On conjugate points and the Leitmann equivalent problem approach

This article extends the Leitmann equivalence method to a class of problems featuring conjugate points. The class is characterised by the requirement that the set of indifference points of a given problem forms a finite stratification.

Semi-global analysis of periodic and quasi-periodic k:1 and k:2 resonances

The present paper investigates a family of nonlinear oscillators at Hopf bifurcation, driven by a small quasi-periodic forcing. In particular, we are interested in the situation that at bifurcation and for vanishing forcing strength, the driving frequency and the normal frequency are in k:1 or k:2 resonance. For small but nonvanishing forcing strength, a semi-global normal form system is found by averaging and applying a van der Pol transformation. The bifurcation diagram is organised by a codimension 3 singularity of nilpotent-elliptic type. A fairly complete analysis of local bifurcations is given; moreover, all the nonlocal bifurcation curves predicted by Dumortier et al. (1991) are found numerically.

Skiba points for small discount rates

The present article uses perturbation techniques to approximate the value function of an economic minimisation problem for small values of the discount rate. This can be used to obtain the approximate location of Skiba states (or indifference thresholds) in the problem; these are states for which there are two distinct optimal state trajectories, converging to different optimal steady states. It is shown that the sets of indifference thresholds are locally smooth manifolds. For a simple example, all relevant quantities are computed explicitely. Moreover, the approximation can be used to obtain parameter-dependent approximatons to indifference manifolds.

A parametrised version of Moser's modifying terms theorem

A sharpened version of Moser's `modifying terms' KAM theorem is derived, and it is shown how this theorem can be used to investigate the persistence of invariant tori in general situations, including those where some of the Floquet exponents of the invariant torus may vanish. The result is `structural' and works for dissipative, Hamiltonian, reversible and symmetric vector fields. These results are derived for the contexts of real analytic, Gevrey regular, ultradifferentiable and finitely differentiable perturbed vector fields. In the first two cases, the conjugacy constructed in the theorem is shown to be Gevrey smooth in the sense of Whitney on the set of parameters satisfying a "Diophantine" non-resonance condition.

Shallow lake economics run deep: Nonlinear aspects of an economic-ecological interest conflict

Outcomes of the shallow lake interest conflict are presented in a number of different contexts: quasi-static and dynamic social planning, and quasi-static one-shot and repeated non-cooperative play. As the underlying dynamics are non-convex, the analysis uses geometrical-numerical methods: the possible kinds of solutions are efficiently classified in bifurcation diagrams

A weak bifurcation theory for discrete time stochastic dynamical systems

This article presents a bifurcation theory of smooth stochastic dynamical systems that are governed by everywhere positive transition densities. The local dependence structure of the unique strictly stationary evolution of such a system can be expressed by the ratio of joint and marginal probability densities; this `dependence ratio' is a geometric invariant of the system. By introducing a weak equivalence notion of these dependence ratios, we arrive at a bifurcation theory for which in the compact case, the set of stable (non-bifurcating) systems is open and dense. The theory is illustrated with some simple examples.

Phenomenological and ratio bifurcations of a class of discrete time stochastic processes

Zeeman proposed a classification of stochastic dynamical systems based on the Morse classification of their invariant probability densities; the associated bifurcations are the ‘phenomenological bifurcations’ of L. Arnold. The classification is however not invariant under diffeomorphisms of the state space. In a recent paper we proposed an alternative classification, based on an invariant that is a ratio of joint and marginal probability density functions, that does not suffer from this defect. This classification entails the concept of what we call ‘ratio bifurcations’. In this note it is shown that for a large class of dynamical systems, ratio bifurcations and phenomenological bifurcations actually coincide. Moreover, we link the ratio invariant to the transformation invariant function that Wagenmakers et al. obtained for stochastic differential equations. The results are illustrated with numerical applications to stochastic dynamical systems.

Bifurcations of optimal vector fields in the shallow lake system

The shallow lake problem is a nonconvex production-pollution dynamic optimisation problem whose solution structure depends nonlinearly on the system parameters. We perform a bifurcation analysis to investigate the consequences of varying the relative cost of pollution and the discount rate.

Markov-perfect Nash equilibria in models with a single capital stock (Revised version, August 2008)

Many economic problems can be formulated as dynamic games in which strategically interacting agents choose actions that determine the current and future levels of a single capital stock. We study necessary conditions that allow us to characterise Markov perfect Nash equilibria for these games. These conditions result in an auxiliary system of ordinary differential equations that helps us to explore stability, continuity and differentiability of these equilibria. The techniques are used to derive detailed properties of Markov-perfect Nash equilibria for several games including the exploitation of a finite resource, the voluntary investment in a public capital stock, and the inter-temporal consumption of a reproductive asset.