Florian Wagener

Skiba points and heteroclinic bifurcations, with applications to the shallow lake system

Abstract Techniques from dynamical systems, specifically from bifurcation theory, are used to investigate the occurrence of Skiba points in one-state, one-co-state control systems, for which the effect of the control has a definite direction. A Skiba point is an initial state for which two different optimal solutions of the control problem exist. It is found that the parameter region for which Skiba points occur is bounded by heteroclinic bifurcation manifolds. A local criterion is given that ensures the existence of Skiba points in systems with small discount rates. The analysis is applied to the shallow lake system investigated by Maler et al. (in: K.-G. Maler, C. Perrings (Eds.), The Economics of Non-Linear Dynamic Systems, Resilience Network, 2000, in preparation). For this system, it is shown that for any given parameter value, there is at most one Skiba point.

CHAPTER 4 – Complex Evolutionary Systems in Behavioral Finance

Publisher Summary This chapter explores some behavioral finance models with evolutionary selection of heterogeneous trading strategies. When strategy selection is driven by short-run realized profits, trend-following strategies may destabilize asset markets. Asset price fluctuations are characterized by phases in which fundamentalists dominate and prices are close to fundamentals, suddenly interrupted by possibly long-lasting phases of price bubbles when trend-following strategies dominate the market and prices deviate persistently from fundamentals. Even in simple heterogeneous belief models, asset prices are difficult to predict and market timing based on the prediction of the start or the collapse of a bubble is extremely difficult and highly sensitive to noise. Simple evolutionary models therefore could provide an explanation. Laboratory experiments using human subjects confirm that coordination on simple trend-following strategies may arise in asset markets and cause persistent deviations from fundamentals. In a heterogeneous beliefs asset-pricing model, as long as prices fluctuate around their fundamental values, fundamentalist strategies do quite well in terms of accumulated profits. If there are no limits to arbitrage and fundamentalists can survive possibly long-lasting bubbles during which they suffer large losses, their strategy performs very well in the long run and may help stabilize markets.

A note on Gevrey regular KAM theory and the inverse approximation lemma

A theorem of the type Kolmogorov-Arnold-Moser holds for small real analytic perturbations of a family of parallel flows on a torus, yielding conjugacies to Diophantine parallel flows that depend Whitney-Gevrey regularly on parameters. This follows from an adaptation of the inverse approximation lemma. The method of proof is generaland extends to a wide range of problems of KAM type.

Resonances in skew and reducible quasi-periodic Hopf bifurcations

Two types of quasi-periodic Hopf bifurcations are known, in which a Whitney smooth family of quasi-periodic attractors loses its hyperbolicity. One is the reducible case, where the normal linear dynamics are trivial. Another is the skew case, where the normal dynamics are topologically non-trivial. There, the dynamics can involve periodicity, quasi-periodicity and chaos, including chaos with a mixed spectrum. This paper investigates a model system where the bifurcating circle supports Morse-Smale (also called resonant) dynamics. First this is done under the assumption of rotational symmetry (in the normal direction) of the system; a local bifurcation analysis is given for this. Then the effects of generic (non-symmetric) perturbations are discussed. It turns out that the bifurcation diagram is organized by several codimension-two bifurcations: saddle-Hopf and degenerate Hopf bifurcations for diffeomorphisms, and Bogdanov-Takens bifurcations of invariant circles. The (local) bifurcation diagram is computed analytically for the model system. We believe that this model is representative for the minimal dynamical complexity which generically occurs whenever the dynamics on the bifurcating circle is resonant.

Equivalence and Bifurcations of Finite Order Stochastic Processes

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.

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

This 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 non-vanishing forcing strength, a semi-global normal form system is found by averaging and applying a van der Pol transformation. The bifurcation diagram is organized by a codimension 3 singularity of nilpotent-elliptic type. A fairly complete analysis of local bifurcations is given; moreover, all the non-local bifurcation curves predicted by Dumortier et al (1991 Bifurcations of Planar Vector Fields (Lecture Notes in Mathematics vol 1480) (Berlin: Springer)), excepting boundary bifurcations, are found numerically.

Bifurcations of Optimal Vector Fields

We study the structure of the solution set of a class of infinite-horizon dynamic programming problems with one-dimensional state spaces, as well as their bifurcations, as problem parameters are varied. The solutions are represented as the integral curves of a multivalued optimal vector field on state space. Generically, there are three types of integral curves: stable points, open intervals that are forward asymptotic to a stable point and backward asymptotic to an unstable point, and half-open intervals that are forward asymptotic to a stable point and backward asymptotic to an indifference point; the latter are initial states to multiple optimal trajectories. We characterize all bifurcations that occur generically in one-and two-parameter families. Most of these are related to global dynamical bifurcations of the state-costate system of the problem.

Consistency of linear forecasts in a nonlinear stochastic economy

The notion of consistent expectations equilibrium is extended to economies that are described by a nonlinear stochastic system. Agents in the model do not know the nonlinear law of motion and use a simple linear forecasting rule to form their expectations. Along a stochastic consistent expectations equilibrium (SCEE), these expectations are correct in a linear statistical sense, i.e., the unconditional mean and autocovariances of the actual (but unknown) nonlinear stochastic process coincide with those of the linear stochastic process on which the agents base their beliefs. In general, the linear forecasts do not coincide with the true conditional expectation, but an SCEE is an ‘approximate rational expectations equilibrium’ in the sense that forecasting errors are unbiased and uncorrelated. Adaptive learning of SCEE is studied in an overlapping generations framework.

In Defense of Trusts: R&D Cooperation in Global Perspective

We present a continuous-time generalization of the seminal R&D model of d’Aspremont and Jacquemin (American Economic Review, Vol. 78, No. 5) to examine the trade-off between the benefits of allowing firms to cooperate in R&D and the corresponding increased potential for product market collusion. We consider all trajectories that are candidates for an optimal solution as well as initial marginal cost levels that exceed the choke price. Firms that collude develop further a wider range of initial technologies, pursue innovations more quickly, and are less likely to abandon a technology. Product market collusion could thus yield higher total surplus.

Research among Copycats: R&D, Spillovers, and Feedback Strategies

We study a stochastic dynamic game of process innovation in which firms can initiate and terminate R&D efforts and production at different times. We discern the impact of knowledge spillovers on the investments in existing markets, as well as on the likely structure of newly forming markets, for all possible asymmetries between firms. While an increase in spillovers may improve the likelihood of a competitive market, it may at the same time reduce the level to which a technology is developed. We show that the relation between spillovers, R&D efforts, and surpluses depends on relative as well as absolute efficiency of firms. High spillovers are not necessarily pro-competitive as they can make it harder for the laggard to catch up with the technology leader.