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March  2019, 18(2): 959-975. doi: 10.3934/cpaa.2019047

Fractal analysis of canard cycles with two breaking parameters and applications

1. 

Hasselt University, Campus Diepenbeek, Agoralaan Gebouw D, 3590 Diepenbeek, Belgium

2. 

University of Zagreb, Faculty of Electrical Engineering and Computing, Department of Applied Mathematics, Unska 3, 10000 Zagreb, Croatia

* Corresponding author

Received  May 2018 Revised  July 2018 Published  October 2018

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.

Citation: Renato Huzak, Domagoj Vlah. Fractal analysis of canard cycles with two breaking parameters and applications. Communications on Pure and Applied Analysis, 2019, 18 (2) : 959-975. doi: 10.3934/cpaa.2019047
References:
[1]

E. Benoit, Équations différentielles: relation entrée-sortie, C. R. Acad. Sci. Paris Sér. I Math., 293 (1981), 293-296. 

[2]

P. De Maesschalck and F. Dumortier, Time analysis and entry-exit relation near planar turning points, J. Differential Equations, 215 (2005), 225-267.  doi: 10.1016/j.jde.2005.01.004.

[3]

P. De Maesschalck and F. Dumortier, Canard cycles in the presence of slow dynamics with singularities, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 265-299.  doi: 10.1017/S0308210506000199.

[4]

P. De Maesschalck and F. Dumortier, Classical Liénard equations of degree n≥ 6 can have $[\frac{n-1} {2} ]+2$ limit cycles, J. Differential Equations, 250 (2011), 2162-2176.  doi: 10.1016/j.jde.2010.12.003.

[5]

P. De Maesschalck and R. Huzak, Slow divergence integrals in classical Liénard equations near centers, J. Dynam. Differential Equations, 27 (2015), 177-185.  doi: 10.1007/s10884-014-9358-1.

[6]

F. Diener and M. Diener, Chasse au canard. I. Les canards, Collect. Math., 32 (1981), 37-74. 

[7]

F. Dumortier, Slow divergence integral and balanced canard solutions, Qual. Theory Dyn. Syst., 10 (2011), 65-85.  doi: 10.1007/s12346-011-0038-9.

[8]

F. DumortierD. Panazzolo and R. Roussarie, More limit cycles than expected in Liénard equations, Proc. Amer. Math. Soc., 135 (2007), 1895-1904.  doi: 10.1090/S0002-9939-07-08688-1.

[9]

F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc., 121 (1996), x+100.  doi: 10.1090/memo/0577.

[10]

F. Dumortier and R. Roussarie, Canard cycles with two breaking parameters, Discrete Contin. Dyn. Syst., 17 (2007), 787-806.  doi: 10.3934/dcds.2007.17.787.

[11]

N. ElezovićV. Županović and D. Žubrinić, Box dimension of trajectories of some discrete dynamical systems, Chaos Solitons Fractals, 34 (2007), 244-252.  doi: 10.1016/j.chaos.2006.03.060.

[12]

K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley and Sons, Ltd., Chichester, (1990). 

[13]

R. Huzak, Box dimension and cyclicity of canard cycles, Qual. Theory Dyn. Syst., 17 (2018), 475-493.  doi: 10.1007/s12346-017-0248-x.

[14]

S. G. Krantz and H. R. Parks, The Geometry of Domains in Space, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-1574-5.

[15]

M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion, J. Differential Equations, 174 (2001), 312-368.  doi: 10.1006/jdeq.2000.3929.

[16]

L. Mamouhdi and R. Roussarie, Canard cycles of finite codimension with two breaking parameters, Qual. Theory Dyn. Syst., 11 (2012), 167-198.  doi: 10.1007/s12346-011-0061-x.

[17]

P. MardešićM. Resman and V. Županović, Multiplicity of fixed points and growth of $\epsilon $-neighborhoods of orbits, J. Differential Equations, 253 (2012), 2493-2514.  doi: 10.1016/j.jde.2012.06.020.

