1,216 research outputs found
Bifurcation Routes to Volatility Clustering under Evolutionary Learning
A simple asset pricing model with two types of adaptively learning traders, fundamentalists and technical analysts, is studied. Fractions of these trader types, which are both boundedly rational, change over time according to evolutionary learning, with technical analysts conditioning their forecasting rule upon deviations from a benchmark fundamental. Volatility clustering arises endogenously in this model. Two mechanisms are proposed as an explanation. The first is coexistence of a stable steady state and a stable limit cycle, which arise as a consequence of a so-called Chenciner bifurcation of the system. The second is intermittency and associated bifurcation routes to strange attractors. Both phenomena are persistent and occur generically. Simple economic intuition why these phenomena arise in nonlinear multi-agent evolutionary systems is provided.
Extended phase diagram of the Lorenz model
The parameter dependence of the various attractive solutions of the three
variable nonlinear Lorenz model equations for thermal convection in
Rayleigh-B\'enard flow is studied. Its bifurcation structure has commonly been
investigated as a function of r, the normalized Rayleigh number, at fixed
Prandtl number \sigma. The present work extends the analysis to the entire
(r,\sigma) parameter plane. An onion like periodic pattern is found which is
due to the alternating stability of symmetric and non-symmetric periodic
orbits. This periodic pattern is explained by considering non-trivial limits of
large r and \sigma. In addition to the limit which was previously analyzed by
Sparrow, we identify two more distinct asymptotic regimes in which either
\sigma/r or \sigma^2/r is constant. In both limits the dynamics is
approximately described by Airy functions whence the periodicity in parameter
space can be calculated analytically. Furthermore, some observations about
sequences of bifurcations and coexistence of attractors, periodic as well as
chaotic, are reported.Comment: 36 pages, 20 figure
Hidden attractors in fundamental problems and engineering models
Recently a concept of self-excited and hidden attractors was suggested: an
attractor is called a self-excited attractor if its basin of attraction
overlaps with neighborhood of an equilibrium, otherwise it is called a hidden
attractor. For example, hidden attractors are attractors in systems with no
equilibria or with only one stable equilibrium (a special case of
multistability and coexistence of attractors). While coexisting self-excited
attractors can be found using the standard computational procedure, there is no
standard way of predicting the existence or coexistence of hidden attractors in
a system. In this plenary survey lecture the concept of self-excited and hidden
attractors is discussed, and various corresponding examples of self-excited and
hidden attractors are considered
Bifurcations of periodic and chaotic attractors in pinball billiards with focusing boundaries
We study the dynamics of billiard models with a modified collision rule: the
outgoing angle from a collision is a uniform contraction, by a factor lambda,
of the incident angle. These pinball billiards interpolate between a
one-dimensional map when lambda=0 and the classical Hamiltonian case of elastic
collisions when lambda=1. For all lambda<1, the dynamics is dissipative, and
thus gives rise to attractors, which may be periodic or chaotic. Motivated by
recent rigorous results of Markarian, Pujals and Sambarino, we numerically
investigate and characterise the bifurcations of the resulting attractors as
the contraction parameter is varied. Some billiards exhibit only periodic
attractors, some only chaotic attractors, and others have coexistence of the
two types.Comment: 30 pages, 17 figures. v2: Minor changes after referee comments.
Version with some higher-quality figures available at
http://sistemas.fciencias.unam.mx/~dsanders/publications.htm
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