Bifurcations in dynamical systems with interior symmetry

We propose a definition of interior symmetry in the context of general dynamical systems. This concept appeared originally in the theory of coupled cell networks, as a generalization of the idea of symmetry of a network. The notion of interior symmetry introduced here can be seen as a special form of forced symmetry breaking of an equivariant system of differential equations. Indeed, we show that a dynamical system with interior symmetry can be written as the sum of an equivariant system and a ‘perturbation term’ which completely breaks the symmetry. Nonetheless, the resulting dynamical system still retains an important feature common to systems with symmetry, namely, the existence of flow-invariant subspaces. We define interior symmetry breaking bifurcations in analogy with the definition of symmetry breaking bifurcation from equivariant bifurcation theory and study the codimension one steady-state and Hopf bifurcations. Our main result is the full analogues of the well-known Equivariant Branching Lemma and the Equivariant Hopf Theorem from the bifurcation theory of equivariant dynamical systems in the context of interior symmetry breaking bifurcations.     

Positively equicontinuous flows

Let G be a positively equicontinuous flow of homeomorphisms of a locally compact metric space E. We show how the dynamics of such a flow are rich. We study when regularly almost periodic elements in G are periodic and we describe orbits and their limit sets. In particular, we show that the limit set L(G?) of G is a closed subset on which G is equicontinuous and that if G moreover has closed orbits, then any w-limit set is a periodic orbit.     

Abundance of elliptic dynamics on conservative three-flows

We consider a compact three-dimensional boundaryless Riemannian manifold M and the set of divergence-free (or zero divergence) vector fields without singularities, then we prove that this set has a C ¹-residual (dense G _δ) such that any vector field inside it is Anosov or else its elliptical orbits are dense in the manifold M. This is the flow-setting counterpart of Newhouse's Theorem 1.3 (S. Newhouse, Quasi-elliptic periodic points in conservative dynamical systems, Am. J. Math. 99 (1977), pp. 1061–1087). Our result follows from two theorems, the first one says that if Λ is a hyperbolic invariant set for some class C ¹ zero divergence vector field X on M, then either X is Anosov, or else Λ has empty interior. The second one says that, if X is not Anosov, then for any open set U ? M there exists Y arbitrarily close to X such that Y ^t has an elliptical closed orbit through U.     

Hopf bifurcations in competitive three-dimensional Lotka–Volterra systems

We study the space of Lotka–Volterra systems modelling three mutually competing species, each of which, in isolation, would exhibit logistic growth. By a theorem of M. W. Hirsch, the compact limit sets of these systems are either fixed points or periodic orbits. We use a geometric analysis of the surfaces ?=0 of a system, to define a combinatorial equivalence relation on the space, in terms of simple inequalities on the parameters. We list the 33 stable equivalence classes, and show that in 25 of these classes all the compact limit sets are fixed points, so we can fully describe the dynamics. We study the remaining eight equivalence classes by finding simple algebraic criteria on the parameters, with which we are able to predict the occurrence of Hopf bifurcations and, consequently, isolated periodic orbits.     

Symmetric and non-symmetric periodic orbits for the digital filter map

We exhibit instances of non-symmetric periodic orbits for the digital filter map, resolving a question posed in the literature as to whether such orbits can exist. This piecewise irrational rotation, depending on a parameter a = 2cos θ, is an isometry of [?1, 1) × [?1, 1) and reflections in the two diagonals are time-reversing symmetries for the map. Symmetric orbits are plentiful and have been much investigated. Each periodic orbit is paired with a symbolic string, from the alphabet {?, 0, +}, arising under iteration of the map because of the presence of a line of discontinuity. We prove the existence of an infinite family of non-symmetric orbits where the period N starts at 29 and increases in steps of 5; they correspond to the strings (+00)⁵(+?)² 0 ^N?19. We describe several computer algorithms to find non-symmetric periodic orbits and their symbolic strings and list non-symmetric strings both for a = 0.5, and for N ≤ 100 across the parameter range. Our evidence suggests that non-symmetric orbits, though not plentiful, are characteristic of the dynamics of the map for all parameter values.     

