On the quasi-periodic d-fold degenerate bifurcation

AbstractThis paper analyses the d-fold degenerate bifurcation of invariant quasi-periodic tori of normal dimension one. It is shown that for integrable families of diffeomorphisms unfolding a d-fold degenerate invariant torus versally, invariant Diophantine tori in the family persist under a small non-integrable perturbation. As a consequence, a subset of positive measure (with respect to an interpolating smooth manifold) of the bifurcation set persists as well. The proof is a consequence of the translated torus theorem of Rüssmann and Herman

More Hedging Instruments may destablize Markets

This paper formalizes the idea that more hedging instruments may destabilize markets when traders are heterogeneous and adapt their behavior according to experience based reinforcement learning. We investigate three different economic settings, a simple mean-variance asset pricing model, a general equilibrium two-period overlapping generations model with heterogeneous expectations and a noisy rational expectations asset pricing model with heterogeneous information signals. In each setting the introduction of additional Arrow securities can destabilize the market, causing a bifurcation of the steady state to multiple steady states, periodic orbits or even chaotic fluctuations

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 ratios of joint and marginal probability densities. The dependence measures are classified topologically using level sets. The corresponding bifurcation theory is illustrated with some simple examples

Normal-normal resonances in a double Hopf bifurcation

We investigate the stability loss of invariant n-dimensional quasi-periodic tori during a double Hopf bifurcation, where at bifurcation the two normal frequencies are in normal-normal resonance. Invariants are used to analyse the normal form approximations in a unified manner. The corresponding dynamics form a skeleton for the dynamics of the original system. Here both normal hyperbolicity and KAM theory are being used.Comment: 22 pages, 6 figure

Normal resonances in a double Hopf bifurcation

We introduce a framework to systematically investigate the resonant double Hopf bifurcation. We use the basic invariants of the ensuing T1-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 weak bifucation 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 (nonbifurcating) systems is open and dense. The theory is illustrated with some simple examples

Bifurcation Routes to Volatility Clustering under Evolutionary Learning

A simple asset pricing model with two types of boundedly rational traders, fundamentalists and chartists, is studied. Fractions of trader types change over time according to evolutionary learning, with chartists conditioning their forecasting rule upon deviations from a benchmark fundamental. Volatility clustering arises endogenously and two generic mechanisms are proposed as an explanation: (1) coexistence of a stable steady state and a stable limit cycle, due to a so-called Chenciner bifurcation of the system and (2) intermittency and associated bifurcation routes to strange attractors. Economic intuition as to why these phenomena arise in nonlinear multi-agent evolutionary systems is provided