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Fractal analysis of canard cycles with two breaking parameters and applications

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  • In previous work [13] we introduced a new box dimension method for computation of the number of limit cycles in planar slow-fast systems, Hausdorff close to balanced canard cycles with one breaking mechanism (the Hopf breaking mechanism or the jump breaking mechanism). This geometric approach consists of a simple iteration method for finding one orbit of the so-called slow relation function and of the calculation of the box dimension of that orbit. Then we read the cyclicity of the balanced canard cycles from the box dimension. The purpose of the present paper is twofold. First, we generalize the box dimension method to canard cycles with two breaking mechanisms. Second, we apply the method from [13] and our generalized method to a number of interesting examples of canard cycles with one breaking mechanism and with two breaking mechanisms respectively.

    Mathematics Subject Classification: Primary: 34E15, 34E17, 34C26.

    Citation:

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  • Figure 1.  The fast subsystem $L_{0,b_0,\mu}$

    Figure 2.  Canard cycles with two breaking parameters, at level $\epsilon = 0$. (a) One jump breaking mechanism, with two jump points $\mathcal{C}_1^1$ and $\mathcal{C}_1^2$, and one Hopf breaking mechanism with a turning point $\mathcal{C}_2$. (b) Two Hopf mechanisms with turning points $\mathcal{C}_1$ and $\mathcal{C}_2$

    Figure 3.  $U_\delta$ has two parts: the nucleus $N_{\delta}$, and the tail $T_{\delta}$. The tail $T_\delta$ contains all $(2\delta)$-intervals of $U_\delta$ before they start to overlap at the point $x_{n_\delta}$

    Figure 4.  The numerical estimate of the box dimension depending on the number of calculated orbit values $M$, in system (11) and test case 3

    Figure 5.  The numerical estimate of the box dimension depending on the number of calculated orbit values $M$, in system (12) and test case 3

    Figure 6.  The numerical estimate of the box dimension depending on the number of calculated orbit values $M$, in system (13) and test case 3

    Table 1.  Factors $\kappa_i$

    test case $i$ $1$ $2$ $3$ $4$ $5$
    factor $\kappa_i$ $1-10^{-16}$ $1-10^{-8}$ $1-10^{-4}$ $1-10^{-2}$ $1-10^{-1}$
     | Show Table
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    Table 2.  Numerically computed box dimensions

    example system (11) (12) (13)
    theoretical box dim. $0$ $1/2$ 0
    num. of digits of prec. 170 60 150
    computed orbit size $M$ 500 10000 2000
    test case $1$ box dim. $0.019946$ $0.499413$ $0.031357$
    test case $2$ box dim. $0.021066$ $0.498836$ $0.033703$
    test case $3$ box dim. $0.021675$ $0.521252$ $0.035013$
    test case $4$ box dim. $0.021993$ $0.532500$ $0.035706$
    test case $5$ box dim. $0.022166$ $0.532658$ $0.036062$
     | Show Table
    DownLoad: CSV
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