Table of contents

Theory of dynamical systems may be split into two parts. The larger one, dealing with multidimensional systems: flows in dim 3 and higher, diffeomorphisms in dim 2 and higher, may be called the realm of chaos. The smaller one, dealing with planar differential equations, may be called the realm of order. The problems below deal with both parts.

This paper shows how the measurement of the stochasticity degree of a finite sequence of real numbers, published by Kolmogorov in Italian in a journal of insurances' statistics, can be usefully applied to measure the objective stochasticity degree of sequences, originating from dynamical systems theory and from number theory.

Namely, whenever the value of Kolmogorov's stochasticity parameter of a given sequence of numbers is too small (or too big), one may conclude that the conjecture describing this sequence as a sample of independent values of a random variables is highly improbable.

Kolmogorov used this strategy fighting (in a paper in 'Doklady', 1940) against Lysenko, who had tried to disprove the classical genetics' law of Mendel experimentally.

Calculating his stochasticity parameter value for the numbers from Lysenko's experiment reports, Kolmogorov deduced, that, while these numbers were different from the exact fulfilment of Mendel's 3 : 1 law, any smaller deviation would be a manifestation of the report's number falsification.

The calculation of the values of the stochasticity parameter would be useful for many other generators of pseudorandom numbers and for many other chaotically looking statistics, including even the prime numbers distribution (discussed in this paper as an example).

The aim of this review is to introduce the reader to some of the physical notions and the mathematical methods that are relevant to the study of nonlinear waves in Bose–Einstein condensates (BECs). Upon introducing the general framework, we discuss the prototypical models that are relevant to this setting for different dimensions and different potentials confining the atoms. We analyse some of the model properties and explore their typical wave solutions (plane wave solutions, bright, dark, gap solitons as well as vortices). We then offer a collection of mathematical methods that can be used to understand the existence, stability and dynamics of nonlinear waves in such BECs, either directly or starting from different types of limits (e.g. the linear or the nonlinear limit or the discrete limit of the corresponding equation). Finally, we consider some special topics involving more recent developments, and experimental setups in which there is still considerable need for developing mathematical as well as computational tools.

The periodic Lorentz gas describes an ensemble of non-interacting point particles in a periodic array of spherical scatterers. We have recently shown that, in the limit of small scatterer density (Boltzmann–Grad limit), the macroscopic dynamics converges to a stochastic process, whose kinetic transport equation is not the linear Boltzmann equation—in contrast to the Lorentz gas with a disordered scatterer configuration. This paper focuses on the two-dimensional set-up and reports an explicit, elementary formula for the collision kernel of the transport equation.

Let T_n be the nilpotent group of real n × n upper-triangular matrices with 1s on the diagonal. The Hamiltonian flow of a left-invariant Hamiltonian on T^*T_n naturally reduces to the Euler flow on , the dual of . This paper shows that the Euler flows of the standard Riemannian and sub-Riemannian structures of T₄ have transverse homoclinic points on all regular coadjoint orbits. As a corollary, left-invariant Riemannian metrics with positive topological entropy are constructed on all quotients D \ T_n where D is a discrete subgroup of T_n and n ⩾ 4.

It is known that the multifractal spectrum of harmonic measure plays a central role in the geometric function theory. Recently Smirnov and Beliaev introduced a new class of random fractals that are called random conformal snowflakes. They proved that the multifractal spectrum of snowflakes is related to the spectral radius of a particular integral operator. In this paper we exploit this connection to estimate the multifractal spectrum of snowflakes.

In this work, we focus on the multifractal centred Hausdorff measure and the multifractal packing measure in . We find that for 0 < s < t < 2, q < 1, then there exist a set and a probability measure μ on such that and, in addition,

Symmetries and Hamiltonian structure are combined with Melnikov's method to show a set of exact solutions to the 2D semi-geostrophic equations in an elliptical tank responding chaotically to gentle periodic forcing of the domain eccentricity (or of the potential vorticity, for that matter) which are sinusoidal in time with nearly any period. A similar approach confirms the chaotic response of the quasi-geostrophic equations to gentle periodic forcing by an external shearing field. Our approach simplifies and strengthens the proof by Bertozzi (upon which it is based) concerning the chaotic response of Kirchoff elliptical vortex patches to gentle shearing in the 2D Euler equations.

