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)
Heteroclinic dynamics in the nonlocal parametrically driven nonlinear Schrödinger equation. (English) Zbl 0988.35155
Summary: Faraday waves are described, under appropriate conditions, by a damped nonlocal parametrically driven nonlinear Schrödinger equation. As the strength of the applied forcing increases this equation undergoes a sequence of transitions to chaotic dynamics. The origin of these transitions is explained using a careful study of a two-mode Galerkin truncation and linked to the presence of heteroclinic connections between the trivial state and spatially periodic standing waves. These connections are associated with cascades of gluing and symmetry-switching bifurcations; such bifurcations are located in the partial differential equations as well.

MSC:
35Q55NLS-like (nonlinear Schrödinger) equations
37J45Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
37J25Stability problems (finite-dimensional Hamiltonian etc. systems)
Software:
AUTO
WorldCat.org
Full Text: DOI
References:
[1] Riecke, H.; Crawford, J. D.; Knobloch, E.: Time-modulated oscillatory convection. Phys. rev. Lett. 61, 1942-1945 (1988)
[2] Rehberg, I.; Rasenat, S.; Fineberg, J.; De La Torre Juárez, M.; Steinberg, V.: Temporal modulation of traveling waves. Phys. rev. Lett. 61, 2449-2452 (1988)
[3] Riecke, H.; Silber, M.; Kramer, L.: Temporal forcing of small-amplitude waves in anisotropic systems. Phys. rev. E 49, 4100-4113 (1994)
[4] Tennakoon, S. G. K.; Andereck, C. D.; Hegseth, J. J.; Riecke, H.: Temporal modulation of traveling waves in the flow between rotating cylinders with broken azimuthal symmetry. Phys. rev. E 54, 5053-5065 (1996)
[5] Miles, J.; Henderson, D.: Parametrically forced surface waves. Ann. rev. Fluid mech. 22, 143-165 (1990)
[6] Vega, J. M.; Knobloch, E.; Martel, C.: Nearly inviscid Faraday waves in annular containers of moderately large aspect ratio. Physica D 154, 313-336 (2001) · Zbl 1029.76017
[7] Martel, C.; Knobloch, E.; Vega, J. M.: Dynamics of counterpropagating waves in parametrically forced systems. Physica D 137, 94-123 (2000) · Zbl 0949.35019
[8] Knobloch, E.; De Luca, J.: Amplitude equations for travelling wave convection. Nonlinearity 3, 975-980 (1990) · Zbl 0717.35070
[9] Knobloch, E.; Gibbon, J. D.: Coupled NLS equations for counter-propagating waves in systems with reflection symmetry. Phys. lett. A 154, 353-356 (1991)
[10] Promislow, K.; Kutz, J. N.: Bifurcation and asymptotic stability in the large detuning limit of the optical parametric oscillator. Nonlinearity 13, 675-698 (2000) · Zbl 0973.78026
[11] E. Martı\acute{}n, C. Martel, J.M. Vega, Drift instability of standing Faraday waves, Preprint, 2001. · Zbl 1035.76020
[12] Duan, J.; Ly, H. V.; Titi, E. S.: The effects of nonlocal interactions on the dynamics of the Ginzburg--Landau equation. Z. angew. Math. phys. 47, 432-455 (1996) · Zbl 0861.35112
[13] Ghidaglia, J. M.: Finite-dimensional behaviour for weakly damped driven Schrödinger equations. Ann. inst. H. Poincaré anal. Non-linéaire 5, 365-405 (1988) · Zbl 0659.35019
[14] Wang, X.: An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attractors. Physica D 88, 167-175 (1995) · Zbl 0900.35372
[15] Goubet, O.: Regularity of attractor for a weakly damped nonlinear Schrödinger equation. Appl. anal. 60, 99-119 (1996) · Zbl 0872.35100
[16] Oliver, M.; Titi, E.: Analyticity of the attractor and the number of determining modes for a weakly damped driven nonlinear Schrödinger equation. Indiana univ. Math. J. 47, 49-73 (1998) · Zbl 0912.35144
[17] E. Knobloch, M.R.E. Proctor, N.O. Weiss, Finite-dimensional description of doubly diffusive convection, in: G.R. Sell, C. Foias, R. Temam (Eds.), Turbulence in Fluid Flows: A Dynamical Systems Approach, IMA Volumes in Mathematics and its Applications, Vol. 55, Springer, New York, 1993, pp. 59--72. · Zbl 0788.76077
[18] Keefe, L. R.: Dynamics of perturbed wavetrain solutions to the Ginzburg--Landau equation. Stud. appl. Math. 73, 91-153 (1985) · Zbl 0575.76055
