Table of contents

In this paper, we present a new method for obtaining lower bounds of the strict invariance entropy by combining an approach from the theory of escape rates and geometric methods used in the dimension theory of dynamical systems. For uniformly expanding systems and for inhomogeneous bilinear systems we can describe the lower bounds in terms of uniform volume growth rates on subbundles of the tangent bundle. In particular, we obtain criteria for positive entropy. We also apply the estimates to bilinear systems on projective space.

This paper deals with the problem of estimation of the topological entropy for non-autonomous systems of differential equations via the second (direct) Lyapunov method. The main result of the paper is illustrated by examples concerning the Lorenz system and Duffing oscillator.

Given a four-dimensional smooth closed manifold, we construct a diffeomorphism that has a homoclinic class whose continuation locally generically satisfies the following condition: it does not admit any kind of dominated splittings whereas any periodic point belonging to it never has index (the dimension of the unstable manifold) one.

We obtain the exact solutions for a family of spin-boson systems. This is achieved through application of the representation theory for polynomial deformations of the su(2) Lie algebra. We demonstrate that the family of Hamiltonians includes, as special cases, known physical models which are the two-site Bose–Hubbard model, the Lipkin–Meshkov–Glick model, the molecular asymmetric rigid rotor, the Tavis–Cummings model and a two-mode generalization of the Tavis–Cummings model.

We discuss spatial dynamics and collapse scenarios of localized waves governed by the nonlinear Schrödinger equation with nonlocal nonlinearity. Firstly, we prove that for arbitrary nonsingular attractive nonlocal nonlinear interaction in arbitrary dimension collapse does not occur. Then we study in detail the effect of singular nonlocal kernels in arbitrary dimension using both Lyapunoff's method and virial identities. We find that in the one-dimensional case, i.e. for n = 1, collapse cannot happen for nonlocal nonlinearity. On the other hand, for spatial dimension n ⩾ 2 and singular kernel ∼1/r^α, no collapse takes place if α < 2, whereas collapse is possible if α ⩾ 2. Self-similar solutions allow us to find an expression for the critical distance (or time) at which collapse should occur in the particular case of ∼1/r² kernels for n = 3. Moreover, different evolution scenarios for the three-dimensional physically relevant case of Bose–Einstein condensates are studied numerically for both the ground state soliton and higher order toroidal states with, and without, an additional local repulsive nonlinear interaction. In particular, we show that the presence of local repulsive nonlinearity can prevent collapse in those cases.

We present four continuations of the critical nonlinear Schrödinger equation (NLS) beyond the singularity: (1) a sub-threshold power continuation, (2) a shrinking-hole continuation for ring-type solutions, (3) a vanishing nonlinear-damping continuation and (4) a complex Ginzburg–Landau (CGL) continuation. Using asymptotic analysis, we explicitly calculate the limiting solutions beyond the singularity. These calculations show that for generic initial data that lead to a loglog collapse, the sub-threshold power limit is a Bourgain–Wang solution, both before and after the singularity, and the vanishing nonlinear-damping and CGL limits are a loglog solution before the singularity, and have an infinite-velocity expanding core after the singularity. Our results suggest that all NLS continuations share the universal feature that after the singularity time T_c, the phase of the singular core is only determined up to multiplication by e^iθ. As a result, interactions between post-collapse beams (filaments) become chaotic. We also show that when the continuation model leads to a point singularity and preserves the NLS invariance under the transformation t → −t and ψ → ψ^*, the singular core of the weak solution is symmetric with respect to T_c. Therefore, the sub-threshold power and the shrinking-hole continuations are symmetric with respect to T_c, but continuations which are based on perturbations of the NLS equation are generically asymmetric.

We study the transport equation with nonlocal velocity introduced in Córdoba et al (2005 Ann. Math.162 1377–89). We prove its global well-posedness under critical and supercritical dissipation, the last case under the smallness condition, in Besov spaces with critical and subcritical regularity indexes, using the Fourier localization method and modulus of continuity.

A renormalization approach has been used in Eckmann et al (1982) and Eckmann et al (1984) to prove the existence of a universal area-preserving map, a map with hyperbolic orbits of all binary periods. The existence of a horseshoe, with positive Hausdorff dimension, in its domain was demonstrated in Gaidashev and Johnson (2009a). In this paper the coexistence problem is studied, and a computer-aided proof is given that no elliptic islands with period less than 18 exist in the domain. It is also shown that less than 1.5% of the measure of the domain consists of elliptic islands. This is proven by showing that the measure of initial conditions that escape to infinity is at least 98.5% of the measure of the domain, and we conjecture that the escaping set has full measure. This is highly unexpected, since generically it is believed that for conservative systems hyperbolicity and ellipticity coexist.

In this paper we discuss the concept of cosymmetries and co-recursion operators for difference equations and present a co-recursion operator for the Viallet equation. We also discover a new type of factorization for the recursion operators of difference equations. This factorization enables us to give an elegant proof that the pseudo-difference operator presented in Mikhailov et al 2011 Theor. Math. Phys.167 421–43 is a recursion operator for the Viallet equation. Moreover, we show that the operator is Nijenhuis and thus generates infinitely many commuting local symmetries. The recursion operator and its factorization into Hamiltonian and symplectic operators have natural applications to Yamilov's discretization of the Krichever–Novikov equation.

In this paper we prove that gradient-like semigroups (in the sense of Carvalho and Langa (2009 J. Diff. Eqns246 2646–68)) are gradient semigroups (possess a Lyapunov function). This is primarily done to provide conditions under which gradient semigroups, in a general metric space, are stable under perturbation exploiting the known fact (see Carvalho and Langa (2009 J. Diff. Eqns246 2646–68)) that gradient-like semigroups are stable under perturbation. The results presented here were motivated by the work carried out in Conley (1978 Isolated Invariant Sets and the Morse Index (CBMS Regional Conference Series in Mathematics vol 38) (RI: American Mathematical Society Providence)) for groups in compact metric spaces (see also Rybakowski (1987 The Homotopy Index and Partial Differential Equations (Universitext) (Berlin: Springer)) for the Morse decomposition of an invariant set for a semigroup on a compact metric space).