[18]

C. Tricot, Curves and Fractal Dimension, With a foreword by Michel Mendès France, Translated from the 1993 French original, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4170-6.

show all references

References:
[1]

E. Benoit, Équations différentielles: relation entrée-sortie, C. R. Acad. Sci. Paris Sér. I Math., 293 (1981), 293-296. 

[2]

P. De Maesschalck and F. Dumortier, Time analysis and entry-exit relation near planar turning points, J. Differential Equations, 215 (2005), 225-267.  doi: 10.1016/j.jde.2005.01.004.

[3]

P. De Maesschalck and F. Dumortier, Canard cycles in the presence of slow dynamics with singularities, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 265-299.  doi: 10.1017/S0308210506000199.

[4]

P. De Maesschalck and F. Dumortier, Classical Liénard equations of degree n≥ 6 can have $[\frac{n-1} {2} ]+2$ limit cycles, J. Differential Equations, 250 (2011), 2162-2176.  doi: 10.1016/j.jde.2010.12.003.

[5]

P. De Maesschalck and R. Huzak, Slow divergence integrals in classical Liénard equations near centers, J. Dynam. Differential Equations, 27 (2015), 177-185.  doi: 10.1007/s10884-014-9358-1.

[6]

F. Diener and M. Diener, Chasse au canard. I. Les canards, Collect. Math., 32 (1981), 37-74. 

[7]

F. Dumortier, Slow divergence integral and balanced canard solutions, Qual. Theory Dyn. Syst., 10 (2011), 65-85.  doi: 10.1007/s12346-011-0038-9.

[8]

F. DumortierD. Panazzolo and R. Roussarie, More limit cycles than expected in Liénard equations, Proc. Amer. Math. Soc., 135 (2007), 1895-1904.  doi: 10.1090/S0002-9939-07-08688-1.

[9]

F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc., 121 (1996), x+100.  doi: 10.1090/memo/0577.

[10]

F. Dumortier and R. Roussarie, Canard cycles with two breaking parameters, Discrete Contin. Dyn. Syst., 17 (2007), 787-806.  doi: 10.3934/dcds.2007.17.787.

[11]

N. ElezovićV. Županović and D. Žubrinić, Box dimension of trajectories of some discrete dynamical systems, Chaos Solitons Fractals, 34 (2007), 244-252.  doi: 10.1016/j.chaos.2006.03.060.

[12]

K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley and Sons, Ltd., Chichester, (1990). 

[13]

R. Huzak, Box dimension and cyclicity of canard cycles, Qual. Theory Dyn. Syst., 17 (2018), 475-493.  doi: 10.1007/s12346-017-0248-x.

[14]

S. G. Krantz and H. R. Parks, The Geometry of Domains in Space, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-1574-5.

[15]

M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion, J. Differential Equations, 174 (2001), 312-368.  doi: 10.1006/jdeq.2000.3929.

[16]

L. Mamouhdi and R. Roussarie, Canard cycles of finite codimension with two breaking parameters, Qual. Theory Dyn. Syst., 11 (2012), 167-198.  doi: 10.1007/s12346-011-0061-x.

[17]

P. MardešićM. Resman and V. Županović, Multiplicity of fixed points and growth of $\epsilon $-neighborhoods of orbits, J. Differential Equations, 253 (2012), 2493-2514.  doi: 10.1016/j.jde.2012.06.020.

[18]

C. Tricot, Curves and Fractal Dimension, With a foreword by Michel Mendès France, Translated from the 1993 French original, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4170-6.

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 $\kappa_i$ $1-10^{-16}$ $1-10^{-8}$ $1-10^{-4}$ $1-10^{-2}$ $1-10^{-1}$
test case $i$ $1$ $2$ $3$ $4$ $5$
factor $\kappa_i$ $1-10^{-16}$ $1-10^{-8}$ $1-10^{-4}$ $1-10^{-2}$ $1-10^{-1}$
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. $0.022166$ $0.532658$ $0.036062$
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. $0.022166$ $0.532658$ $0.036062$
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