Table of contents

We consider endomorphisms of a compact manifold which are expanding except for a finite number of points and prove the existence and uniqueness of a physical measure and its stochastic stability. We also characterize the zero-noise limit measures for a model of the intermittent map and obtain stochastic stability for some values of the parameter. The physical measures are obtained as zero-noise limits which are shown to satisfy Pesin's entropy formula.

We investigate a multidimensional nonisentropic hydrodynamic model for semiconductors, where the energy-conserved equation with nonzero thermal conductivity coefficient is contained. We establish the global existence of smooth solutions for the Cauchy–Neumann problem with small perturbed initial data and Neumann boundary values. We prove that the solutions converge to the stationary solutions of the corresponding drift-diffusion equations; that is, the solutions tend to the stationary solution exponentially fast as t → +∞. Moreover, the existence and uniqueness of the stationary solutions to the corresponding drift-diffusion equations are obtained.

We present a priori estimates of the Boltzmann equation for Maxwellian molecules with an angular cut-off, which lead to the L¹-stability of mild solutions when the initial datum is the small perturbation of a vacuum. For this, we employ Bony-type dispersion estimates and construct an L¹-equivalent nonlinear functional satisfying a quasi-Lyapunov estimate.

Coupled cell systems are systems of ordinary differential equations or ODEs, defined by 'admissible' vector fields, associated with a network whose nodes represent variables and whose edges specify couplings between nodes. It is known that non-isomorphic networks can correspond to the same space of admissible vector fields. Such networks are said to be 'ODE-equivalent'. We prove that two networks are ODE-equivalent if and only if they determine the same space of linear vector fields; moreover, the variable associated with each node may be assumed one-dimensional for that purpose. We briefly discuss the combinatorics of the resulting linear algebra problem.

In this paper we prove that the finite time blow-up of classical solutions of the three-dimensional homogeneous incompressible Euler equations are controlled by the Besov space, , norm of the two components of the vorticity. For the axisymmetric flows with swirl we deduce that the blow-up of solution is controlled by the same Besov space norm of the angular component of the vorticity. For a proof of these results we use the Beale–Kato–Majda criterion, and the special structure of the vortex stretching term in the vorticity formulation of the Euler equations.

The influence of a conserved quantity on an oscillatory pattern-forming instability is examined in one space dimension. Amplitude equations are derived which are not only generic for systems with a pseudoscalar conserved quantity (e.g. rotating convection, magnetoconvection) but also applicable to systems with a scalar conserved quantity. The stability properties of both travelling and standing waves are analysed, with particular progress being possible in the limit of long-wavelength perturbations. For both forms of waves, the corresponding modulational stability boundaries are significantly altered by the presence of the conserved quantity; also, new instabilities are generated. For general perturbations, the full stability regions are found numerically. Simulations of the nonlinear governing equations are performed using a pseudo-spectral code; a variety of stable attractors are thus found of varying degrees of complexity. Previously unseen, highly localized, solutions are observed.

We study synchronization in an array of coupled identical nonlinear dynamical systems where the coupling topology is expressed as a directed graph and give synchronization criteria related to properties of a generalized Laplacian matrix of the directed graph. In particular, we extend recent results by showing that the array synchronizes for sufficiently large cooperative coupling if the underlying graph contains a spanning directed tree. This is an intuitive yet nontrivial result that can be paraphrased as follows: if there exists a dynamical system which influences directly or indirectly all other systems, then synchronization is possible for strong enough coupling. The converse is also true in general.

We study the relation between the Jacobian conjecture and the so-called jolt map representation. We define jolt maps as any map that is symplectic-conjugate to a shear map. In particular, we study a family of homogeneous-symplectic maps and conjecture that all homogeneous-symplectic maps are jolt maps. We prove the result in the plane.

We investigate the semi-classical properties of a two-parameter family of piece-wise linear maps on the torus known as the Casati–Prosen or triangle map. This map is weakly chaotic and has zero Lyapunov exponent. A correspondence between classical and quantum observables is established, leading to an appropriate statement regarding equidistribution of eigenfunctions in the semi-classical limit. We then give a full description of our numerical study of the eigenvalues and eigenvectors of this family of maps. For generic choices of parameters, the spectral and eigenfunction statistics are seen to follow the predictions of the random matrix theory conjecture.

We study in detail complex structures of homoclinic bifurcations in a three-dimensional rate-equation model of a semiconductor laser receiving optically injected light of amplitude K and frequency detuning ω. Specifically, we find and follow in the (K, ω)-plane curves of n-homoclinic bifurcations, where a saddle-focus is connected to itself at the nth return to a neighbourhood of the saddle. We reveal an intricate interplay of codimension-two double-homoclinic and T-point bifurcations. Furthermore, we study how the bifurcation diagram changes with an additional parameter, the so-called linewidth enhancement factor α of the laser. In particular, we find folds (minima) of T-point bifurcation and double-homoclinic bifurcation curves, which are accumulated by infinitely many changes of the bifurcation diagram due to transitions through singularities of surfaces of homoclinic bifurcations.

