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Pseudospectral reduction to compute Lyapunov exponents of delay differential equations

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    * Corresponding author 

The first author is a member of INdAM Research group GNCS and is supported by INdAM GNCS projects "Analisi numerica di sistemi dinamici infinito-dimensionali e non regolari" (2015) and "Analisi numerica di certi tipi non classici di equazioni di evoluzione" (2016) and by the project PSD 2015 2017 DIMA PRID 2017 ZANOLIN "SIDIA – SIstemi DInamici e Applicazioni" (UNIUD)

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  • A recent pseudospectral collocation is used to reduce a nonlinear delay differential equation to a system of ordinary differential equations. Standard methods are then applied to compute Lyapunov exponents. The validity of this simple approach is shown experimentally. Matlab codes are also included.

    Mathematics Subject Classification: Primary: 37M25, 65L03, 65L07.

    Citation:

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  • Figure 1.  First five rightmost characteristic roots ($\times$) of (14) and first five dominant Lyapunov exponents computed with the current method ($\bullet$) and with the method in [14] ($\circ$), both for $M = 20$ and $T = 10^{5}$ (left); relevant absolute errors for increasing $M$ (right): current method (solid $\bullet$) and method in [14] (dashed $\circ$).

    Figure 2.  Absolute error with respect to $1$ of the largest exponent of (14) plotted against the final truncation time $T$, computed with the current method for varying $M = 10,15,20$ (solid $\bullet$, top-to-bottom) and with the method in [14] (dashed $\circ$) for $M = 20$.

    Figure 3.  Projection of the attractor of (15) for $a = 0.2$, $b = 0.1$, $c = 10$ and $\tau = 14$ (left), $\tau = 17$ (right).

    Table 1.  First six exponents of (15) for $a = 0.2$, $b = 0.1$, $c = 10$ and $\tau = 50$ computed with $M = 20$ (first column), from [14] (second column) and from [39] (third column); the reference solution corresponds to the initial function of constant value $\varphi\equiv2$ in (2).

    $5.85\times10^{-3}$ $5.76\times10^{-3}$ $5.83\times10^{-3}$
    $3.29\times10^{-3}$ $3.02\times10^{-3}$ $3.15\times10^{-3}$
    $0.53\times10^{-3}$ $0.65\times10^{-3}$ $0.01\times10^{-3}$
    $-0.92\times10^{-3}$ $-0.85\times10^{-3}$ $-0.29\times10^{-3}$
    $-5.17\times10^{-3}$ $-4.78\times10^{-3}$ $-5.08\times10^{-3}$
    $-9.56\times10^{-3}$ $-9.85\times10^{-3}$ $-9.78\times10^{-3}$
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    Table 2.  First three exponents of (16) for $a = b = 0.1$, $c = 14$ and $\epsilon = 0.5$ and varying coupling delay $\tau = 1$ (first column), $1.5$ (second column) and $2$ (third column), computed with the current method for $M = 5$ and $T = 10^3$ (first three rows) and with the method in [14] for $M = 20$ and $T = 10^{4}$ (second three rows); the reference solution of (2) corresponds to the initial function of constant value a (pseudo)random vector in $\mathbb{R}^{6}$.

    $-2.20\times10^{-2}$ $-6.74\times10^{-2}$ $-1.51\times10^{-2}$
    $-2.21\times10^{-2}$ $-6.77\times10^{-2}$ $-1.49\times10^{-2}$
    $-3.65\times10^{-1}$ $-1.19\times10^{-1}$ $-1.21\times10^{-1}$
    $-2.28\times10^{-2}$ $-6.81\times10^{-2}$ $-1.55\times10^{-2}$
    $-2.30\times10^{-2}$ $-6.89\times10^{-2}$ $-1.57\times10^{-2}$
    $-3.59\times10^{-1}$ $-1.11\times10^{-1}$ $-1.14\times10^{-1}$
     | Show Table
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