American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A nonlinear model for marble sulphation including surface rugosity: Theoretical and numerical results
  • CPAA Home
  • This Issue
  • Next Article
    Semi-hyperbolic patches of solutions to the two-dimensional compressible magnetohydrodynamic equations
March  2019, 18(2): 959-975. doi: 10.3934/cpaa.2019047

Fractal analysis of canard cycles with two breaking parameters and applications

Renato Huzak 1,, and  Domagoj Vlah 2,

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.

Keywords: Box dimension, canard cycles, singular perturbation theory, slow divergence integral, Tricot method.
Mathematics Subject Classification: Primary: 34E15, 34E17, 34C26.
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. Dumortier, D. 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. Elezović, V. 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. Mardešić, M. 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. Dumortier, D. 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. Elezović, V. 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. Mardešić, M. 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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Factors $\kappa_i$
test case $i$ $1$ $2$ $3$ $4$ $5$
factor $\kappa_i$ $1-10^{-16}$ $1-10^{-8}$ $1-10^{-4}$ $1-10^{-2}$ $1-10^{-1}$
Table Options

Download as excel

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. $0.022166$ $0.532658$ $0.036062$
Table Options

Download as excel

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. $0.022166$ $0.532658$ $0.036062$
[1]

P. De Maesschalck, Freddy Dumortier. Detectable canard cycles with singular slow dynamics of any order at the turning point. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 109-140. doi: 10.3934/dcds.2011.29.109
[2]

Ilya Schurov. Duck farming on the two-torus: Multiple canard cycles in generic slow-fast systems. Conference Publications, 2011, 2011 (Special) : 1289-1298. doi: 10.3934/proc.2011.2011.1289
[3]

Renato Huzak, P. De Maesschalck, Freddy Dumortier. Primary birth of canard cycles in slow-fast codimension 3 elliptic bifurcations. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2641-2673. doi: 10.3934/cpaa.2014.13.2641
[4]

Zainidin Eshkuvatov. Homotopy perturbation method and Chebyshev polynomials for solving a class of singular and hypersingular integral equations. Numerical Algebra, Control and Optimization, 2018, 8 (3) : 337-350. doi: 10.3934/naco.2018022
[5]

Freddy Dumortier, Robert Roussarie. Birth of canard cycles. Discrete and Continuous Dynamical Systems - S, 2009, 2 (4) : 723-781. doi: 10.3934/dcdss.2009.2.723
[6]

Freddy Dumortier, Robert Roussarie. Canard cycles with two breaking parameters. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 787-806. doi: 10.3934/dcds.2007.17.787
[7]

Fabio Camilli, Annalisa Cesaroni. A note on singular perturbation problems via Aubry-Mather theory. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 807-819. doi: 10.3934/dcds.2007.17.807
[8]

Nathan Glatt-Holtz, Mohammed Ziane. Singular perturbation systems with stochastic forcing and the renormalization group method. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1241-1268. doi: 10.3934/dcds.2010.26.1241
[9]

Stéphane Chrétien, Sébastien Darses, Christophe Guyeux, Paul Clarkson. On the pinning controllability of complex networks using perturbation theory of extreme singular values. application to synchronisation in power grids. Numerical Algebra, Control and Optimization, 2017, 7 (3) : 289-299. doi: 10.3934/naco.2017019
[10]

Morten Brøns. An iterative method for the canard explosion in general planar systems. Conference Publications, 2013, 2013 (special) : 77-83. doi: 10.3934/proc.2013.2013.77
[11]

Lana Horvat Dmitrović. Box dimension and bifurcations of one-dimensional discrete dynamical systems. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1287-1307. doi: 10.3934/dcds.2012.32.1287
[12]

Raphaël Danchin, Piotr B. Mucha. Divergence. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1163-1172. doi: 10.3934/dcdss.2013.6.1163
[13]

Luis Barreira. Dimension theory of flows: A survey. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3345-3362. doi: 10.3934/dcdsb.2015.20.3345
[14]

Eduard Marušić-Paloka, Igor Pažanin. Homogenization and singular perturbation in porous media. Communications on Pure and Applied Analysis, 2021, 20 (2) : 533-545. doi: 10.3934/cpaa.2020279
[15]

Jianhe Shen, Maoan Han. Bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Liénard systems. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 3085-3108. doi: 10.3934/dcds.2013.33.3085
[16]

Amal Attouchi, Eero Ruosteenoja. Gradient regularity for a singular parabolic equation in non-divergence form. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5955-5972. doi: 10.3934/dcds.2020254
[17]

Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881
[18]

Li Wang, Yang Li, Liwei Zhang. A differential equation method for solving box constrained variational inequality problems. Journal of Industrial and Management Optimization, 2011, 7 (1) : 183-198. doi: 10.3934/jimo.2011.7.183
[19]

Walter Allegretto, Yanping Lin, Shuqing Ma. On the box method for a non-local parabolic variational inequality. Discrete and Continuous Dynamical Systems - B, 2001, 1 (1) : 71-88. doi: 10.3934/dcdsb.2001.1.71
[20]

José Luis Bravo, Manuel Fernández, Armengol Gasull. Stability of singular limit cycles for Abel equations. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1873-1890. doi: 10.3934/dcds.2015.35.1873

2021 Impact Factor: 1.273

Tools

Metrics

  • PDF downloads (266)
  • HTML views (188)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy