Abstract
We consider a singularly perturbed system of differential equations of the form ε u′=g(u,v,λ), v′=f(u,v,λ), where (u,v)∈R 3, 0<ε≪1, and λ is a set of parameters. Such a system describes a modified Chua’s circuit with mixed-mode oscillations (MMOs). MMOs consist of a series of small-amplitude oscillations (canard solutions) and large-amplitude relaxations. In the paper we provide a series of both numerical and analytical analyses of the singularly perturbed system for the modified Chua’s circuit with nonlinear f and g. In particular, we analyze the occurrence of the Farey sequence \(\it L^{s}\), where \(\it L\) and \(\it s\) are the numbers of large and small oscillations, respectively.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Abshagen, J.M. Lopez, F. Marques, G. Pfister, Bursting dynamics due to a homoclinic cascade in Taylor–Couette flow. J. Fluid Mech. 613, 357–384 (2008)
M. Brøns, M. Krupa, M. Wechselberger, Mixed mode oscillations due to the generalized canard phenomenon. Fields Inst. Commun. 49, 39–63 (2006)
D. Cafagna, G. Grassi, Generation of chaotic beats in a modified Chua’s circuit, Part I: Dynamical behavior & Part II: Circuit design. Nonlinear Dyn. 44, 91–108 (2005)
Focus issue, Mixed mode oscillations: experiment, computation, and analysis. Chaos 18 (2008)
M.T.M. Koper, Bifurcation of mixed-mode oscillations in a three-variable autonomous Van der Pol-Duffing model with a cross-shaped phase diagram. Physica D 80, 72–94 (1995)
M. Krupa, P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions. SIAM J. Math. Anal. 33, 286–314 (2001)
M. Krupa, P. Szmolyan, Relaxation oscillation and canard explosion. J. Differ. Equ. 174, 312–368 (2001)
M. Krupa, N. Popovic, N. Kopell, Mixed-mode oscillations in three time-scale systems: a prototypical example. SIAM J. Appl. Dyn. Syst. 7, 361–420 (2008)
R. Li, Z. Duan, B. Wang, G. Chen, A modified Chua’s circuit with an attraction-repulsion function. Int. J. Bifurc. Chaos (IJBC) 18, 1865–1888 (2008)
W. Marszalek, Fold points and singularities in Hall MHD differential-algebraic equations. IEEE Trans. Plasma Sci. 37, 254–260 (2009)
W. Marszalek, Z.W. Trzaska, New solutions of resistive MHD systems with singularity induced bifurcations. IEEE Trans. Plasma Sci. 35, 509–515 (2007)
W. Marszalek, T. Amdeberhan, R. Riaza, Singularity crossing phenomena in DAEs: a two-phase fluid flow application case study. Comput. Math. Appl. 49, 303–319 (2005)
R. Riaza, S.L. Campbell, W. Marszalek, On singular equilibria of index-1 DAEs. Circuits Syst. Signal Process. 19, 131–157 (2000)
P. Szmolyan, M. Wechselberger, Canards in R 3. J. Differ. Equ. 177, 419–453 (2001)
F. Tang, L. Wang, An adaptive active control for the modified Chua’s circuit. Phys. Lett. A 346, 342–346 (2005)
Q. Wang, Y. Chen, Generalized Q-S (lag, anticipated and complete) synchronization in modified Chua’s circuit and Hindmarsh-Rose systems. Appl. Math. Comput. 181, 48–56 (2006)
M. Wechselberger, Existence and bifurcations of canards in R 3 in the case of a folded node. SIAM J. Appl. Dyn. Syst. 4, 101–139 (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Marszalek, W., Trzaska, Z. Mixed-Mode Oscillations in a Modified Chua’s Circuit. Circuits Syst Signal Process 29, 1075–1087 (2010). https://doi.org/10.1007/s00034-010-9190-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00034-010-9190-8