Understanding of spatial and temporal behaviour of interacting species or reactants in ecological or chemical systems has become a central issue, and rigorously determining the formation of patterns in models from various mechanisms is of particular interest to applied mathematicians. In this paper, we study a bimolecular autocatalytic reaction–diffusion model with saturation law and are mainly concerned with the corresponding steady-state problem subject to the homogeneous Neumann boundary condition. In particular, we derive some results for the existence and non-existence of non-constant stationary solutions when the diffusion rate of a certain reactant is large or small. The existence of non-constant stationary solutions implies the possibility of pattern formation in this system. Our theoretical analysis shows that the diffusion rate of this reactant and the size of the reactor play decisive roles in leading to the formation of stationary patterns.

The initial- and boundary-value problem for the Kawahara equation, a fifth-order KdV type equation, is studied in weighted Sobolev spaces. This functional framework is based on the dual-Petrov–Galerkin algorithm, a numerical method proposed by Shen (2003 SIAM J. Numer. Anal.41 1595–619) to solve third and higher odd-order partial differential equations. The theory presented here includes the existence and uniqueness of a local mild solution and of a global strong solution in these weighted spaces. If the L²-norm of the initial data is sufficiently small, these solutions decay exponentially in time. Numerical computations are performed to complement the theory.

We present a class of systems of ordinary differential equations (ODEs), which we call 'pod systems', that offers a new perspective on dynamical systems defined on a spatial domain. Such systems are typically studied as partial differential equations, but pod systems bring the analytic techniques of ODE theory to bear on the problems, and are thus able to study behaviours and bifurcations that are not easily accessible to the standard methods. In particular, pod systems are specifically designed to study spatial dynamical systems that exhibit multi-modal solutions.

A pod system is essentially a linear combination of parametrized functions in which the coefficients and parameters are variables whose dynamics are specified by a system of ODEs. That is, pod systems are concerned with the dynamics of functions of the form Ψ(s, t) = y₁(t) ϕ(s; x₁(t)) + ··· + y_N(t) ϕ(s; x_N(t)), where s ∊ Rⁿ is the spatial variable and ϕ: Rⁿ × R^d → R. The parameters x_i ∊ R^d and coefficients y_i ∊ R are dynamic variables which evolve according to some system of ODEs, and , for i = 1, ..., N. The dynamics of Ψ in function space can then be studied through the dynamics of the x and y in finite dimensions.

A vital feature of pod systems is that the ODEs that specify the dynamics of the x and y variables are not arbitrary; restrictions on G_i and H_i are required to guarantee that the dynamics of Ψ in function space are well defined (that is, that trajectories are unique). One important restriction is symmetry in the ODEs which arises because Ψ is invariant under permutations of the indices of the (x_i, y_i) pairs. However, this is not the whole story, and the primary goal of this paper is to determine the necessary structure of the ODEs explicitly to guarantee that the dynamics of Ψ are well defined.

The existence of inertial manifolds for a Smoluchowski equation—a nonlinear and nonlocal Fokker–Planck equation which arises in the modelling of colloidal suspensions—is investigated. The difficulty due to first-order derivatives in the nonlinearity is circumvented by a transformation.

We show that generic Hopf–Neĭmark–Sacker bifurcations occur in the dynamics of a large class of feed-forward coupled cell networks. To this end we present a framework for studying such bifurcations in parametrized families of perturbed forced oscillators near weak resonance points. Our approach is based on fine-tuning existing normal form techniques. We then apply this framework to show that certain cells in the feed-forward networks exhibit this forced oscillator dynamics and, hence, undergo generic Hopf–Neĭmark–Sacker bifurcations. These bifurcations correspond to the occurrence of resonance tongues in parameter space with a 'standard' geometry.

