Suppose that a dynamical system possesses an invariant submanifold, and the restriction of the system to this submanifold has a chaotic attractor A. Under which conditions is A an attractor for the original system, and in what sense? We characterize the transverse dynamics near A in terms of the normal Liapunov spectrum of A. In particular, we emphasize the role of invariant measures on A. Our results identify the points at which A: (1) ceases to be asymptotically stable, possibly developing a locally riddled basin; (2) ceases to be an attractor; (3) becomes a transversely repelling chaotic saddle. We show, in the context of what we call 'normal parameters' how these transitions can be viewed as being robust. Finally, we discuss some numerical examples displaying these transitions.
Breather solutions are time-periodic and space-localized solutions of nonlinear dynamical systems. We show that the concept of anticontinuous limit, which was used before for proving an existence theorem on breathers and multibreather solutions in arrays of coupled nonlinear oscillators, can be used constructively as a high-precision numerical method for finding these solutions. The method is based on the continuation of breather solutions which trivially exist at the anticontinuous limit. It is quite universal and applicable to a wide class of nonlinear models which can be of arbitrary dimension, periodic or random, with or without a driving force plus damping, etc. The main advantage of our method compared with other available methods is that we can distinguish unambiguously the different breather (or multibreather) solutions by their coding sequence. Another advantage is that we can obtain the corresponding solutions whether they are linearly stable or not. These solutions can be calculated in their full domain of existence. We illustrate the techniques with examples of breather calculations in several models. We mostly consider arrays of coupled anharmonic oscillators in one dimension, but we also test the method in two dimensions. Our method allows us to show that the breather solution can be continued while its frequency enters the phonon band (it then superposes to a band edge phonon with a finite amplitude). We also test that our method works when introducing an extra time-periodic driving force plus damping. Our method is applied for the calculation of breathers in so-called Fermi-Pasta-Ulam (FPU) chains, that is, one-dimensional chains of atoms with anharmonic nearest-neighbour coupling without on-site potential. The breather and multibreather solutions are then obtained by continuation from the anticontinuous limit of an extended model containing an extra parameter. Finally, we show that we can also calculate 'rotobreathers' in arrays of coupled rotators, which correspond to solutions with one or several rotators rotating while the remaining rotators are only oscillating. The linear stability analysis of the obtained time periodic solutions (Floquet analysis) of all these models will be done in a forthcoming paper.
It is proved here that minimizing measures of a Lagrangian Row are invariant and the Lagrangian is cohomologous to a constant on the support of their ergodic components. Moreover, it is shown that generic Lagrangians have a unique minimizing measure which is uniquely ergodic and is a limit of invariant probabilities supported on periodic orbits of the Lagrangian Rows.
We present two new criteria for studying the nonexistence, existence and uniqueness of limit cycles of planar vector fields. We apply these criteria to some families of quadratic and cubic polynomial vector fields, and to compute an explicit formula for the number of limit cycles which bifurcate out of the linear centre x = -y, y, = x, when we deal with the system x = -y + epsilon Sigma(i+j=1)(n) a(ij)x(i)y(j), y = x + epsilon Sigma(i+j=1)(n) b(ij)x(i)y(j). Moreover, by using the second criterion we present a method to derive the shape of the bifurcated limit cycles from a centre.
We present a straightforward and reliable continuous method for computing the full or partial Lyapunov spectrum associated with a dynamical system specified by a set of differential-equations. We do this by introducing a stability parameter beta > 0 and augmenting the dynamical system with an orthonormal k-dimensional frame and a Lyapunov vector such that the frame is continuously Gram-Schmidt orthonormalized and at most linear growth of the dynamical variables is involved. We prove that the method is strongly stable when beta > -lambda(k) where lambda(k) is the kth Lyapunov exponent in descending order and we show through examples how the method is implemented. It extends many previous results.
