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  • 1995-1999  (2)
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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1572-9575
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract Different quasiperiodically and parametrically driven nonlinear oscillators with quadratic and cubic nonlinearities are considered, and the corresponding homoclinic bifurcation sets in a five-dimensional parameter space are explicitly calculated. We classify all these cases into two basic types of homoclinic bifurcation sets: the first one corresponds to quasiperiodically driven oscillators and the second one corresponds to parametrically driven oscillators.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Woodbury, NY : American Institute of Physics (AIP)
    Chaos 7 (1997), S. 125-138 
    ISSN: 1089-7682
    Source: AIP Digital Archive
    Topics: Physics
    Notes: Standard dynamical systems theory is based on the study of invariant sets. However, when noise is added, there are no bounded invariant sets. Our goal is then to study the fractal structure that exists even with noise. The problem we investigate is fluid flow past an array of cylinders. We study a parameter range for which there is a periodic oscillation of the fluid, represented by vortices being shed past each cylinder. Since the motion is periodic in time, we can study a time-1 Poincaré map. Then we add a small amount of noise, so that on each iteration the Poincaré map is perturbed smoothly, but differently for each time cycle. Fix an x coordinate x0 and an initial time t0. We discuss when the set of initial points at a time t0 whose trajectory (x(t),y(t)) is semibounded (i.e., x(t)〉x0 for all time) has a fractal structure called an indecomposable continuum. We believe that the indecomposable continuum will become a fundamental object in the study of dynamical systems with noise. © 1997 American Institute of Physics.
    Type of Medium: Electronic Resource
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