For spiking neural networks we consider the stability problem of global synchrony, arguably the simplest non-trivial collective dynamics in such networks. We find that even this simplest dynamical problem—local stability of synchrony—is non-trivial to solve and requires novel methods for its solution. In particular, the discrete mode of pulsed communication together with the complicated connectivity of neural interaction networks requires a non-standard approach. The dynamics in the vicinity of the synchronous state is determined by a multitude of linear operators, in contrast to a single stability matrix in conventional linear stability theory. This unusual property qualitatively depends on network topology and may be neglected for globally coupled homogeneous networks. For generic networks, however, the number of operators increases exponentially with the size of the network.

We present methods to treat this multi-operator problem exactly. First, based on the Gershgorin and Perron–Frobenius theorems, we derive bounds on the eigenvalues that provide important information about the synchronization process but are not sufficient to establish the asymptotic stability or instability of the synchronous state. We then present a complete analysis of asymptotic stability for topologically strongly connected networks using simple graph-theoretical considerations.

For inhibitory interactions between dissipative (leaky) oscillatory neurons the synchronous state is stable, independent of the parameters and the network connectivity. These results indicate that pulse-like interactions play a profound role in network dynamical systems, and in particular in the dynamics of biological synchronization, unless the coupling is homogeneous and all-to-all. The concepts introduced here are expected to also facilitate the exact analysis of more complicated dynamical network states, for instance the irregular balanced activity in cortical neural networks.

In this paper notions of sensitive sets (S-sets) and regionally proximal sets (Q-sets) are introduced. It is shown that a transitive system is sensitive if and only if there is an S-set with Card(S) ⩾ 2, and for a transitive system each S-set is a Q-set. Moreover, the converse holds when (X, T) is minimal. It turns out that each transitive (X, T) has a maximal almost equicontinuous factor.

According to the cardinalities of the S-sets, transitive systems are divided into several classes. Characterizations and examples are given for this classification both in minimal and transitive non-minimal settings. It is proved that for a transitive system any entropy set is an S-set, and consequently, a transitive system which has no uncountable S-sets has zero topological entropy. Moreover, it is shown that a transitive, non-minimal system with dense set of minimal points has an infinite S-set, and there exists a Devaney chaotic system which has no uncountable S-set. Finally, a non-minimal sensitive E-system is constructed such that each of its S-set has cardinality at most 4.

Let Ω be a simply connected, bounded, smooth domain of or . We consider the equation of steady motion of a third grade fluid in Ω with homogeneous Dirichlet boundary conditions. We prove that the monotonicity technique used by Paicu (2008 Physica D at press) in the full space for unsteady motion allows us to obtain the existence of a solution provided that the forcing belongs to . The size of the forcing is arbitrary.

We prove that there exists an open and dense subset of the incompressible 3-flows of class C² such that, if a flow in this set has a positive volume regular invariant subset with dominated splitting for the linear Poincaré flow, then it must be an Anosov flow. With this result we are able to extend the dichotomies of Bochi–Mañé (see Bessa 2007 Ergod. Theory Dyn. Syst.27 1445–72, Bochi 2002 Ergod. Theory Dyn. Syst.22 1667–96, Mañé 1996 Int. Conf. on Dynamical Systems (Montevideo, Uruguay, 1995) (Harlow: Longman) pp 110–9) and of Newhouse (see Newhouse 1977 Am. J. Math.99 1061–87, Bessa and Duarte 2007 Dyn. Syst. Int. J. submitted Preprint0709.0700) for flows with singularities. That is, we obtain for a residual subset of the C¹ incompressible flows on 3-manifolds that: (i) either all Lyapunov exponents are zero or the flow is Anosov and (ii) either the flow is Anosov or else the elliptic periodic points are dense in the manifold.