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We introduce a framework to systematically investigate the resonant double Hopf bifurcation. We use the basic invariants of the ensuing T 1 -action to analyse the approximating normal form truncations in a unified manner. In this way we obtain a global description of the parameter space and thus find the organising resonance droplet, which is the present analogue of the resonant gap. The dynamics of the normal form yields a skeleton for the dynamics of the original system. In the ensuing perturbation theory both normal hyperbolicity (centre manifold theory) and kam theory are being used.

A system that is at a steady state responds most of the time gradually to external changes. But in exceptional circumstances, it can exhibit a sudden ‘catastrophic’ shift to a different regime. It is of great practical interest to develop early warning indicators that signal the imminence of such a shift. A promising class of such indicators uses the universal fact that the average return time to a stable steady state after a small disturbance increases sharply close to a catastrophic shift. It is however important to realize that there are classes of dynamic regime shifts that cannot be predicted in this way. After reviewing the mathematical ideas behind these indicators, this article discusses their scope and their limitations.

Economic consequences of critical transitions or regime shifts of ecological systems are a source of considerable concern. Though the underlying mechanisms are well-known, only since the advent of electronic computers the analysis of economic models exhibiting critical transitions can be attacked systematic- ally. In particular bifurcation theory allows to present the qualitative effects of parameter changes in a convenient way. This article gives an overview of recent economic literature dealing with potential environmental regime shifts with a focus on the lake model

A system that is at a steady state responds most of the time gradually to external changes. But in exceptional circumstances, it can exhibit a sudden 'catastrophic' shift to a different regime. It is of great practical interest to develop early warning indicators that signal the imminence of such a shift. A promising class of such indicators uses the universal fact that the average return time to a stable steady state after a small disturbance increases sharply close to a catastrophic shift. It is however important to realise that there are classes of dynamic regime shifts that cannot be predicted in this way. After reviewing the mathematical ideas behind these indicators, this article discusses their scope and their limitations.

Financiële instrumenten maken de spreiding van financiële risico’s mogelijk, en daardoor welvaartswinsten. Op markten met begrensd rationele agenten schuilt in financiële vernieu- wing echter ook een gevaar: effecten van kuddegedrag worden versterkt, wat uiteindelijk tot instabiliteit en grote welvaartsverliezen kan leiden.

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.

This article investigates the discrete time indifference-attractor bifurcation of infinite horizon continous-time optimal control problems with a single state variable. We show that these bifurcations are linked to a heteroclinic bifurcation scenario of the state-costate equations, and we analyse the consequences for the optimal solutions. In particular, we can characterise the bifurcation value at which indifference thresholds appear by a geometric condition, and we find that for certain parameter values, there are countably infinitely many indifference points. We apply our results to a modified version of the shallow lake pollution management problem.

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.

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 as well as sufficient conditions that allow us to characterise Markov-perfect Nash equilibria for these games. These conditions can be translated into 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 voluntary investment in a public capital stock, the inter-temporal consumption of a reproductive asset, and the pollution of a shallow lake.

Traditional finance is built on the rationality paradigm. This chapter discusses simple models from an alternative approach in which financial markets are viewed as complex evolutionary systems. Agents are boundedly rational and base their investment decisions upon market forecasting heuristics. Prices and beliefs about future prices co-evolve over time with mutual feedback. Strategy choice is driven by evolutionary selection, so that agents tend to adopt strategies that were successful in the past. Calibration of "simple complexity models" with heterogeneous expectations to real financial market data and laboratory experiments with human subjects are also discussed.

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.

The purpose of this note is to show how Leitmann’s equivalent problem approach ties in with the classical notions of the Calculus of Variations, and how it can be exploited to give a rapid and elegant approach to Weierstrass’ theory of sufficient conditions. Both fixed and free endpoint conditions are considered.

The shallow lake optimal management problem is one of the simplest ecological-economic interest conflict models for which several qualitatively different long run outcomes are possible. We extend the original model by adding the capital stock of an industry as a second state variable. A government can mitigate the effects of pollution arising from industrial activities by imposing the requirement to abate emissions. Within this framework two scenarios are examined. In the social optimal benchmark, the social planner optimally allocates investment. In the competitive equilibrium, market forces determine the investment in capital, but the social planner can still abate emissions. We show that in the case of irreversibilities catastrophes are avoided in the competitive equilibrium when it is socially optimal to do so. However, in the competitive equilibrium, either the catastrophe is avoided in an inefficient way or the catastrophe is badly managed. In case of hysteresis, catastrophes are almost always avoided. Moreover, the decision to avoid catastrophes does not depend on long-term considerations.

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.

This article presents an equivalence notion of finite order stochastic processes. Local dependence measures are defined in terms of joint and marginal densities. The dependence measures are classified topologically using level sets. The corresponding bifurcation theory is illustrated with some simple examples.

This paper develops the notion of a Large Type Limit (LTL) describing the dynamical behavior of heterogeneous markets with many trader types. It is shown that generic and persistent features of adaptive evolutionary systems with many trader types are well described by the large type limit. Stability and bifurcation routes to instability and strange attractors in a simple evolutionary financial market model are studied. An increase in the ``intensity of adaption'' or in the diversity of beliefs may lead to deviations from an unstable RE fundamental benchmark and excess volatility. A large evolutionary system may thus become unstable and complicated dynamics may arise when agents become sensitive to small differences in fitness.

This paper develops the notion of a Large Type Limit (LTL) describing the average behavior of adaptive evolutionary systems with many trader types. It is shown that generic and persistent features of adaptive evolutionary systems with many trader types are well described by the large type limit. Stability and bifurcation routes to instability and strange attractors are studied. An increase in the ``intensity of adaption'' or in the diversity of beliefs may lead to deviations from the RE fundamental benchmark and excess volatility. Simple examples of LTL are able to generate important stylized facts, such as volatility clustering andlong memory, observed in real financial data.