zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
On hyperjerky systems. (English) Zbl 1148.37026
Summary: Hyperjerky systems have been recently investigated by {\it K. E. Clouverakis} and {\it J. C. Sprott} [Chaos Solitons Fractals 28, No. 3, 739--746 (2006; Zbl 1106.37024)]. These authors have focused on the appearance of chaos in rather simple functional forms of such scalar, nonlinear ordinary differential equations of fourth order. We discuss the connection between externally driven nonlinear oscillators and specific uni- and bidirectionally coupled systems of two autonomous oscillators. This offers an interesting reinterpretation of simple chaotic forms of hyperjerky systems. We also provide some criteria that exclude chaotic behavior in some classes of hyperjerky systems.

37D45Strange attractors, chaotic dynamics
34C28Complex behavior, chaotic systems (ODE)
37C10Vector fields, flows, ordinary differential equations
34C15Nonlinear oscillations, coupled oscillators (ODE)
Full Text: DOI
[1] Chlouverakis, K. E.; Sprott, J. C.: Chaotic hyperjerk systems, Chaos, solitons & fractals 28, 739-746 (2006) · Zbl 1106.37024
[2] Linz, S. J.: Nonlinear dynamical models and jerky motion, Am J phys 65, 523-526 (1997)
[3] Sprott, J. C.: Some simple chaotic Jerk functions, Am J phys 65, 537-543 (1997)
[4] Sprott, J. C.: Simplest dissipative chaotic flow, Phys lett A 228, 271-274 (1998) · Zbl 1043.37504 · doi:10.1016/S0375-9601(97)00088-1
[5] Linz, S. J.: Newtonian jerky dynamics: some general properties, Am J phys 66, 1109-1114 (1998)
[6] Eichhorn, R.; Linz, S. J.; Hänggi, P.: Transformations of nonlinear dynamical systems to jerky motion and its application to minimal chaotic flows, Phys rev E 58, 7151-7164 (1998)
[7] Eichhorn, R.; Linz, S. J.; Hänggi, P.: Simple polynomial classes of chaotic jerky dynamics, Chaos, solitons & fractals 13, 1-15 (2002) · Zbl 0993.37019 · doi:10.1016/S0960-0779(00)00237-X
[8] Sprott, J. C.; Linz, S. J.: Algebraically simple chaotic flows, Int J chaos theory appl 5, 3-22 (2000)
[9] Linz, S. J.; Sprott, J. C.: Elementary chaotic flow, Phys lett A 259, 240-245 (1999) · Zbl 0935.37008 · doi:10.1016/S0375-9601(99)00450-8
[10] Linz, S. J.: No-chaos criteria for certain jerky dynamics, Phys lett A 275, 204-210 (2000) · Zbl 1115.70309 · doi:10.1016/S0375-9601(00)00576-4
[11] Pais, A.; Uhlenbeck, G. E.: On field theories with non-localized action, Phys rev 19, 145-165 (1950) · Zbl 0040.13203
[12] Eichhorn, R.; Linz, S. J.; Hänggi, P.: Transformation invariance of Lyapunov exponents, Chaos, solitons & fractals 12, 1377-1383 (2001) · Zbl 1024.37024
[13] Mateos, J. L.: Chaotic transport and current reversal in deterministic ratchets, Phys rev lett 84, 258-261 (2000)
[14] Thompson, J. M. T.; Stewart, H. B.: Nonlinear dynamics and chaos, (1986) · Zbl 0601.58001
[15] Ueda, Y.: Randomly transitional phenomena in the system governed by Duffing’s equation, J stat phys 20, 181-196 (1979)
[16] Gottlieb, H. P. W.; Sprott, J. C.: Simplest driven conservative chaotic oscillator, Phys lett A 291, 385-389 (2001) · Zbl 1010.70020 · doi:10.1016/S0375-9601(01)00765-4
[17] Linz, S. J.: No-chaos criteria for certain classes of driven nonlinear oscillators, Acta phys Pol 34, 3741-3749 (2003)
[18] Linz, S. J.: A simple non-linear system: the oscillator with quadratic friction, Eur J phys 16, 67-72 (1995)
[19] Gilmore, R.: Topological analysis of chaotic dynamical systems, Rev mod phys 70, 1455-1530 (1998) · Zbl 1205.37002 · doi:10.1103/RevModPhys.70.1455
[20] Linz SJ., in preparation.