This paper studies limit measures of stationary measures of stochastic ordinary differential equations on the Euclidean space and tries to determine which invariant measures of an unperturbed system will survive. Under the assumption for SODEs to admit the Freidlin-Wentzell or Dembo-Zeitouni large deviations principle with weaker compactness condition, we prove that limit measures are concentrated away from repellers which are topologically transitive, or equivalent classes, or admit Lebesgue measure zero. We also preclude concentrations of limit measures on acyclic saddle or trap chains. This illustrates that limit measures are concentrated on Liapunov stable compact invariant sets. Applications are made to the Morse-Smale systems, the Axiom A systems including structural stability systems and separated star systems, the gradient or gradient-like systems, those systems possessing the Poincare-Bendixson property with a finite number of limit sets to obtain that limit measures live on Liapunov stable critical elements, Liapunov stable basic sets, Liapunov stable equilibria and Liapunov stable limit sets including equilibria, limit cycles and saddle or trap cycles, respectively. A number of nontrivial examples admitting a unique limit measure are provided, which include monostable, multistable systems and those possessing infinite equivalent classes.
The aim of this text is to extend the theory of generalized ordinary differential equations to the setting of metric spaces. We present existence and uniqueness theorems that significantly improve previous results even when restricted back to Euclidean spaces.
Inverse Approach In The Study Of Ordinary Differential Equations
本文推广了1952年发表的“Construction of the whole set of常微分方程with a given积分曲线“一文中的Eruguin结果，构造了一个在$\Bbb{R}^N$中允许一组给定的偏积分的微分系统，特别地，我们研究了当这些函数是多项式时的情形.本文构造了一个非达布可积的平面多项式系统$n$，它具有一条不变的不可约代数曲线$g(x,y)=0$。对这个系统，我们分析了代数极限环的Darboux可积性、Poincare问题和第16代Hilbert问题.
In this Letter we identify special systems of (an arbitrary number) N of first-order Ordinary Differential Equations with homogeneous polynomials of arbitrary degree M on their right-hand sides, which feature very simple explicit solutions; as well as variants of these systems--with right-hand sides no more homogeneous--which feature periodic solutions. A novelty of these findings is to consider special systems characterized by constraints involving both their parameters and their initial data.
The paper analyzes the structure and the inner long-term dynamics of the invariant compact sets for the skewproduct flow induced by a family of time-dependent ordinary differential equations of nonhomogeneous linear dissipative type. The main assumptions are made on the dissipative term and on the homogeneous linear term of the equations. The rich casuistic includes the uniform stability of the invariant compact sets, as well as the presence of Li-Yorke chaos and Auslander-Yorke chaos inside the attractor.
We prove that measure-preserving symmetries of an $n$-dimensional differential system preserve its divergence and the divergence derivatives along the solutions. Also, we prove that measure-preserving reversibilities preserve odd-order divergence derivatives along the solutions, and that even-order derivatives are multiplied by $-1$. We apply such results to find all the area-preserving symmetries and reversibilities of planar Lotka-Volterra and Liénard systems.
We discuss existence, non-uniqueness and regularity of one- and two-sided solutions of initial value problems for scalar quasi-linear ordinary differential equations where the initial condition corresponds to an impasse point of the equation. With a differential geometric approach, we reduce the problem to questions in dynamical systems theory. As an application, we discuss in detail second-order equations of the form $g(x)u''=f(x,u,u')$ with an initial condition imposed at a simple zero of $g$. This generalises results by Liang and also makes them more transparent via our geometric approach.
Properties of partial integrals such as real and complex-valued polynomial, multiple polynomial, exponential, and conditional for ordinary differential systems are studied. The possibilities of constructing first integrals and last multipliers by known partial integrals are considered. Applications of partial integrals to solve the Darboux problem and to the extended Darboux problem are given. Integral basis of the Jacobi system is built. And the inverse problem of constructing differential systems on the base of their partial integrals is solved.
This evocative essay focuses on the mathematical activities witnessed by the author along 1962-64 at IMPA. The list of research problems proposed in September 1962 by Mauricio Peixoto at the Seminar on the Qualitative Theory of Differential Equations is pointed out as a landmark for the genesis of the research interest in the Qualitative Theory of Differential Equations and Dynamical Systems in Brazil.
A system of 4 nonlinearly-coupled Ordinary Differential Equations has been recently introduced to investigate the evolution of human respiratory virus epidemics. In this paper we point out that some explicit solutions of that system can be obtained by algebraic operations, provided the parameters of the model satisfy certain constraints.
A novel way of using neural networks to learn the dynamics of time delay systems from sequential data is proposed. A neural network with trainable delays is used to approximate the right hand side of a delay differential equation. We relate the delay differential equation to an ordinary differential equation by discretizing the time history and train the corresponding neural ordinary differential equation (NODE) to learn the dynamics. An example on learning the dynamics of the Mackey-Glass equation using data from chaotic behavior is given. After learning both the nonlinearity and the time del…
Traditional solvers for delay differential equations (DDEs) are designed around only a single method and do not effectively use the infrastructure of their more-developed ordinary differential equation (ODE) counterparts. In this work we present DelayDiffEq, a Julia package for numerically solving delay differential equations (DDEs) which leverages the multitude of numerical algorithms in OrdinaryDiffEq for solving both stiff and non-stiff ODEs, and manages to solve challenging stiff DDEs. We describe how compiling the ODE integrator within itself, and accounting for discontinuity propagation,…
