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1.
Traveling waves in the complex Ginzburg-Landau equation 总被引:1,自引:0,他引:1
A. Doelman 《Journal of Nonlinear Science》1993,3(1):225-266
Summary In this paper we consider a modulation (or amplitude) equation that appears in the nonlinear stability analysis of reversible
or nearly reversible systems. This equation is the complex Ginzburg-Landau equation with coefficients with small imaginary
parts. We regard this equation as a perturbation of the real Ginzburg-Landau equation and study the persistence of the properties
of the stationary solutions of the real equation under this perturbation. First we show that it is necessary to consider a
two-parameter family of traveling solutions with wave speedυ and (temporal) frequencyθ; these solutions are the natural continuations of the stationary solutions of the real equation. We show that there exists
a two-parameter family of traveling quasiperiodic solutions that can be regarded as a direct continuation of the two-parameter
family of spatially quasi-periodic solutions of the integrable stationary real Ginzburg-Landau equation. We explicitly determine
a region in the (wave speedυ, frequencyθ)-parameter space in which the weakly complex Ginzburg-Landau equation has traveling quasi-periodic solutions. There are two
different one-parameter families of heteroclinic solutions in the weakly complex case. One of them consists of slowly varying
plane waves; the other is directly related to the analytical solutions due to Bekki & Nozaki [3]. These solutions correspond
to traveling localized structures that connect two different periodic patterns. The connections correspond to a one-parameter
family of heteroclinic cycles in an o.d.e. reduction. This family of cycles is obtained by determining the limit behaviour
of the traveling quasi-periodic solutions as the period of the amplitude goes to ∞. Therefore, the heteroclinic cycles merge
into the stationary homoclinic solution of the real Ginzburg-Landau equation in the limit in which the imaginary terms disappear. 相似文献
2.
Xiao-Biao LinIgnacio B. Vivancos 《Journal of Differential Equations》2002,182(1):219-265
Vivancos and Minzoni (New Choatic behaviour in a singularly perturbed model, preprint) proposed a singularly perturbed rotating convection system to model the Earth's dynamo process. Numerical simulation shows that the perturbed system is rich in chaotic and periodic solutions. In this paper, we show that if the perturbation is sufficiently small, the system can only have simple heteroclinic solutions and two types of periodic solutions near the simple heteroclinic solutions. One looks like a figure “Delta” and the other looks like a figure “Eight”. Due to the fast - slow characteristic of the system, the reduced slow system has a relay nonlinearity (“Asymptotic Method in Singularly Perturbed Systems,” Consultants Bureau, New York and London, 1994) - solutions to the slow system are continuous but their derivative changes abruptly at certain junction surfaces. We develop new types of Melnikov integral and Lyapunov-Schmidt reduction methods which are suitable to study heteroclinic and periodic solutions for systems with relay nonlinearity. 相似文献
3.
Localized planar patterns arise in many reaction-diffusion models. Most of the paradigm equations that have been studied so
far are two-component models. While stationary localized structures are often found to be stable in such systems, travelling
patterns either do not exist or are found to be unstable. In contrast, numerical simulations indicate that localized travelling
structures can be stable in three-component systems. As a first step towards explaining this phenomenon, a planar singularly
perturbed three-component reaction-diffusion system that arises in the context of gas-discharge systems is analysed in this
paper. Using geometric singular perturbation theory, the existence and stability regions of radially symmetric stationary
spot solutions are delineated and, in particular, stable spots are shown to exist in appropriate parameter regimes. This result
opens up the possibility of identifying and analysing drift and Hopf bifurcations, and their criticality, from the stationary
spots described here. 相似文献
4.
H. Uecker 《Journal of Nonlinear Science》2001,11(2):89-121
Summary. We show the existence and stability of modulating multipulse solutions for a class of bifurcation problems given by a dispersive
Swift-Hohenberg type of equation with a spatially periodic forcing. Equations of this type arise as model problems for pattern
formation over unbounded weakly oscillating domains and, more specifically, in laser optics. As associated modulation equation,
one obtains a nonsymmetric Ginzburg-Landau equation which possesses exponentially stable stationary n—pulse solutions. The modulating multipulse solutions of the original equation then consist of a traveling pulselike envelope
modulating a spatially oscillating wave train. They are constructed by means of spatial dynamics and center manifold theory.
In order to show their stability, we use Floquet theory and combine the validity of the modulation equation with the exponential
stability of the n—pulses in the modulation equation. The analysis is supplemented by a few numerical computations.
In addition, we also show, in a different parameter regime, the existence of exponentially stable stationary periodic solutions
for the class of systems under consideration.
Received November 30, 1999; accepted December 4, 2000 Online publication March 23, 2001 相似文献
5.
M Balachandra 《Journal of Mathematical Analysis and Applications》1975,49(2):302-316
Asymptotic results are obtained for an initial-value problem for singularly perturbed systems. Existence of bounded solutions to singularly perturbed systems is deduced from the results of a previous paper [9]. These results significantly enlarge the class of limiting asymptotic solutions of singularly perturbed systems inasmuch as the limiting solutions satisfy equations more general than the classical reduced system. These results generalize those of Tikhonov [3] for the initial value problem, Flatto and Levinson [6] for the existence of periodic solutions and Hale and Seifert [7] for the existence of almost-periodic solutions. 相似文献
6.
We study a singularly perturbed system of partial di erential equations that models a one-dimensional array of coupled Chua's circuits. The PDE system is a natural generalization to the FitzHugh-Nagumo equation. In part I of the paper, we show that similar to the FitzHugh-Nagumo equation, the system has periodic traveling wave solutions formed alternatively by fast and slow flows. First, asymptotic method is used on the singular limit of the fast/slow systems to construct a formal periodic solution. Then, dynamical systems method is used to obtain an exact solution near the formal periodic soluion. In part II, we show that the system can have more complicated periodic and chaotic traveling wave solutions that do not exist in the FitzHugh-Nagumos equation. 相似文献
7.
In order to describe slow modulations in time and space of stable or slightly unstable spatially periodic stationary solutions
of pattern forming reaction–diffusion systems, so-called phase diffusion equations and Cahn–Hilliard equations can be derived
via multiple scaling analysis as formal approximation equations. In the case that these equations degenerate, waiting time
phenomena are well known to occur. In this paper, we prove that such waiting time phenomena can also occur approximately in
the original reaction–diffusion systems by proving estimates between the formal approximations and the exact solutions of
the original systems. 相似文献
8.
The mathematical model considered here of a neuron system is a chain of an arbitrary number m ≥ 2 of diffusion-coupled singularly perturbed nonlinear delay differential equations with Neumann-type conditions at the
endpoints. We study the existence, asymptotic behavior, and stability of relaxation periodic solutions of this system. 相似文献
9.
A. Ya. Kazakov 《Journal of Mathematical Sciences》1996,79(4):1212-1216
A singularly perturbed system can have points of quasi-intersection, where different stable and unstable branches meet. The
behavior of solutions in the vicinity of such points is considered. A criterion of “breaking down” is determined for a model
problem. Corresponding problems of bifurcation theory are discussed. Bibliography: 8 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 203, 1992, pp. 83–91.
Translated by A. Ya. Kazakov. 相似文献
10.
Asymmetric spike patterns are constructed for the two-component Schnakenburg reaction–diffusion system in the singularly perturbed limit of a small diffusivity of one of the components. For a pattern with k spikes, the construction yields k 1 spikes that have a common small amplitude and k 2 = k − k 1 spikes that have a common large amplitude. A k -spike asymmetric equilibrium solution is obtained from an arbitrary ordering of the small and large spikes on the domain. Explicit conditions for the existence and linear stability of these asymmetric spike patterns are determined using a combination of asymptotic techniques and spectral properties associated with a certain nonlocal eigenvalue problem. These asymmetric solutions are found to bifurcate from symmetric spike patterns at certain critical values of the parameters. Two interesting conclusions are that asymmetric patterns can exist for a reaction–diffusion system with spatially homogeneous coefficients under Neumann boundary conditions and that these solutions can be linearly stable on an O (1) time scale. 相似文献
11.
