Table of contents

The nonlinear dynamics of biochemical reactions in a small-sized system on the order of a cell are stochastic. Assuming spatial homogeneity, the populations of n molecular species follow a multi-dimensional birth-and-death process on . We introduce the Delbrück–Gillespie process, a continuous-time Markov jump process, whose Kolmogorov forward equation has been known as the chemical master equation, and whose stochastic trajectories can be computed via the Gillespie algorithm. Using simple models, we illustrate that a system of nonlinear ordinary differential equations on emerges in the infinite system size limit. For finite system size, transitions among multiple attractors of the nonlinear dynamical system are rare events with exponentially long transit times. There is a separation of time scales between the deterministic ODEs and the stochastic Markov jumps between attractors. No diffusion process can provide a global representation that is accurate on both short and long time scales for the nonlinear, stochastic population dynamics. On the short time scale and near deterministic stable fixed points, Ornstein–Uhlenbeck Gaussian processes give linear stochastic dynamics that exhibit time-irreversible circular motion for open, driven chemical systems. Extending this individual stochastic behaviour-based nonlinear population theory of molecular species to other biological systems is discussed.

Recently, there has been a wide interest in the study of aggregation equations and Patlak–Keller–Segel (PKS) models for chemotaxis with degenerate diffusion. The focus of this paper is the unification and generalization of the well-posedness theory of these models. We prove local well-posedness on bounded domains for dimensions d ⩾ 2 and in all of space for d ⩾ 3, the uniqueness being a result previously not known for PKS with degenerate diffusion. We generalize the notion of criticality for PKS and show that subcritical problems are globally well-posed. For a fairly general class of problems, we prove the existence of a critical mass which sharply divides the possibility of finite time blow-up and global existence. Moreover, we compute the critical mass for fully general problems and show that solutions with smaller mass exists globally. For a class of supercritical problems we prove finite time blow-up is possible for initial data of arbitrary mass.

In this paper we introduce Hamiltonian dynamics, inspired by zero-sum games (best response and fictitious play dynamics). The Hamiltonian functions we consider are continuous and piecewise affine (and of a very simple form). It follows that the corresponding Hamiltonian vector fields are discontinuous and multi-valued. Differential equations with discontinuities along a hyperplane are often called 'Filippov systems', and there is a large literature on such systems, see for example (di Bernardo et al 2008 Theory and applications Piecewise-Smooth Dynamical Systems (Applied Mathematical Sciences vol 163) (London: Springer); Kunze 2000 Non-Smooth Dynamical Systems (Lecture Notes in Mathematics vol 1744) (Berlin: Springer); Leine and Nijmeijer 2004 Dynamics and Bifurcations of Non-smooth Mechanical Systems (Lecture Notes in Applied and Computational Mechanics vol 18) (Berlin: Springer)). The special feature of the systems we consider here is that they have discontinuities along a large number of intersecting hyperplanes. Nevertheless, somewhat surprisingly, the flow corresponding to such a vector field exists, is unique and continuous. We believe that these vector fields deserve attention, because it turns out that the resulting dynamics are rather different from those found in more classically defined Hamiltonian dynamics. The vector field is extremely simple: outside codimension-one hyperplanes it is piecewise constant and so the flow ϕ_t piecewise a translation (without stationary points). Even so, the dynamics can be rather rich and complicated as a detailed study of specific examples show (see for example theorems 7.1 and 7.2 and also (Ostrovski and van Strien 2011 Regular Chaotic Dynf.16 129–54)). In the last two sections of the paper we give some applications to game theory, and finish with posing a version of the Palis conjecture in the context of the class of non-smooth systems studied in this paper.

For a bounded domain whose boundary contains a number of flat pieces Γ_i, i = 1, ..., l we consider a family of non-symmetric billiards Ω constructed by patching several copies of Ω⁰ along Γ_is. It is demonstrated that the length spectrum of the periodic orbits in Ω is degenerate with the multiplicities determined by a matrix group G. We study the energy spectrum of the corresponding quantum billiard problem in Ω and show that it can be split into a number of uncorrelated subspectra corresponding to a set of irreducible representations α of G. Assuming that the classical dynamics in Ω⁰ are chaotic, we derive a semiclassical trace formula for each spectral component and show that their energy level statistics are the same as in standard random matrix ensembles. Depending on whether α is real, pseudo-real or complex, the spectrum has either Gaussian orthogonal, Gaussian symplectic or Gaussian unitary types of statistics, respectively.

We establish the existence of travelling wave solutions for delayed cooperative recursions that are allowed to have more than two equilibria. We define an important extended real number that is used to determine the speeds of travelling wave solutions. The results can be applied to a large class of delayed cooperative reaction–diffusion models. We show that for a delayed Lotka–Volterra reaction–diffusion competition model, there exists a finite positive number that can be characterized as the slowest speed of travelling wave solutions connecting two mono-culture equilibria or connecting a mono-culture with the coexistence equilibrium.

In this paper we study the differentiability of Mather's average action, or β-function, on closed surfaces, with particular attention to its relation to the integrability of the system.

In this paper, we study the structure of radially symmetric standing waves for the nonlinear Schrödinger equation with harmonic potential, which arises in a wide variety of applications and is known as the Gross–Pitaevskii equation in the context of Bose–Einstein condensates with parabolic traps. Both global and local bifurcation behaviour are determined showing the existence of infinitely symmetric localized states. In particular, our theory provides a theoretical proof of the existence of a solution with prescribed numbers of zeros depending on the frequency of the wave. After a few remarks concerning the critical case, numerical computations are finally presented in order to provide an illustration of the theoretical results that have been obtained and also to investigate the supercritical case for which only few results are known.

We show that the Novikov–Veselov equation (an analogue of KdV in dimension 2 + 1) does not have exponentially localized solitons at negative energy.

Dynamical systems that describe the escape from the basins of attraction of stable invariant sets are presented and analysed. It is shown that the stable fixed points of such dynamical systems are the index-1 saddle points. Generalizations to high index saddle points are discussed. Both gradient and non-gradient systems are considered. Preliminary results on the nature of the dynamical behaviour are presented.

We use asymptotic analysis and numerical simulations to study peak-type singular solutions of the supercritical biharmonic nonlinear Schrödinger equation. These solutions have a quartic-root blowup rate, and collapse with a quasi-self-similar universal profile, which is a zero-Hamiltonian solution of a fourth-order nonlinear eigenvalue problem.

We investigate a steady flow of incompressible fluid in the plane. The motion is governed by the Navier–Stokes equations with prescribed velocity u_∞ at infinity. The main result shows the existence of unique solutions for arbitrary force, provided sufficient largeness of u_∞. Furthermore a spatial structure of the solution is obtained in comparison with the Oseen flow. A key element of our new approach is based on a setting which treats the direction of the flow as the time direction. The analysis is done in the framework of the Fourier transform taken in one (perpendicular) direction and a special choice of function spaces which take into account the inhomogeneous character of the symbol of the Oseen system. From that point of view our technique can be used as an effective tool in examining spatial asymptotics of solutions to other systems modelled by elliptic equations.

We consider the exit problem for small white noise perturbation of a smooth dynamical system on the plane in the neighbourhood of a hyperbolic critical point. We show that if the distribution of the initial condition has a scaling limit then the exit distribution and exit time also have a joint scaling limit as the noise intensity goes to zero. The limiting law is computed explicitly. The result completes the theory of noisy heteroclinic networks in two dimensions. The analysis is based on normal forms theory.