Transitive sets for quasi-periodically forced monotone maps

There has been considerable interest in recent years in quasi-periodically forced systems, partly due to the fact that these commonly exhibit strange non-chaotic attractors. Relatively little is known rigorously about such systems. In this paper we concentrate on investigating the structure of the simplest possible invariant sets for a particular class of quasi-periodically forced maps, namely those that are monotone in each fibre. Due to the quasi-periodic nature of the forcing, periodic orbits cannot occur, and their role is played by various types of invariant graph. Any compact invariant set is bounded by two invariant graphs, which are respectively upper and lower semi-continuous. If the set is minimal, these two graphs intersect on a residual set, on which both are continuous. Any transitive set Ω contains either one or two minimal sets, which must be the closure of one or the other of the boundaries of Ω. If Ω contains only one minimal set, then again its upper and lower boundaries intersect on a residual set. This case contains the original example of Grebogi et al. and its generalizations by Keller and by Glendinning. If Ω contains two minimal sets, then its upper and lower boundaries cannot intersect, though as far as we are aware, there is no known example where the existence of such a transitive set has been proven rigorously.     

Geodesic finite elements on simplicial grids

We introduce geodesic finite elements as a conforming way to discretize partial differential equations for functions v : Ω → M, where Ω is an open subset of and M is a Riemannian manifold. These geodesic finite elements naturally generalize standard first‐order finite elements for Euclidean spaces. They also generalize the geodesic finite elements proposed for d = 1 in a previous publication of the author. Our formulation is equivariant under isometries of M and, hence, preserves objectivity of continuous problem formulations. We concentrate on partial differential equations that can be formulated as minimization problems. Discretization leads to algebraic minimization problems on product manifolds Mⁿ. These can be solved efficiently using a Riemannian trust‐region method. We propose a monotone multigrid method to solve the constrained inner problems with linear multigrid speed. As an example, we numerically compute harmonic maps from a domain in to S². Copyright © 2012 John Wiley & Sons, Ltd.     

Dense distribution of chaotic maps in continuous map spaces

This article is concerned with distribution of several kinds of chaotic maps in a continuous map space, in which the maps are defined in a closed bounded set of a Banach space. It is shown that the map space contains a dense set of maps that are strictly coupled-expanding, have nondegenerate and regular snap-back repellers, have nondegenerate and regular homoclinic orbits to repellers, and consequently that are chaotic in the sense of Devaney as well as in the original sense of Li–Yorke, and have the topological entropy larger than any given positive constant. Further, in the finite-dimensional case, there exists a dense residual set of the map space such that every map?f in the set is strictly coupled-expanding in k pairwise disjoint compact sets for any given integer k?≥?2, is chaotic in the sense of Li–Yorke and has the infinite topological entropy and a nontrivial invariant measure.     

Transverse intersection of invariant manifolds in perturbed multi-symplectic systems

A multi-symplectic system is a PDE with a Hamiltonian structure in both temporal and spatial variables. This article considers spatially periodic perturbations of symmetric multi-symplectic systems. Due to their structure, unperturbed multi-symplectic systems often have families of solitary waves or front solutions, which together with the additional symmetries lead to large invariant manifolds. Periodic perturbations break the translational symmetry in space and might break some of the other symmetries as well. In this article, periodic perturbations of a translation invariant PDE with a one-dimensional symmetry group are considered. It is assumed that the unperturbed PDE has a three-dimensional invariant manifold associated with a solitary wave or front connection of multi-symplectic relative equilibria. Using the momentum associated with the symmetry group, sufficient conditions for the persistence of invariant manifolds and their transversal intersection are derived. In the equivariant case, invariance of the momentum under the perturbation gives the persistence of the full three-dimensional manifold. In this case, there is also a weaker condition for the persistence of a two-dimensional submanifold with a selected value of the momentum. In the non-equivariant case, the condition leads to the persistence of a one-dimensional submanifold with a seleceted value of the momentum and a selected action of the symmetry group. These results are applicable to general Hamiltonian systems with double zero eigenvalue in the linearization due to continuous symmetry. The conditions are illustrated on the example of the defocussing non-linear Schrödinger equations with perturbations which illustrate the three cases. The perturbations are: an equivariant Hamiltonian perturbation which keeps the momentum level sets invariant; an equivariant damped, driven perturbation; and a perturbation which breaks the rotational symmetry.     

Linking invariants for smooth minimal solenoids

In this paper we consider smooth diffeomorphisms of the 2-disk which are the identity on the boundary. We assume that the dynamics of any such a diffeomorphism φ restricted to a φ-invariant Cantor set is minimal and uniquely ergodic. Then the average linking number of the orbits of φ can be computed in two standard ways. We prove that the asymptotic average of the diagonal component of the Calabi invariant coincides with the Ruelle invariant of the minimal Cantor system.     

