Table of contents

We give a simple necessary and sufficient condition for the dynamical equivalence of two coupled cell networks. The results are applicable to both continuous and discrete dynamical systems and are framed in terms of what we term input and output equivalence. We also give an algorithm that allows explicit construction of the cells in a system with a given network architecture in terms of the cells from an equivalent system with different network architecture. Details of proofs are provided for the case of cells with asymmetric inputs—details for the case of symmetric inputs are provided in a companion paper.

We show that two networks of coupled dynamical systems are dynamically equivalent if and only if they are output equivalent. We also obtain necessary and sufficient conditions for two dynamically equivalent networks to be input equivalent. These results were previously described in the companion paper 'Dynamical equivalence of networks of coupled dynamical systems: I. Asymmetric inputs' but only proved there for the case of asymmetric inputs. In this paper, we allow for symmetric inputs. We also provide a number of examples to illustrate the main results in the case when there are both symmetric and asymmetric inputs.

A Lorenz map is a Poincaré map for a three-dimensional Lorenz flow. We describe the theory of renormalization for Lorenz maps with a critical point and prove that a restriction of the renormalization operator acting on such maps has a hyperbolic fixed point. The proof is computer assisted and we include a detailed exposition on how to make rigorous estimates using a computer as well as the implementation of the estimates.

Phase reduction is a commonly used techinque for analysing stable oscillators, particularly in studies concerning synchronization and phase lock of a network of oscillators. In a widely used numerical approach for obtaining phase reduction of a single oscillator, one needs to obtain the gradient of the phase function, which essentially provides a linear approximation of the local isochrons. In this paper, we extend the method for obtaining partial derivatives of the phase function to arbitrary order, providing higher order approximations of the local isochrons. In particular, our method in order 2 can be applied to the study of dynamics of a stable oscillator subjected to stochastic perturbations, a topic that will be discussed in a future paper. We use the Stuart–Landau oscillator to illustrate the method in order 2.

The parabolic resonance instability emerges in diverse applications ranging from optical systems to simple mechanical ones. It appears persistently in p-parameter families of near-integrable Hamiltonian systems with n degrees of freedom provided n + p ⩾ 3. Here we study the simplest (n = 2, p = 1) symmetric case. The structure and the phase-space volume of the corresponding instability zones are characterized. It is shown that the symmetric case has six distinct non-degenerate normal forms, and two degenerate ones. In the regular cases, the instability zone has the usual extent in the action direction. However, the phase-space volume of this zone is found to be polynomial in the perturbation parameter ε (and not exponentially small as in the elliptic resonance case). Finally, the extent of the instability zone in some of the degenerate cases is explored. Three applications in which the symmetric parabolic resonance arises are presented and analysed.

We study a class of quasi-linear Schrödinger equations arising in the theory of superfluid film in plasma physics. Using gauge transforms and a derivation process we solve, under some regularity assumptions, the Cauchy problem. Then, by means of variational methods, we study the existence, the orbital stability and instability of standing waves which minimize some associated energy.

We deal with a two-component system of reaction–diffusion equations with conservation of mass in a bounded domain with the Neumann or periodic boundary conditions. This system is proposed as a conceptual model for cell polarity. Since the system has conservation of mass, the steady state problem is reduced to that of a scalar reaction–diffusion equation with a nonlocal term. That is, there is a one-to-one correspondence between an equilibrium solution of the system with a fixed mass and a solution of the scalar equation. In particular, we consider the case when the reaction term is linear in one variable. Then the equations are transformed into the same equations as the phase-field model for solidification. We thereby show that the equations allow a Lyapunov function. Moreover, by investigating the linearized stability of a nonconstant equilibrium solution, we prove that given a nondegenerate stable equilibrium solution of the nonlocal scalar equation, the corresponding equilibrium solution of the system is stable. We also exhibit global bifurcation diagrams for equilibrium solutions to specific model equations by numerics together with a normal form near a bifurcation point.

A delayed reaction–diffusion model of the Fisher type with a single discrete delay and zero-Dirichlet boundary conditions on a general bounded open spatial domain with a smooth boundary is considered. The stability of a spatially heterogeneous positive steady state solution and the existence of Hopf bifurcation about this positive steady state solution are investigated. In particular, by using the normal form theory and the centre manifold reduction for partial functional differential equations, the stability of bifurcating periodic solutions occurring through Hopf bifurcations is investigated. It is demonstrated that the bifurcating periodic solution occurring at the first bifurcation point is orbitally asymptotically stable while those occurring at the other bifurcation points are unstable.

We consider a model of thermal explosion in porous media which is a natural generalization of the well-known problem of self-ignition introduced by (Gelfand 1963 Am. Math. Soc. Trans.29 295–381). We rigorously prove that, similar to the Gelfand–Barenblatt problem, the thermal explosion (finite time blow-up of all solutions for the problem with non-negative initial data) occurs exclusively due to the absence of a weak solution of the corresponding stationary problem.

We consider the random wavelet series built from Gibbs measures, and study the Hausdorff dimension of the graph and range of these functions restricted to their iso-Hölder sets. To obtain the Hausdorff dimension of these sets, we apply the potential theoretic method to families of Gibbs measures defined on a sequence of topologically transitive subshift of finite type whose Hausdorff distance to the set of zeros of the mother wavelet tends to 0.

We classify the dynamics of (orientation-preserving) flat spot maps on the circle, and derive explicit expressions for the function counting the first entrance time into the flat spot. Metric properties of first entrance time functions for the standard flat spot family are analysed in detail, via a computation of conditional expectation with respect to the orbit partition. This facilitates investigation of the median of the entrance time function, proving the surprising result that its first entrance time is constrained to equal either 1, 2, 4, 5 or 12, provided the rotation number of the flat spot map does not equal the exceptional values 0, ±2/7, ±3/10, ±1/3, ±3/8.