limit, considering them as weakly dissipative Hamiltonian
oscillators with very weakly coupling. This provides a derivation of example phase
equations discussed in chapter 2. Applications are shown for two van der Pol-Duffing
oscillators in the case of a twin-well potential.
Finally, we turn our attention to the Kuramoto-Sivashinsky equation. Chapter 6
starts by discussing flame front equations in general, and non-linear models in particular.
The Kuramoto-Sivashinsky equation on a rectangular domain with simple
boundary conditions is found to be an example of a large class of systems whose linear
behaviour gives rise to arbitrarily high order mode interactions.
Chapter 7 presents computation of some of these mode interactions using competerised
Liapunov-Schmidt reduction onto the kernel of the linearisation, and investigates
the bifurcation diagrams in two parameters
Mathematical frameworks for oscillatory network dynamics in neuroscience
The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear—for example, heteroclinic network attractors. In this review we present a set of mathemat- ical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical frame- work for further successful applications of mathematics to understanding network dynamics in neuroscience
Local Causal States and Discrete Coherent Structures
Coherent structures form spontaneously in nonlinear spatiotemporal systems
and are found at all spatial scales in natural phenomena from laboratory
hydrodynamic flows and chemical reactions to ocean, atmosphere, and planetary
climate dynamics. Phenomenologically, they appear as key components that
organize the macroscopic behaviors in such systems. Despite a century of
effort, they have eluded rigorous analysis and empirical prediction, with
progress being made only recently. As a step in this, we present a formal
theory of coherent structures in fully-discrete dynamical field theories. It
builds on the notion of structure introduced by computational mechanics,
generalizing it to a local spatiotemporal setting. The analysis' main tool
employs the \localstates, which are used to uncover a system's hidden
spatiotemporal symmetries and which identify coherent structures as
spatially-localized deviations from those symmetries. The approach is
behavior-driven in the sense that it does not rely on directly analyzing
spatiotemporal equations of motion, rather it considers only the spatiotemporal
fields a system generates. As such, it offers an unsupervised approach to
discover and describe coherent structures. We illustrate the approach by
analyzing coherent structures generated by elementary cellular automata,
comparing the results with an earlier, dynamic-invariant-set approach that
decomposes fields into domains, particles, and particle interactions.Comment: 27 pages, 10 figures;
http://csc.ucdavis.edu/~cmg/compmech/pubs/dcs.ht
Bifurcation to heteroclinic cycles and sensitivity in three and four coupled phase oscillators
Copyright © 2008 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 237, Issue 4, 2008, DOI: 10.1016/j.physd.2007.09.015We study the bifurcation and dynamical behaviour of the system of N globally coupled identical phase oscillators introduced by Hansel, Mato and Meunier, in the cases N=3 and N=4. This model has been found to exhibit robust ‘slow switching’ oscillations that are caused by the presence of robust heteroclinic attractors. This paper presents a bifurcation analysis of the system in an attempt to better understand the creation of such attractors. We consider bifurcations that occur in a system of identical oscillators on varying the parameters in the coupling function. These bifurcations preserve the permutation symmetry of the system. We then investigate the implications of these bifurcations for the sensitivity to detuning (i.e. the size of the smallest perturbations that give rise to loss of frequency locking).
For N=3 we find three types of heteroclinic bifurcation that are codimension-one with symmetry. On varying two parameters in the coupling function we find three curves giving (a) an S3-transcritical homoclinic bifurcation, (b) a saddle–node/heteroclinic bifurcation and (c) a Z3-heteroclinic bifurcation. We also identify several global bifurcations with symmetry that organize the bifurcation diagram; these are codimension-two with symmetry.
For N=4 oscillators we determine many (but not all) codimension-one bifurcations with symmetry, including those that lead to a robust heteroclinic cycle. A robust heteroclinic cycle is stable in an open region of parameter space and unstable in another open region. Furthermore, we verify that there is a subregion where the heteroclinic cycle is the only attractor of the system, while for other parts of the phase plane it can coexist with stable limit cycles. We finish with a discussion of bifurcations that appear for this coupling function and general N, as well as for more general coupling functions
Unstable Attractors: Existence and Robustness in Networks of Oscillators With Delayed Pulse Coupling
We consider unstable attractors; Milnor attractors such that, for some
neighbourhood of , almost all initial conditions leave . Previous
research strongly suggests that unstable attractors exist and even occur
robustly (i.e. for open sets of parameter values) in a system modelling
biological phenomena, namely in globally coupled oscillators with delayed pulse
interactions.
In the first part of this paper we give a rigorous definition of unstable
attractors for general dynamical systems. We classify unstable attractors into
two types, depending on whether or not there is a neighbourhood of the
attractor that intersects the basin in a set of positive measure. We give
examples of both types of unstable attractor; these examples have
non-invertible dynamics that collapse certain open sets onto stable manifolds
of saddle orbits.
