Abstract
We consider numerical one-step approximations of ordinary dierential equations and present two results on the persistence of attractors appearing in the numerical system. First, we show that the upper limit of a sequence of numerical attractors for a sequence of vanishing time step is an attractor for the approximated system if and only if for all these time steps the numerical one-step schemes admit attracting sets which approximate this upper limit set and attract with a uniform rate. Second, we show that if these numerical attractors themselves attract with a uniformly rate, then they converge to some set if and only if this set is an attractor for the approximated system. In this case, we can also give an estimate for the rate of convergence depending on the rate of attraction and on the order of the numerical scheme. AMS Classication: 65L20, 65L06, 34D45, 34E10
Highlights
Attractors play an important role in the understanding of the behaviour of complex dynamical systems
For one-step discretizations of ordinary differential equations (ODEs) the basic result in that direction has been obtained by Kloeden and Lorenz in 1986 [11]
In this paper it is shown that if the ODE possesses an attractor the numerical approximations possess absorbing sets nearby, which converge to the attractor as the time step tends to 0
Summary
Attractors play an important role in the understanding of the behaviour of complex dynamical systems. It is shown that convergence holds if both the continuous and the numerical attractors attract exponentially (in this case an estimate for the rate of convergence is given), or if the continuous time attractor consists of the unstable manifolds of finitely many hyperbolic equilibrium points (which is shown to be true for gradient systems with a bounded set of hyperbolic equilibria). These conditions have in common that certain assumptions on the dynamics of the approximated ODE are made.
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