[19] Rodriguez, J. D.; Schell, M.: Global bifurcations into chaos in systems with $SO(2)$ symmetry. Phys. lett. A 146, 25-31 (1990)
[20] Doelman, A.: Finite-dimensional models of the Ginzburg--Landau equation. Nonlinearity 4, 231-250 (1991) · Zbl 0841.35007
[21] Bishop, A. R.; Forest, M. G.; Mclaughlin, D. W.; Ii, E. A. Overman: A modal representation of chaotic attractors for the driven, damped pendulum chain. Phys. lett. A 144, 17-25 (1990)
[22] Kovačič, G.; Wiggins, S.: Orbits homoclinic to resonances, with an application to chaos in a model of the forced and damped sine-Gordon equation. Physica D 57, 185-225 (1992) · Zbl 0755.35118
[23] Haller, G.; Wiggins, S.: Orbits homoclinic to resonances: the Hamiltonian case. Physica D 66, 298-346 (1993) · Zbl 0791.34041
[24] Haller, G.; Wiggins, S.: N-pulse homoclinic orbits in perturbations of resonant Hamiltonian systems. Arch. rat. Mech. anal. 130, 25-101 (1995) · Zbl 0829.58016
[25] Haller, G.; Wiggins, S.: Multi-pulse jumping orbits and homoclinic trees in a modal truncation of the damped-forced nonlinear Schrödinger equation. Physica D 85, 311-347 (1995) · Zbl 0890.58048
[26] Howard, L. N.; Krishnamurti, R.: Large-scale flow in turbulent convection: a mathematical model. J. fluid mech. 170, 385-410 (1986) · Zbl 0606.76053
[27] Rucklidge, A. M.; Matthews, P. C.: Analysis of the shearing instability in nonlinear convection and magnetoconvection. Nonlinearity 9, 311-351 (1996) · Zbl 0888.34035
[28] Knobloch, E.: Onset of zero Prandtl number convection. J. phys. II France 2, 995-999 (1992)
[29] Jolly, M. S.; Kevrekidis, I. G.; Titi, E. S.: Approximate inertial manifolds for the Kuramoto--Sivashinsky equation: analysis and computations. Physica D 44, 38-60 (1990) · Zbl 0704.58030
[30] E.J. Doedel, A.R. Champneys, T.F. Fairgrieve, Y. Kuznetsov, B. Sandstede, X.J. Wang, AUTO 97: Continuation and bifurcation software for ordinary differential equations, 1997, pub/doedel/auto at ftp.cs.concordia.ca.
[31] B. Ermentrout, XPPAUT, Dynamical systems software with continuation and bifurcation capabilities, 2000, /pub/bardware at ftp.math.pit.edu.
[32] Knobloch, E.; Weiss, N. O.: Bifurcations in a model of double-diffusive convection. Phys. lett. A 85, 127-130 (1981)
[33] Glendinning, P.; Sparrow, C.: Local and global behavior near homoclinic orbits. J. stat. Phys. 35, 645-696 (1984) · Zbl 0588.58041
[34] S. Wiggins, Global Bifurcations and Chaos: Analytical Methods, Springer, New York, 1988. · Zbl 0661.58001
[35] Swift, J. W.; Wiesenfeld, K.: Suppression of period doubling in symmetric systems. Phys. rev. Lett. 52, 705-708 (1984)
[36] Glendinning, P.: Bifurcations near homoclinic orbits with symmetry. Phys. lett. A 103, 163-166 (1984)
[37] Tresser, C.: About some theorems by L.P. Shil’nikov. Ann. inst. H. Poincaré anal. Non-linéaire 40, 440-461 (1994)
[38] C.T. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Springer, New York, 1982. · Zbl 0504.58001
[39] Ashwin, P.; Buescu, J.; Stewart, I.: From attractor to saddle: a tale of transverse instability. Nonlinearity 9, 703-738 (1996) · Zbl 0887.58034
[40] Fowler, A. C.; Sparrow, C. T.: Bifocal homoclinic orbits in four dimensions. Nonlinearity 4, 1159-1183 (1991) · Zbl 0741.34017
[41] Arnéodo, A.; Coullet, P.; Tresser, C.: A possible new mechanism for the onset of turbulence. Phys. lett. A 81, 197-201 (1981)
[42] Lyubimov, D. V.; Zaks, M. A.: Two mechanisms of the transition to chaos in finite-dimensional models of convection. Physica D 9, 52-64 (1983) · Zbl 0598.58029
[43] D.W. McLaughlin, E.A. Overman II, S. Wiggins, C. Xiong, Homoclinic orbits in a four-dimensional model of a perturbed NLS equation: a geometric singular perturbation study, in: Dynamics Reported, Vol. 5, Springer, New York, 1996, pp. 190--287. · Zbl 0848.35125
[44] Bishop, A. R.; Flesch, R.; Forest, M. G.; Mclaughlin, D. W.; Ii, E. A. Overman: Correlations between chaos in a perturbed sine-Gordon equation and a truncated model system. SIAM J. Math. anal. 21, 1511-1536 (1990) · Zbl 0718.35086