zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Hyperbolic chaos in a system of resonantly coupled weakly nonlinear oscillators. (English) Zbl 1242.34064
Summary: We show that a hyperbolic chaos can be observed in resonantly coupled oscillators near a Hopf bifurcation, described by normal-form-type equations for complex amplitudes. The simplest example consists of four oscillators, comprising two alternatively activated, due to an external periodic modulation, pairs. In terms of the stroboscopic Poincaré map, the phase differences change according to an expanding Bernoulli map that depends on the coupling type. Several examples of hyperbolic chaos for different types of coupling are illustrated numerically.
MSC:
34C15Nonlinear oscillations, coupled oscillators (ODE)
34C28Complex behavior, chaotic systems (ODE)
34C23Bifurcation (ODE)
References:
[1]Pikovsky, A.; Rosenblum, M.; Kurths, J.: Synchronization: A universal concept in nonlinear science, (2001)
[2]Kapitaniak, T.; Steeb, W. H.: Phys. lett. A, Phys. lett. A 152, 33 (1991)
[3]Katok, A.; Hasselblatt, B.: Introduction to the modern theory of dynamical systems, (1995)
[4]Smale, S.: Bull. amer. Math. soc., Bull. amer. Math. soc. 13 (1967)
[5]Plykin, R.: Math. USSR sbornik, Math. USSR sbornik 23, 233 (1974)
[6]Ruelle, D.; Takens, F.: Comm. math. Phys., Comm. math. Phys. 20, 167 (1971)
[7]Kuznetsov, S. P.: Phys. rev. Lett., Phys. rev. Lett. 95, 144101 (2005)
[8]Kuznetsov, S. P.; Sataev, I. R.: Phys. lett. A, Phys. lett. A 365, 97 (2007)
[9]Kuznetsov, S. P.; Seleznev, E. P.: J. exp. Theor. phys., J. exp. Theor. phys. 102, 355 (2006)
[10]Isaeva, O. B.; Jalnine, A. Y.; Kuznetsov, S. P.: Phys. rev. E, Phys. rev. E 74, 046207 (2006)
[11]Kuznetsov, S. P.; Pikovsky, A.: Physica D, Physica D 232, 87 (2007)
[12]Bogoliubov, N. N.; Mitropolsky, Y. A.: Asymptotic methods in the theory of nonlinear oscillations, (1961) · Zbl 0151.12201
[13]Guckenheimer, J.; Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, (1986)
[14]Kaplan, J. L.; Yorke, J. A.: Functional differential equations and approximations of fixed point, Lecture in mathematics 730, 204 (1979)
[15]Grassberger, P.; Procaccia, I.: Physica D, Physica D 9, 189 (1983)
[16]Cross, M. C.; Zumdieck, A.; Lifshitz, R.; Rogers, J. L.: Phys. rev. Lett., Phys. rev. Lett. 93, 224101 (2004)
[17]Zalalutdinov, M.; Zehnder, A.; Olkhovets, A.; Turner, S.; Sekaric, L.; Ilic, B.; Czaplewski, D.; Parpia, J. M.; Craighead, H. G.: Appl. phys. Lett., Appl. phys. Lett. 79, 695 (2001)
[18]Ott, E.; Antonsen, Th.: Chaos, Chaos 18, 037113 (2008)