title: Chaos in high-dimensional dissipative dynamical systems.
abstract: For dissipative dynamical systems described by a system of ordinary differential equations, we address the question of how the probability of chaotic dynamics increases with the dimensionality of the phase space. We find that for a system of d globally coupled ODE's with quadratic and cubic non-linearities with randomly chosen coefficients and initial conditions, the probability of a trajectory to be chaotic increases universally from ~10(-5)- 10(-4) for d = 3 to essentially one for d ~ 50. In the limit of large d, the invariant measure of the dynamical systems exhibits universal scaling that depends on the degree of non-linearity, but not on the choice of coefficients, and the largest Lyapunov exponent converges to a universal scaling limit. Using statistical arguments, we provide analytical explanations for the observed scaling, universality, and for the probability of chaos.
authors.affiliation: Departamento de Fisica, Universidad de Santiago de Chile, Santiago, Chile.
authors.forename: Iaroslav
authors.initials: I
authors.lastname: Ispolatov
authors.affiliation: Department of Zoology and Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z4 Canada.
authors.forename: Vaibhav
authors.initials: V
authors.lastname: Madhok
authors.affiliation: Departamento de Fisica, Universidad de Santiago de Chile, Santiago, Chile.
authors.forename: Sebastian
authors.initials: S
authors.lastname: Allende
authors.affiliation: Department of Zoology and Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z4 Canada.
authors.forename: Michael
authors.initials: M
authors.lastname: Doebeli