Determination of the area of exponential attraction in one-dimensional finite-time systems using meshless collocation

Peter Giesl; James McMichen

doi:10.3934/dcdsb.2018094

Article Contents

2018, Volume 23, Issue 4: 1835-1850. Doi: 10.3934/dcdsb.2018094

This issue Previous Article Singular perturbed renormalization group theory and its application to highly oscillatory problems Next Article Stability and robustness analysis for a multispecies chemostat model with delays in the growth rates and uncertainties

Determination of the area of exponential attraction in one-dimensional finite-time systems using meshless collocation

Peter Giesl^, and
James McMichen

Department of Mathematics, University of Sussex, Falmer BN1 9QH, United Kingdom

* Corresponding author
* Corresponding author

Received: November 30, 2016

Revised: October 31, 2017

Early access: March 2018

Published: April 2018

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Abstract

We consider a non-autonomous ordinary differential equation over a finite time interval $[T_1,T_2]$. The area of exponential attraction consists of solutions such that the distance to adjacent solutions exponentially contracts from $T_1$ to $T_2$. One can use a contraction metric to determine an area of exponential attraction and to provide a bound on the rate of attraction.

In this paper, we will give the first method to algorithmically construct a contraction metric for finite-time systems in one spatial dimension. We will show the existence of a contraction metric, given by a function which satisfies a second-order partial differential equation with boundary conditions. We then use meshless collocation to approximately solve this equation, and show that the resulting approximation itself defines a contraction metric, if the collocation points are sufficiently dense. We give error estimates and apply the method to an example.

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Mathematics Subject Classification: Primary: 37C60, 34D20; Secondary: 65P40, 65N15.

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Figure 1. The collocation points $X_1$ and $X_2$ as well as some numerically computed solutions of the system (21) with $T_2 = 2$.

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Figure 2. The function $L_m(t,x)$, using the approximation $w$ with $T_2 = 2$.

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Figure 3. Some level sets of $L_m(t,x)$. The 0-level set of $L_m$ crosses the $x$-axis at $\pm0.1789$ and is an approximation of the area of exponential attraction.

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Figure 4. Zero level sets of $L_m(t,x)$ for different values of $T_2$, namely $0.4$ (black), $0.8$ (blue), $1.2$ (red), $1.6$ (green), $2$ (magenta) and $2.4$ (cyan). The 0-level set of $L_m$ is an approximation of the area of exponential attraction. The size of the area of exponential attraction in $x$-direction shrinks until $T_2 = 1.5$ and then grows again.

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