Nonreversible homoclinic snaking

Homoclinic snaking refers to the sinusoidal ‘snaking’ continuation curve of homoclinic orbits near a heteroclinic cycle connecting an equilibrium E and a periodic orbit P. Along this curve the homoclinic orbit performs more windings about the periodic orbit. Typically, this behaviour appears in reversible Hamiltonian systems. Here we discuss this phenomenon in systems without any particular structure. We give a rigorous analytical verification of homoclinic snaking under certain assumptions on the behaviour of the stable and unstable manifolds of E and P. We show how the snaking behaviour depends on the signs of the Floquet multipliers of P. Further we present a nonsnaking scenario. Finally, we show numerically that these assumptions are fulfilled in a model equation.     

Separation of Orbits under Group Actions with an Application to Quantum Systems

Motivated by the task to decide whether two quantum states are equally entangled we consider the orbits under the action of the group of all one-qubit operations. To investigate the orbit structure of this group of local unitary operations we propose to use methods from classical invariant theory as well as new results. Two approaches are presented. The first uses the orbit separation property of invariant rings to distinguish among nonequivalent quantum states. In this context we study the Molien series which describes the structure of the invariant ring as a graded ring. We give a closed formula for the Molien series of the group of one-qubit operations. Our second approach makes use of an equivalence relation, the so-called graph of the action, which relates two points iff they are on the same orbit. For finite groups which factor, are synchronous direct sums, or tensor products we analyze the structure of the graph of the action. This yields new algorithms for the computation of the graph of the action. Received: December 23, 1998; revised version August 27, 1999     

Random and deterministic perturbation of a class of skew-product systems

Abstract. This paper is concerned with the stability properties of skew-products T (x,y) = (f(x), g(x,y)) in which (f,X,mu) is an ergodic map of a compact metric space X and g: Xx Rn Rn is continuous. We assume that the skew-product has a negative maximal Lyapunov exponent in the fibre. We study the orbit stability and stability of mixing of T(x,y) = (f(x), g(x,y)) under deterministic and random perturbation of g. We show that such systems are stable in the sense that for any > 0 there is a pairing of orbits of the perturbed and unperturbed system such that paired orbits stay within a distance of each other except for a fraction of the time. Furthermore, we show that the invariant measure for the perturbed system is continuous (in the Hutchinson metric) as a function of the size of the perturbation to g (Lipschitz topology) and the noise distribution. Our results have applications to the stability of Iterated Function Systems which 'contract on average'.     

Time shift between unstable periodic orbits of coupled chaotic oscillators

The shift Δt between unstable periodic orbits of coupled oscillators occurring in the chaotic synchronization regime has been studied. It is shown that this time shift is the same for all equiphase orbits with various topological parameters and depends on the coupling parameter ε. This dependence obeys the universal power law Δt ∼ εⁿ with an exponent of n = −1.     

Midpoint rule for variational integrators on Lie groups

The midpoint rule provides a standard method to obtain symmetric, symplectic, and second‐order accurate variational integrators for mechanical systems whose configuration manifold is the vector space ?ⁿ. In this work, we discuss how to extend this rule to a generic finite‐dimensional Lie group G while retaining the same properties. We show that the function κ_G(g)=exp(½log(g)), g∈G plays a special role in the theory and, for G=SO(3), we give a compact formula to compute it. We also discuss sufficient conditions for the method to conserve momentum maps associated with left (or right) group actions. As an example, the variational integrator obtained from the midpoint rule is applied to simulating rigid body dynamics. The resulting integrator is compared with state‐of‐the‐art symmetric and second‐order accurate integrators for rigid body motion. Copyright © 2009 John Wiley & Sons, Ltd.     

Torus square tilings

We consider periodic square tilings of the plane. By extending a formalism introduced in 1940 for tiling of rectangles by squares we build a correspondence between periodic plane maps endowed with a periodic harmonic vector and periodic square tilings satisfying a regularity condition. The space of harmonic vectors is isomorphic to the first homology group of a torus. So, periodic plane square tilings are described by two parameters and the set of parameters is split into angular sectors. The correspondence between symmetry of the square tiling and symmetry of the corresponding plane map and harmonic vector is discussed and a method for enumerating the regular periodic plane square tilings having $r$ orbits of squares is outlined.