We prove existence and practical stability of breathers in chains of weakly coupled anharmonic oscillators. Precisely, for a large class of chains, we prove that there exist periodic solutions exponentially localized in space, with the property that, given an initial datum O(epsilon(a)) (with a greater than or equal to 1/2) close to the phase space trajectory of the breather, then the corresponding solution remains at a distance O(epsilon(a) + root/t/exp(-epsilon(-1/6))) from the above trajectory, up to times growing exponentially with the inverse of epsilon, epsilon being a parameter measuring the size of the interaction among the particles. This result is deduced from a general normal form theorem for abstract Hamiltonian systems in Banach spaces, which we think could be interesting in itself.
Montgomery has conjectured that the non-trivial zeros of the Riemann zeta-function are pairwise distributed like the eigenvalues of matrices in the Gaussian unitary ensemble (GUE) of random matrix theory (RMT). In this respect, they provide an important model for the statistical properties of the energy levels of quantum systems whose classical limits are strongly chaotic. We generalize this connection by showing that for all n greater than or equal to 2 the n-point correlation function of the zeros is equivalent to the corresponding GUE result in the appropriate asymptotic limit. Our approach is based on previous demonstrations for the particular cases n = 2, 3, 4. It relies on several new combinatorial techniques, first for evaluating the multiple prime sums involved using a Hardy-Littlewood prime-correlation conjecture, and second for expanding the GUE correlation-function determinant. This constitutes the first complete demonstration of RMT behaviour for all orders of correlation in a simple, deterministic model.
We consider a forced harmonic oscillator at resonance with a nonlinear perturbation and obtain a sharp condition for the existence of unbounded motions. Such a condition is extended to the case of a semilinear vibrating string.
In this paper we prove a theorem on the uniqueness of limit cycles surrounding one or more singularities for Lienard equations. By using this theorem we give a positive answer to the conjecture in Dumortier and Rousseau (1990 Nonlinearity 3 1015-39), completing the classification of the cubic Lienard equations with linear damping. It also finishes the study of the generic three-parameter unfoldings of the nilpotent focus in the plane.
In this paper and its sequel we study arrays of coupled identical cells that possess a 'global' symmetry group g, and in which the cells possess their own 'internal' symmetry group L. We focus on general existence conditions for symmetry-breaking steady-state and Hopf bifurcations. The global and internal symmetries can combine in two quite different ways, depending on how the internal symmetries affect the coupling. Algebraically, the symmetries either combine to give the wreath product L (sic) g of the two groups or the direct product L x g. Here we develop a theory for the wreath product: we analyse the direct product case in the accompanying paper (henceforth referred to as II). The wreath product case occurs when the coupling is invariant under internal symmetries. The main objective of the paper is to relate the patterns of steady-state and Hopf bifurcation that occur in systems with the combined symmetry group L (sic) g to the corresponding bifurcations in systems with symmetry L or g. This organizes the problem by reducing it to simpler questions whose answers can often be read off from known results. The basic existence theorem for steady-state bifurcation is the equivariant branching lemma, which states that under appropriate conditions there will be a symmetry-breaking branch of steady states for any isotropy subgroup with a one-dimensional fixed-point subspace, We call such an isotropy subgroup axial. The analogous result for equivariant Hopf bifurcation involves isotropy subgroups with a two-dimensional fixed-point subspace, which we call C-axial because of an analogy involving a natural complex structure. Our main results are classification theorems for axial and C-axial subgroups in wreath products. We study some typical examples, rings of cells in which the internal symmetry group is O(2) and the global symmetry group is dihedral. As these examples illustrate, one striking consequence of our general results is that systems with wreath product coupling often have states in which some cells are performing nontrivial dynamics, while others remain quiescent. We also discuss the common occurrence of heteroclinic cycles in wreath product systems.
A possibility that in the FPU problem the critical energy for chaos goes to zero when the number of particles in the chain increases is discussed. The distribution for long linear waves in this regime is found and an estimate for the new border of the transition to energy equipartition is given.