In the second part we give the first rigorous demonstration of existence and
robust occurrence of unstable attractors in a network of oscillators with
delayed pulse coupling. Although such systems are technically hybrid systems of
delay differential equations with discontinuous `firing' events, we show that
their dynamics reduces to a finite dimensional hybrid system system after a
finite time and hence we can discuss Milnor attractors for this reduced finite
dimensional system. We prove that for an open set of phase resetting functions
there are saddle periodic orbits that are unstable attractors.Comment: 29 pages, 8 figures,submitted to Nonlinearit
From travelling waves to mild chaos: a supercritical bifurcation cascade in pipe flow
We study numerically a succession of transitions in pipe Poiseuille flow that
leads from simple travelling waves to waves with chaotic time-dependence. The
waves at the origin of the bifurcation cascade possess a shift-reflect symmetry
and are both axially and azimuthally periodic with wave numbers {\kappa} = 1.63
and n = 2, respectively. As the Reynolds number is increased, successive
transitions result in a wide range of time dependent solutions that includes
spiralling, modulated-travelling, modulated-spiralling,
doubly-modulated-spiralling and mildly chaotic waves. We show that the latter
spring from heteroclinic tangles of the stable and unstable invariant manifolds
of two shift-reflect-symmetric modulated-travelling waves. The chaotic set thus
produced is confined to a limited range of Reynolds numbers, bounded by the
occurrence of manifold tangencies. The states studied here belong to a subspace
of discrete symmetry which makes many of the bifurcation and path-following
investigations presented technically feasible. However, we expect that most of
the phenomenology carries over to the full state-space, thus suggesting a
mechanism for the formation and break-up of invariant states that can sustain
turbulent dynamics.Comment: 38 pages, 35 figures, 1 tabl
Nonlinear Dynamics Of Coupled Capillary-Surface Oscillators
The nonlinear dynamics of coupled liquid droplets and bridges are examined. By restricting droplet and bridge shapes to equilibrium states, the quasi-static dynamics of such systems may be studied using ordinary differential equations, and the techniques of nonlinear dynamics may be applied. For example, liquid droplets are restricted to spherical-caps, whose shapes may be deduced solely from their volume. Networks of liquid droplets are first considered. Static solutions are grouped into families, each with some p droplets large and some q = n [-] p small. The twodroplet system is modeled as a conservative second-order oscillator and fixed points undergo a pitchfork bifurcation as the total volume is increased; furthermore, when subjected to periodic forcing, chaotic dynamics are possible. Bounds for chaotic dynamics are investigated by using Melnikov's method and calculating Lyapunov exponents. Results are compared qualitatively with experimental results, thereby confirming the existence of chaotic motions. The two-droplet model is then extended to a n-droplet frictionless Sn symmetric model that consists of n [-] 1 second-order differential equations. Symmetry of the system is fundamental. In particular, independent of the equations, fixedpoints may be grouped into families by the number of small and large droplets. Within the families, stability is invariant and hence significantly reduces the number of equilibria to be considered. All equilibria, and their associated stability, are calculated analytically for an arbitrary number of droplets. For three droplets, the system is fourth order and thus trajectories are (in general) quasi-periodic or chaotic. Because the equations are S3 symmetric, trajectories may also possess S3 , or one of the three flip symmetries. Since there is no dissipation there are no asymptotically stable attractors. As such, trajectories of interest are away from equilibrium. In particular, trajectories with no initial velocity are analyzed to ascertain their symmetry as well as their dynamic nature. Both these determinations may be done in an automated fashion through the use of symmetry detectives and Lyapunov exponents, respectively. For this system, the results of these two methods reflect a strong correlation between symmetry and nonlinear dynamics; chaotic trajectories are S3 symmetric while quasi-periodic trajectories possess one of the three flip symmetries. Next, a non-smooth switching bridge-droplet system is considered. The system has two states: droplet-droplet and bridge-droplet. The switching system can be obtained from the two droplet system by introducing a planar substrate below one of the droplets. As the system oscillates, it may transition between states if the droplet impacts the wall or the liquid bridge breaks. The two transitions occur at different places in state space which results in a region for which the system is multiply defined. In addition, transitions are assumed to be instantaneous with no loss of velocity. The two states are first analyzed separately. The bridgedroplet state undergoes a cusp bifurcation in a two parameter expansion. Boundary equilibrium bifurcations also occur when an equilibrium point collides with a nonsmooth boundary. If the bridge-droplet and droplet-droplet states are combined, a two parameter bifurcation diagram for the switching system is realized. Switching trajectories are of particular interest because each switching cycle dampens the system until it no longer switches. These trajectories are mapped into a semi- infinite cylindrical space in which long-term behavior can be described solely by the dynamics in the multiply defined region. In the final chapter models for pull-off adhesive failure are considered. Recognizing engineering applications (i.e. a capillary adhesion device) as well as a phenomenon found in nature (i.e. defense mechanism of palm beetle), models for pull-off adhesive failure are developed for different loading conditions and compared with available observations. In particular, array geometry and the relationship of adhesive failure to the instabilities of a single liquid bridge are emphasized
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