59 research outputs found
Combinatorial approach to detection of fixed points, periodic orbits, and symbolic dynamics
We present a combinatorial approach to rigorously show the existence of fixed
points, periodic orbits, and symbolic dynamics in discrete-time dynamical
systems, as well as to find numerical approximations of such objects. Our
approach relies on the method of `correctly aligned windows'. We subdivide the
`windows' into cubical complexes, and we assign to the vertices of the cubes
labels determined by the dynamics. In this way we encode the dynamics
information into a combinatorial structure. We use a version of the Sperner
Lemma saying that if the labeling satisfies certain conditions, then there
exist fixed points/periodic orbits/orbits with prescribed itineraries. Our
arguments are elementary
On Wesner's method of searching for chaos on low frequency
An alternative to Wesner's method of detecting deterministic behavior and chaos in small sample sets is presented. This new method is applied to analyze the dynamics of several stock prices.
Critical Transitions In a Model of a Genetic Regulatory System
We consider a model for substrate-depletion oscillations in genetic systems,
based on a stochastic differential equation with a slowly evolving external
signal. We show the existence of critical transitions in the system. We apply
two methods to numerically test the synthetic time series generated by the
system for early indicators of critical transitions: a detrended fluctuation
analysis method, and a novel method based on topological data analysis
(persistence diagrams).Comment: 19 pages, 8 figure
Geometry of Weak Stability Boundaries
The notion of a weak stability boundary has been successfully used to design
low energy trajectories from the Earth to the Moon. The structure of this
boundary has been investigated in a number of studies, where partial results
have been obtained. We propose a generalization of the weak stability boundary.
We prove analytically that, in the context of the planar circular restricted
three-body problem, under certain conditions on the mass ratio of the primaries
and on the energy, the weak stability boundary about the heavier primary
coincides with a branch of the global stable manifold of the Lyapunov orbit
about one of the Lagrange points
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