208 research outputs found
Deterministic continutation of stochastic metastable equilibria via Lyapunov equations and ellipsoids
Numerical continuation methods for deterministic dynamical systems have been
one of the most successful tools in applied dynamical systems theory.
Continuation techniques have been employed in all branches of the natural
sciences as well as in engineering to analyze ordinary, partial and delay
differential equations. Here we show that the deterministic continuation
algorithm for equilibrium points can be extended to track information about
metastable equilibrium points of stochastic differential equations (SDEs). We
stress that we do not develop a new technical tool but that we combine results
and methods from probability theory, dynamical systems, numerical analysis,
optimization and control theory into an algorithm that augments classical
equilibrium continuation methods. In particular, we use ellipsoids defining
regions of high concentration of sample paths. It is shown that these
ellipsoids and the distances between them can be efficiently calculated using
iterative methods that take advantage of the numerical continuation framework.
We apply our method to a bistable neural competition model and a classical
predator-prey system. Furthermore, we show how global assumptions on the flow
can be incorporated - if they are available - by relating numerical
continuation, Kramers' formula and Rayleigh iteration.Comment: 29 pages, 7 figures [Fig.7 reduced in quality due to arXiv size
restrictions]; v2 - added Section 9 on Kramers' formula, additional
computations, corrected typos, improved explanation
Relatively Coherent Sets as a Hierarchical Partition Method
Finite time coherent sets [8] have recently been defined by a measure based
objective function describing the degree that sets hold together, along with a
Frobenius-Perron transfer operator method to produce optimally coherent sets.
Here we present an extension to generalize the concept to hierarchially defined
relatively coherent sets based on adjusting the finite time coherent sets to
use relative mesure restricted to sets which are developed iteratively and
hierarchically in a tree of partitions. Several examples help clarify the
meaning and expectation of the techniques, as they are the nonautonomous double
gyre, the standard map, an idealized stratospheric flow, and empirical data
from the Mexico Gulf during the 2010 oil spill. Also for sake of analysis of
computational complexity, we include an appendic concerning the computational
complexity of developing the Ulam-Galerkin matrix extimates of the
Frobenius-Perron operator centrally used here
Computation and verification of Lyapunov functions
Lyapunov functions are an important tool to determine the basin of attraction of equilibria in Dynamical Systems through their sublevel sets. Recently, several numerical construction methods for Lyapunov functions have been proposed, among them the RBF (Radial Basis Function) and CPA (Continuous Piecewise Affine) methods. While the first method lacks a verification that the constructed function is a valid Lyapunov function, the second method is rigorous, but computationally much more demanding. In this paper, we propose a combination of these two methods, using their respective strengths: we use the RBF method to compute a potential Lyapunov function. Then we interpolate this function by a CPA function. Checking a finite number of inequalities, we are able to verify that this interpolation is a Lyapunov function. Moreover, sublevel sets are arbitrarily close to the basin of attraction. We show that this combined method always succeeds in computing and verifying a Lyapunov function, as well as in determining arbitrary compact subsets of the basin of attraction. The method is applied to two examples
Estimating long term behavior of flows without trajectory integration: the infinitesimal generator approach
The long-term distributions of trajectories of a flow are described by
invariant densities, i.e. fixed points of an associated transfer operator. In
addition, global slowly mixing structures, such as almost-invariant sets, which
partition phase space into regions that are almost dynamically disconnected,
can also be identified by certain eigenfunctions of this operator. Indeed,
these structures are often hard to obtain by brute-force trajectory-based
analyses. In a wide variety of applications, transfer operators have proven to
be very efficient tools for an analysis of the global behavior of a dynamical
system.
The computationally most expensive step in the construction of an approximate
transfer operator is the numerical integration of many short term trajectories.
In this paper, we propose to directly work with the infinitesimal generator
instead of the operator, completely avoiding trajectory integration. We propose
two different discretization schemes; a cell based discretization and a
spectral collocation approach. Convergence can be shown in certain
circumstances. We demonstrate numerically that our approach is much more
efficient than the operator approach, sometimes by several orders of magnitude
A Descent Method for Equality and Inequality Constrained Multiobjective Optimization Problems
In this article we propose a descent method for equality and inequality
constrained multiobjective optimization problems (MOPs) which generalizes the
steepest descent method for unconstrained MOPs by Fliege and Svaiter to
constrained problems by using two active set strategies. Under some regularity
assumptions on the problem, we show that accumulation points of our descent
method satisfy a necessary condition for local Pareto optimality. Finally, we
show the typical behavior of our method in a numerical example
Studying Attractor Symmetries by Means of Cross Correlation Sums
We use the cross correlation sum introduced recently by H. Kantz to study
symmetry properties of chaotic attractors. In particular, we apply it to a
system of six coupled nonlinear oscillators which was shown by Kroon et al. to
have attractors with several different symmetries, and compare our results with
those obtained by ``detectives" in the sense of Golubitsky et al.Comment: LaTeX file, 16 pages and 16 postscript figures; tarred, gzipped and
uuencoded; submitted to 'Nonlinearity
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