The injection laser emerges as a system that allows one to study codimension-two bifurcations of n-homoclinic orbits in a concrete vector field. At the same time, the bifurcation diagram in the (K, ω)-plane is of physical relevance. An example is the identification of regions, and their dependence on the parameter α, of multi-pulse excitability where the laser reacts to a single small perturbation by sending out n pulses.

For a chain transitive flow, the averages produced by the invariant probability measures are approximated by multi-valued averages of an inflated flow. The results are applied to singularly perturbed differential equations with slow and fast trajectories. Here, the multi-valued character of the averaged system is due to the presence of many invariant probability measures, often caused by resonances. Hence, the inflation of a chain transitive flow gives information on the relevance of resonances. The Takens equation modelling strong elastic constraints is treated in detail.

The purpose of this work is to give precise estimates for the size of the remainder of the normalized Hamiltonian around a non-semi-simple 1: −1 resonant periodic orbit, as a function of the distance to the orbit.

We consider a periodic orbit of a real analytic three-degrees of freedom Hamiltonian system having a pairwise collision of its non-trivial characteristic multipliers on the unit circle. Under generic hypotheses of non-resonance and non-degeneracy of the collision, we present a constructive methodology to reduce the Hamiltonian around the orbit to its (integrable) normal form, up to any given order. This constructive process allows to obtain quantitative estimates for the size of the remainder of the normal form, as a function of the normalizing order. By selecting appropriately this order in terms of the distance R to the resonant orbit (measured using suitable coordinates), r = r_opt(R) := 2+⌊exp(W(log(1/R^1/(τ+1+ε))))⌋, we have proved that the size of the remainder can be bounded (for small R) by . Here, W(·) stands for Lambert's W function and verifies that W(z)exp(W(z)) = z, τ ⩾ 1 is the exponent of the required Diophantine condition and ε > 0 is any small quantity. The reasons leading to this bound instead of classical exponentially small estimates are also discussed.

We study critical invariant circles of several noble rotation numbers at the edge of break-up for an area-preserving map of the cylinder, which violates the twist condition.

These circles admit essentially unique parametrizations by rotational coordinates. We present a high accuracy computation of about 10⁷ Fourier coefficients. This allows us to compute the regularity of the conjugating maps and to show that, to the extent of numerical precision, it only depends on the tail of the continued fraction expansion.

In higher order model equations such as the Swift–Hohenberg equation and the nonlinear beam equation, different length scales may be distinguished, depending on the parameters in the equation. In this paper, we discuss this phenomenon for stationary solutions of the Swift–Hohenberg equation and show that when the scales are very different a multi-scale analysis can be used to yield asymptotic expressions for multi-bump periodic solutions and the bifurcation diagram of such solutions with prescribed qualitative properties, such as the number of bumps.

We study the limiting behaviour of the Cauchy problem for a class of Carleman-like models in the diffusive scaling with data in the spaces L^p, 1 ⩽ p ⩽ ∞. We show that, in the limit, the solution of such models converges towards the solution of a nonlinear diffusion equation with initial values determined by the data of the hyperbolic system. When the data belong to L¹, a condition of conservation of mass is needed to uniquely identify the solution in some cases, whereas the solution may disappear in the limit in other cases.

In this paper, we demonstrate, through asymptotic expansions, the convergence of a phase field formulation to model surfaces minimizing the mean curvature energy with volume and surface area constraints. Under the assumption of the existence of a smooth limiting surface, it is shown that the interface of a phase field, which is a critical point of the elastic bending energy, converges to a critical point of the surface energy. Further, the elastic bending energy of the phase field converges to the surface energy and the Lagrange multipliers associated with the volume and surface area constraints remain uniformly bounded. This paper is a first step to analytically justify the numerical simulations performed by Du, Liu and Wang in 2004 to model equilibrium configurations of vesicle membranes.

The holomorphic flow of Riemann's xi function is considered. Phase portraits are plotted and the following results, suggested by the portraits, proved: all separatrices tend to the positive and/or negative real axes. There are an infinite number of crossing separatrices. In the region between each pair of crossing separatrices—a band—there is at most one zero on the critical line. All zeros on the critical line are centres or have all elliptic sectors. The flows for ξ(z) and cosh(z) are linked with a differential equation. Simple zeros on the critical line and Gram points never coincide. The Riemann hypothesis is equivalent to all zeros being centres or multiple together with the non-existence of separatrices which enter and leave a band in the same half plane.

