• 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

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 & 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

Eduard Marušić-Paloka, Igor Pažanin. Homogenization and singular perturbation in porous media. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020279

[2]

Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305

[3]

Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344

[4]

Manxue You, Shengjie Li. Perturbation of Image and conjugate duality for vector optimization. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020176

[5]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[6]

Nguyen Anh Tuan, Donal O'Regan, Dumitru Baleanu, Nguyen H. Tuan. On time fractional pseudo-parabolic equations with nonlocal integral conditions. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020109

[7]

Zi Xu, Siwen Wang, Jinjin Huang. An efficient low complexity algorithm for box-constrained weighted maximin dispersion problem. Journal of Industrial & Management Optimization, 2021, 17 (2) : 971-979. doi: 10.3934/jimo.2020007

[8]

Chungang Shi, Wei Wang, Dafeng Chen. Weak time discretization for slow-fast stochastic reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021019

[9]

Jianfeng Huang, Haihua Liang. Limit cycles of planar system defined by the sum of two quasi-homogeneous vector fields. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 861-873. doi: 10.3934/dcdsb.2020145

[10]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

[11]

Lisa Hernandez Lucas. Properties of sets of Subspaces with Constant Intersection Dimension. Advances in Mathematics of Communications, 2021, 15 (1) : 191-206. doi: 10.3934/amc.2020052

[12]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[13]

Kung-Ching Chang, Xuefeng Wang, Xie Wu. On the spectral theory of positive operators and PDE applications. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3171-3200. doi: 10.3934/dcds.2020054

[14]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[15]

Craig Cowan, Abdolrahman Razani. Singular solutions of a Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 621-656. doi: 10.3934/dcds.2020291

[16]

Xinfu Chen, Huiqiang Jiang, Guoqing Liu. Boundary spike of the singular limit of an energy minimizing problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3253-3290. doi: 10.3934/dcds.2020124

[17]

Ziang Long, Penghang Yin, Jack Xin. Global convergence and geometric characterization of slow to fast weight evolution in neural network training for classifying linearly non-separable data. Inverse Problems & Imaging, 2021, 15 (1) : 41-62. doi: 10.3934/ipi.2020077

[18]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[19]

Sabira El Khalfaoui, Gábor P. Nagy. On the dimension of the subfield subcodes of 1-point Hermitian codes. Advances in Mathematics of Communications, 2021, 15 (2) : 219-226. doi: 10.3934/amc.2020054

[20]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (119)
  • HTML views (186)
  • Cited by (0)

Other articles
by authors

[Back to Top]