Coupled map lattices have been introduced recently for studying systems with spatial complexity. We consider simple examples of such systems generated by expanding maps of the unit interval (or circle) with some kind of diffusion coupling. It is shown that such systems have a symbolic representation by two-dimensional lattice models of statistical mechanics. The main result states that the Z(2) dynamical system generated by space translations and dynamics has a unique invariant mixing Gibbs measure with absolutely continuous finite-dimensional projections. This measure is an analogy of the BRS measure constructed for finite-dimensional hyperbolic transformations.
Persistent trajectories of the n-dimensional system x(over dot)(i) = x(i)N(i)(x(1),..., x(n)), x(i) >= 0, are studied under the assumptions that the system is competitive and dissipative with irreducible community matrices [partial derivative N-i/partial derivative x(j)]. The main result is that there is a canonically defined countable (generically finite) family of disjoint invariant open (n-1) cells which attract all non-convergent persistent trajectories. These cells are Lipschitz submanifolds and are transverse to positive rays. In dimension 3 this implies that an omega limit set of a persistent orbit is either an equilibrium, a cycle bounding an invariant disc, or a one-dimensional continuum having a non-trivial first tech cohomology group and containing an equilibrium. Thus the existence of a persistent trajectory in the three-dimensional case implies the existence of a positive equilibrium. In any dimension it is shown that if the community matrices are strictly negative then there is a closed invariant (n-1) cell which attracts every persistent trajectory. This shows that a seemingly special construction by Smale of certain competitive systems is in fact close to the general case.
The complex Ginzburg-Landau equation in one spatial dimension with periodic boundary conditions is studied from the viewpoint of effective low-dimensional behaviour by three distinct methods. Linear stability analysis of a class of exact solutions establishes lower bounds on the dimension of the universal, or global, attractor and the Fourier spanning dimension, defined here as the number of Fourier modes required to span the universal attractor. We use concepts from the theory of inertial manifolds to determine rigorous upper bounds on the Fourier spanning dimension, which also establishes the finite dimensionality of the universal attractor. Upper bounds on the dimension of the attractor itself are obtained by bounding (or, for some parameter values, computing exactly) the Lyapunov dimension and invoking a recent theorem that asserts that the Lyapunov dimension, defined by the Kaplan-Yorke formula with the universal (global) Lyapunov exponents, is an upper bound on the Hausdorff dimension. This study of low dimensionality in the complex Ginzburg-Landau equation allows for an examination of the current techniques used in the rigorous investigation of finite-dimensional behaviour. Contact is made with some recent results for fluid turbulence models, and we discuss some unexplored directions in the area of low-dimensional behaviour in the complex Ginzburg-Landau equation.
By pretending that the imaginary parts E-m of the Riemann zeros are eigenvalues of a quantum Hamiltonian whose corresponding classical trajectories are chaotic and without time-reversal symmetry, it is possible to obtain by asymptotic arguments a formula for the mean square difference V (L; x) between the actual and average number of zeros near the xth zero in an interval where the expected number is L. This predicts that when L > L-max, V will have quasirandom oscillations about the mean value pi(-2)(ln In(E/2 pi) + 1.4009). Comparisons with V(L; x) computed by Odlyzko from 10(5) zeros E-m near x = 10(12)' confirm all details of the semiclassical predictions to within the limits of graphical precision.
A study is made of solutions of the Yang-Mills equations over a quaternionic Kahler manifold. The corresponding notion of self-duality is interpreted in terms of holomorphic geometry on a twistor space. Self-dual connections are constructed on various vector bundles over quaternionic projective spaces.
Four identical nonlinear oscillators, coupled with the symmetry of a square, can undergo a symmetric version of the standard Hopf bifurcation. Golubitsky and Stewart have studied the case of N oscillators coupled in a ring with nearest-neighbour coupling. Their results are incomplete for the square case (N = 4) because they only considered periodic solutions which have 'maximal' symmetry. Here we study the dynamics of all possible square-symmetric Hopf bifurcations; these codimension-one bifurcations are parametrised by the three complex cubic coefficients in the normal form. We find that invariant tori (quasiperiodic solutions with two frequencies) and periodic solutions with 'minimal' symmetry bifurcate from the origin for open regions of the parameter space of cubic coefficients. The Coefficients can be chosen so that the invariant ton are the only asymptotically stable solutions near the origin. Thus a direct transition from a stable fixed point to flow on a stable invariant torus is expected in certain laboratory experiments with only one adjustable parameter (provided the square symmetry is accurate enough). Furthermore, it is conjectured that there are attracting chaotic solutions arbitrarily close to the bifurcation in a certain codimension-two case.
The recursively spiralling patterns drawn in the complex plane by the values of S-L(tau) = Sigma(L)(n=1) exp(i pi tau n(2)) as L -> infinity with tau fixed in the range 0 0 (semiclassical), T -> infinity (long time). The intensity of light diffracted by a grating with many slits and detected on a distant screen depends on vertical bar S-L(tau)vertical bar(2). In principle the effects of the curlicues are observable, but experiments are likely to be difficult.
A new definition of the stability of ordinary differential equations is proposed as an alternative to structural stability. It is particularly aimed at dissipative nonlinear systems, including those with chaos or strange attractors. The definition is as follows. Given a vector field v on an oriented manifold X, and given epsilon > 0, let u be the steady state of the Fokker-Planck equation for v with epsilon-diffusion. The existence, uniqueness and global attraction of v is proved in the case when Xis compact (in the non-compact case a suitable boundary condition on v is required for the existence of U). Vector fields are defined to be equivalent, or stable, according to whether their steady states are. A similar theory is developed for diffeomorphisms. The new definition has a number of advantages over structural stability. Stable systems are dense, and therefore most strange attractors are stable, including non-hyperbolic ones. The equivalence extends the Thom classification of gradient systems to non-gradient systems. The theory is closely related to applications, because the steady state v is an epsilon-smoothing of the measure on the attractors of the flow of v, and therefore in numerical and physical experiments v can be used to model the data with epsilon-error.
We study the paths of individual fluid particles moving in velocity fields which model Taylor vortices close to the onset of the wavy instability. In particular, we consider the possibility of particle transport between vortices. By studying the flow in the context of dynamical systems theory, we show that this arises through the destruction of invariant surfaces which form the vortex boundaries in the absence of the wave. Particles able to pass between vortices follow chaotic trajectories (in the sense of showing sensitive dependence on initial conditions). This results in a mixing process that has some properties in common with diffusion.
The chiral equation, for maps into a non-Abelian group, is only integrable in two-dimensional spacetime. If, however, one adds a torsion term, then integrability in higher dimensions can be achieved. But the Painleve test indicates that dimension four is as far as one can go.
We study the periodic orbits which can occur in a neighbourhood of a codimension-two gluing bifurcation involving two trajectories bi-asymptotic to the same stationary point. Provided some simple conditions are satisfied we prove that there are either zero, one or two closed curves and that these hove a specific symbolic form which, in particular, allows us to associate a rotation number with each of them. Furthermore, pairs of orbits which can coexist are identified: the two rotation numbers must be Farey neighbours.
Strange attractors in dynamical systems that go to chaos via quasiperiodicity are considered. It is shown that there exists an infinite number of points in parameter space where the topology of the strange attractors is universal. At such points the periodic points belonging to unstable periodic orbits can be organised on ternary trees which are pruned by local rules. The grammar is universal, and thus the topological entropy is universal at each of these points in parameter space. The complete understanding of the topology is used to calculate systematically the metric properties of the attractors. The spectrum of scaling indices f(alpha) is computed. It is found that there is no metric universality, although some aspects of the metric properties are universal. Experiments to test some of the predictions of this theory are suggested.
Two-dimensional systems (x)over dot = P(x, y) (y)over dot = Q(x, y) in which P and Q are cubic polynomials, are considered, and a number of classes with several limit cycles are described. Examples of systems with six small-amplitude limit cycles are given. Other classes of systems with several limit cycles are obtained by considering simultaneous bifurcation from a finite critical point and infinity. Simultaneous bifurcation from several critical points is investigated.
We find universal scaling behaviour for the period-doubling tree in two-parameter families of bimodal maps of the interval. A renormalisation group explanation is given in terms of a horseshoe with a Cantor set of two-dimensional unstable manifolds instead of the usual fixed point with one unstable direction.
We investigate a modification of the Weiss-Tabor-Carnevale procedure that enables one to obtain Lax pairs and Backlund transformations for systems of ordinary differential equations. This method can yield both auto-Backlund transformations and, where necessary, Backlund transformations between different equations. In the latter case we investigate the circumstances under which the general Backlund transformations reduce to auto-Backlunds. In addition, special solution families for the second and fourth Poinleve transcendents are obtained.
Previous results about universality in phase space have been local in nature and only concern scaling about a single point. In this paper I prove that a much stronger global result holds for three important examples: period-doubling cascades, golden critical circle maps and certain diffeomorphisms of the circle. In each case I prove that the conjugacy between the appropriate phase space structures of two systems in the same stable manifold of the appropriate renormalisation transformation is or can be extended to a C1+alpha diffeomorphism. This means that the structures are globally geometrically equivalent and have the same global quantitative scaling properties. These results are corollaries of a general theory for Markov families and the general techniques and results should be applicable to a much wider class of problems.
By reducing the Ward correspondence, we show that there is a correspondence between stationary axisymmetric solutions of the vacuum Einstein equations and a class of holomorphic vector bundles over a reduced twistor space, which is a compact one-dimensional, but non-Hausdorff, complex manifold. We show that the solutions generated by Ward's ansatze correspond to bundles which have a simple behaviour on the 'real axis' in the reduced space. We identify the Geroch group (Kinnersley and Chitre's 'group K') with a subgroup of the loop group of GL(2, C) and we describe its orbits. We also identify some of the subgroups which preserve asymptotic flatness.
A symmetry-breaking Hopf bifurcation in an O(2)-equivariant system generically produces a branch of standing waves and two branches of oppositely propagating travelling waves. This generic bifurcation assumes three non-degeneracy conditions on the cubic terms of the Poincarh-Birkhoff normal form. When these conditions fail more complicated behaviour accompanies the bifurcation; in particular one finds secondary bifurcations of quasiperiodic waves. For these degenerate bifurcations, the effects of perturbations which break the reflection symmetry are considered. The perturbed system retains a residual SO(2) symmetry. Qualitatively these perturbations have three effects: (1) they split the double multiplicity eigenvalues so that the travelling waves bifurcate separately, (2) they perturb the primary standing wave branches to secondary branches of modulated waves and (3) they produce new steady-state bifurcations along the modulated wave branches. For experiments in which the symmetry-breaking perturbation is externally applied, there are various possibilities for inducing transitions among the various branches of travelling and modulated waves.
Considering first return maps, a most natural renormalisation group fixed point is determined. From it a simple presentation function is constructed, immediately leading to the thermodynamics of critical rotation. The rotation number is encoded in the topological action of the presentation function and the algebraic singularity of criticality in that function's derivatives at its fixed points. Any such presentation function determines a circle map dynamics of that rotation number and index of criticality. These functions are naturally parametrised by a trajectory scaling function. The requirement that the dynamics be smooth leads to a prescription for the calculation of the scaling function and hence the dynamics. The theory is highly constrained and suffers in finite-order approximation from the extra constraint of commutativity, which howevever can be overcome.