In previous work we have shown that the quantum potential can be derived from the classical kinetic equations both for particles with and without spin. Here, we extend these mappings to the relativistic case. The essence of the analysis consists of Fourier transforming the momentum coordinate of the distribution function. This procedure introduces a natural parameter η with units of angular momentum. In the non-relativistic case the ansatz of either separability, or separability and additivity, imposed on the probability distribution function produces mappings onto the Schrödinger equation and the Pauli equation, respectively. The former corresponds to an irrotational flow, the latter to a fluid with non-zero vorticity. In this work we show that the relativistic mappings lead to the Klein–Gordon equation in the irrotational case and to the second-order Dirac equation in the rotational case. These mappings are irreversible; an approximate inverse is the Wigner function. Taken together, these results provide a statistical interpretation of quantum mechanics.

In this paper, we study the Cauchy problem of a two-dimensional model for a moving ferromagnetic continuum and prove global existence and uniqueness of solutions. In addition, equivalence to the coupled system of nonlinear Schrödinger equations interacting with a Chern–Simons gauge field is established.

We consider a class of multi-degree-of-freedom Hamiltonian systems having saddle-centres at which all eigenvalues are purely imaginary except a pair of positive and negative ones, and to which there are homoclinic orbits. In our situation, there exist whiskered invariant tori near the saddle-centres. We develop a Melnikov-type technique for detecting the existence of orbits transversely homoclinic or heteroclinic to the invariant tori. We also show that the systems are nonintegrable in an appropriate meaning and Arnold diffusion type motions occur if such homoclinic or heteroclinic orbits exist. Our theory is applied to systems with potentials and a concrete example is given.

We consider the evolution of ultra-short optical pulses in linear and nonlinear media. For the linear case, we first show that the initial-boundary value problem for Maxwell's equations in which a pulse is injected into a quiescent medium at the left endpoint can be approximated by a linear wave equation which can then be further reduced to the linear short-pulse equation (SPE). A rigorous proof is given that the solution of the SPE stays close to the solutions of the original wave equation over the time scales expected from the multiple scales derivation of the SPE. For the nonlinear case we compare the predictions of the traditional nonlinear Schrödinger equation (NLSE) approximation with those of the SPE. We show that both equations can be derived from Maxwell's equations using the renormalization group method, thus bringing out the contrasting scales. The numerical comparison of both equations with Maxwell's equations shows clearly that as the pulse length shortens, the NLSE approximation becomes steadily less accurate, while the SPE provides a better and better approximation.

For a two-segmental complete chaotic map F: [0, 1] → [0, 1] that preserves an invariant density φ and has a partitioning point at x_c, its opposite map is defined to possess the following four characteristics: (i) has the same metric structure; (ii) preserves an invariant density ; (iii) both F and have the same degree of chaoticity in the sense of identical Lyapunov exponent and (iv) the partitioning point of is at . An approach for constructing opposite maps analytically for all four types of two-segmental complete chaotic maps is provided. Meanwhile, a mutual implication relationship that is invariant with respect to conjugation (metric equivalence) is defined for all two-segmental complete chaotic maps that share an identical invariant measure, an identical Lyapunov exponent and an identical partitioning point. Through this relationship, a unique implied family of chaotic maps is formed so that as long as any member of this family is identified, the rest can be constructed analytically, which makes it possible for all known statistical properties originally established for a particular class of chaotic maps to be generalized to all two-segmental chaotic maps. Numerical simulations conducted are in good agreement with theoretical results.

We consider a Hamiltonian system with slow and fast motions, one degree of freedom corresponding to fast motion, and the other degrees of freedom corresponding to slow motion. Suppose that at frozen values of the slow variables there is a non-degenerate saddle point and a separatrix on the phase plane of the fast variables. In the process of variation of the slow variables, the projection of a phase trajectory onto the phase plane of the fast variables may repeatedly cross the separatrix. These crossings are described by the crossing parameter called the pseudo-phase. We obtain an asymptotic formula for the pseudo-phase dependence on the initial conditions, and calculate the change of the pseudo-phase between two subsequent separatrix crossings.

We show that three families of relative periodic solutions bifurcate out of the Eight solution of the equal-mass three-body problem: the planar Hénon family, the spatial Marchal P₁₂ family and a new spatial family. The Eight, considered as a spatial curve, is invariant under the action of the 24-element group D₆ × Z₂. The three families correspond to symmetry breakings where the invariance group becomes isomorphic to D₆, the three D₆s being embedded in the larger group in different ways. The proof of the existence of these three families relies on writing down the action integral in a rotating frame, viewing the angular velocity of the frame as a parameter, exploiting the invariance of the action under a group action which acts on the angular velocities as well as the curves and, finally, checking numerically the non-degeneracy of the Eight. Pictures and numerical evidence of the three families are presented at the end.