
-
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
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 |
In previous work [
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. |






test case |
|
||||
factor |
test case |
|
||||
factor |
example system | (11) | (12) | (13) |
theoretical box dim. | 0 | ||
num. of digits of prec. | 170 | 60 | 150 |
computed orbit size |
500 | 10000 | 2000 |
test case |
|||
test case |
|||
test case |
|||
test case |
|||
test case |
example system | (11) | (12) | (13) |
theoretical box dim. | 0 | ||
num. of digits of prec. | 170 | 60 | 150 |
computed orbit size |
500 | 10000 | 2000 |
test case |
|||
test case |
|||
test case |
|||
test case |
|||
test case |
[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] |
Raphaël Danchin, Piotr B. Mucha. Divergence. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1163-1172. doi: 10.3934/dcdss.2013.6.1163 |
[12] |
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 |
[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] |
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 |
[19] |
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 |
[20] |
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 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]