Random ODE and application to statistics. Michail Zak Senior Research Scientist (Emeritus) Jet Propulsion Laboratory California Institute of Technology Pasadena, CA 91109 Abstract. It is demonstrated that any statistics can be represented by an attractor of the solution to the corresponding system of ODE coupled with its Liouville equation. Such a non-Newtonian representation allows one to redu...

Coexistence The special case when all the independent solutions of a linear, T -periodic ODE are T -periodic. Hill’s equation A second order ODE of the form ẍ + p(t)x = 0, with p(t) T -periodic. Instability pockets Finite domains, usually intersections of instability tongues, where the trivial solution of linear, T -periodic ODEs is unstable. Instability tongues Domains in parameter space where...

We suggest multiscale methods for the integration of systems of second-order ordinary differential equations (ODEs) whose solutions include components that oscillate with large frequencies and small amplitudes. The methods do not need to integrate completely the fast oscillations and may employ step-sizes determined by the rate of change of the slow motions of the system. The technique may be u...

We consider a switching system with time delay composed of a finite number of linear delay differential equations (DDEs). Each DDE consists of a sum of a linear ODE part and a linear DDE part. We study two particular cases: (a) all the ODE parts are stable and (b) all the ODE parts are unstable and determine conditions for delay independent stability. For case (a), we extend a standard result o...

This paper considers the stability problem of a linear time invariant system in feedback with a string equation. A new Lyapunov functional candidate is proposed based on the use of augmented states which enriches and encompasses the classical Lyapunov functional proposed in the literature. It results in tractable stability conditions expressed in terms of linear matrix inequalities. This method...

We consider the adaptive strategies applicable to a simple model describing the phase lock of two coupled oscillators. This model has been used to show an instance of failure of the ODE45 RungeKutta-Felberg solver implemented within the MATLAB ODE suite, see [J. D. Skufca. Analysis still matters: a surprising instance of failure of Runge-KuttaFelberg ODE solvers. SIAM Review, 46:729-737, 2004]....

Ordinary differential equations (ODEs) are the primary means to modelling dynamical systems in many natural and engineering sciences. The number of equations required to describe a system with high heterogeneity limits our capability of effectively performing analyses. This has motivated a large body of research, across many disciplines, into abstraction techniques that provide smaller ODE syst...

We study accelerated mirror descent dynamics in continuous and discrete time. Combining the original continuous-time motivation of mirror descent with a recent ODE interpretation of Nesterov’s accelerated method, we propose a family of continuous-time descent dynamics for convex functions with Lipschitz gradients, such that the solution trajectories converge to the optimum at a O(1/t2) rate. We...

This paper discusses the discrete analogue of the gradient of a function and shows how discrete gradients can be used in the numerical integration of ordinary diieren-tial equations (ODE's). Given an ODE and one or more rst integrals (i.e., constants of the motion) and/or Lyapunov functions, it is shown that the ODE can be rewritten as a `linear-gradient system.' Discrete gradients are used to ...

A systematic algorithm for building integrating factors of the form μ(x, y), μ(x, y) or μ(y, y) for second order ODEs is presented. The algorithm can determine the existence and explicit form of the integrating factors themselves without solving any differential equations, except for a linear ODE in one subcase of the μ(x, y) problem. Examples of ODEs not having point symmetries are shown to be...