We present the first purely semiclassical calculation of the resonance spectrum in the diamagnetic Kepler problem (DKP), a hydrogen atom in a constant magnetic field with L(z) = 0. The classical system is unbound and completely chaotic for a scaled energy epsilon similar to EB(-2/3) larger than a critical value epsilon(c) > 0. The quantum mechanical resonances can in semiclassical approximation be expressed as the zeros of the semiclassical zeta function, a product over all the periodic orbits of the underlying classical dynamics. Intermittency originating from the asymptotically separable limit of the potential at large electron-nucleus distance causes divergences in the periodic orbit formula. Using a regularization technique introduced in (Tanner G and Wintgen D 1995 Phys. Rev. Dtt. 75 2928) together with a modified cycle expansion, we calculate semiclassical resonances, both position and width, which are in good agreement with quantum mechanical results obtained by the method of complex rotation. The method also provides good estimates for the bound state spectrum obtained here from the classical dynamics of a scattering system. A quasi-Einstein-Brillouin-Keller (QEBK) quantization is derived that allows for a description of the spectrum in terms of approximate quantum numbers and yields the correct asymptotic behaviour of the Rydberg-like series converging towards the different Landau thresholds.
A general procedure to construct a generating partition in 2D symplectic maps is introduced. The implementation of the method, specifically discussed with reference to the standard map, can be easily extended to any model where chaos originates from a horseshoe-type mechanism. Symmetries arising from the symplectic structure of the dynamics are exploited to eliminate the remaining ambiguities of the encoding procedure, so that the resulting symbolic dynamics possesses the same symmetry as that of the original model. Moreover, the dividing line of the partition turns out to pass through the stability islands, in such a way as to yield a proper representation of the quasiperiodic dynamics as well as of the chaotic component. As a final confirmation of the correctness of our approach, we construct the associated pruning front and show that it is monotonous.
We study the effect of edge diffraction on the semiclassical analysis of two-dimensional quantum systems by deriving a trace formula which incorporates paths hitting any number of vertices embedded in an arbitrary potential. This formula is used to study the cardioid billiard, which has a single vertex. The formula works well for most of the short orbits we analysed but fails for a few diffractive orbits due to a breakdown-in the formalism for certain geometries. We extend the symbolic dynamics to account for diffractive orbits and use it to show that in the presence of parity symmetry the trace formula decomposes in an elegant manner such that for the cardioid billiard the diffractive orbits have no effect on the odd spectrum. Including diffractive orbits helps resolve peaks in the density of even states but does not appear to affect their positions. An analysis of the level statistics shows no significant difference between spectra with and without diffraction.
We study a system of two conservation laws which is strictly hyperbolic, but not genuinely nonlinear. We solve the Riemann problem for the inviscid system in a unique way and find explicit travelling wave solutions for the viscous system. It is established that the solutions must be searched for in the space of bounded Radon measures.
We continue the study of arrays of coupled identical cells that possess both global and internal symmetries, begun in part I. Here we concentrate on the 'direct product' case, for which the symmetry group of the system decomposes as the direct product L x g of the internal group L and the global group g. Again, the main aim is to find general existence conditions for symmetry-breaking steady-state and Hopf bifurcations by reducing the problem to known results for systems with symmetry L or g separately. Unlike the wreath product case, the theory makes extensive use of the representation theory of compact Lie groups. Again the central algebraic task is to classify axial and C-axial subgroups of the direct product and to relate them to axial and C-axial subgroups of the two groups L and g. We demonstrate how the results lead to efficient classification by studying both steady state and Hopf bifurcation in rings of coupled cells, where L = O(2) and g = D-n. In particular we show that for Hopf bifurcation the case n = 4 module 4 is exceptional, by exhibiting two extra types of solution that occur only for those values of n.
The Poincare-Melnikov-Arnold method for planar maps gives rise to a Melnikov function defined by an infinite and (a priori) analytically uncomputable sum. Under an assumption of meromorphicity, residues theory can be applied to provide an equivalent finite sum. Moreover, the Melnikov function turns out to be an elliptic function and a general criterion about non-integrability is provided. Several examples are presented with explicit estimates of the splitting angle. In particular, the non-integrability of non-trivial symmetric entire perturbations of elliptic billiards is proved, as well as the non-integrability of standard-like maps.
We investigate the bulk scaling of the convective heat transport in the Boussinesq equations. An a priori bound on the scaling exponent is obtained without making any assumptions on the turbulent fluctuations. If horizontal gradients of temperature fluctuations are relatively small near the boundary then different scaling regimes are obtained.