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

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  • 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.

    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 $T_2 = 2$.

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

    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.

    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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