Explicitly Solvable Nonlocal Eigenvalue Problems and the Stability of Localized Stripes in Reaction‐Diffusion Systems 下载免费PDF全文
The transverse stability of localized stripe patterns for certain singularly perturbed two‐component reaction‐diffusion (RD) systems in the asymptotic limit of a large diffusivity ratio is analyzed. In this semi‐strong interaction regime, the cross‐sectional profile of the stripe is well‐approximated by a homoclinic pulse solution of the corresponding 1‐D problem. The linear instability of such homoclinic stripes to transverse perturbations is well known from numerical simulations to be a key mechanism for the creation of localized spot patterns. However, in general, owing to the difficulty in analyzing the associated nonlocal and nonself‐adjoint spectral problem governing stripe stability for these systems, it has not previously been possible to provide an explicit analytical characterization of these instabilities, including determining the growth rate and the most unstable mode within the band of unstable transverse wave numbers. Our focus is to show that such an explicit characterization of the transverse instability of a homoclinic stripe is possible for a subclass of RD system for which the analysis of the underlying spectral problem reduces to the study of a rather simple algebraic equation in the eigenvalue parameter. Although our simplified theory for stripe stability can be applied to a class of RD system, it is illustrated only for homoclinic stripe and ring solutions for a subclass of the Gierer–Meinhardt model and for a three‐component RD system modeling patterns of criminal activity in urban crime. 相似文献
12.
Wenjie Hu 《Mathematical Methods in the Applied Sciences》2019,42(12):4184-4196
The aim of this paper is to study the stability and Hopf bifurcation in a general class of differential equation with nonlocal delayed feedback that models the population dynamics of a two age structured spices. The existence of Hopf bifurcation is firstly established after delicately analyzing the eigenvalue problem of the linearized nonlocal equation. The direction of the Hopf bifurcation and stability of the bifurcated periodic solutions are then investigated by means of center manifold reduction. Subsequently, we apply our main results to explore the spatial‐temporal patterns of the nonlocal Mackey‐Glass equation. We obtain both spatially homogeneous and inhomogeneous periodic solutions and numerically show that the former is stable while the latter is unstable. We also show that the inhomogeneous periodic solutions will eventually tend to homogeneous periodic solutions after transient oscillations and increasing of the immature mobility constant will shorten the transient oscillation time. 相似文献
13.
In this paper, the problems of the nonlocal initial conditions for the singularly perturbed reaction diffusion systems are considered. Under suitable conditions, using the comparison theorem the asymptotic behavior of solutions for the initial boundary value problems are studied. 相似文献
14.
We consider a singularly perturbed system depending on two parameters with two (possibly the same) normally hyperbolic center
manifolds. We assume that the unperturbed system has an orbit that connects a hyperbolic fixed point on one center manifold
to a hyperbolic fixed point on the other. Then we prove some old and new results concerning the persistence of these connecting
orbits and apply the results to find examples of systems in dimensions greater than three that possess Sil’nikov saddle-focus
homoclinic orbits.
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 1, pp. 28–55, January, 2008. 相似文献
15.
K. R. Schneider 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》1983,34(2):236-240
We show that the existence of wave trains with high velocity of generalized reaction-diffusion equations can be easily established by using a theorem of D. V. Anosov on the existence of periodic solutions of singularly perturbed differential systems. 相似文献
16.
In this paper the singularly perturbed initial boundary value problems for a nonlocal reaction diffusion system are considered. Using the iteration method and the comparison theorem, the existence and asymptotic behavior of solutions for the problem are studied. 相似文献
17.
In this paper we study nonlinear second-order differential inclusions involving the ordinary vectorp-Laplacian, a multivalued maximal monotone operator and nonlinear multivalued boundary conditions. Our framework is general
and unifying and incorporates gradient systems, evolutionary variational inequalities and the classical boundary value problems,
namely the Dirichlet, the Neumann and the periodic problems. Using notions and techniques from the nonlinear operator theory
and from multivalued analysis, we obtain solutions for both the ‘convex’ and ‘nonconvex’ problems. Finally, we present the
cases of special interest, which fit into our framework, illustrating the generality of our results. 相似文献
18.
数学物理学报 《数学物理学报(B辑英文版)》1999,19(3):1
The
problems of the nonlocal boundary conditions for the singularly perturbed reaction
diffusion systems are considered. Under suitable conditions, using the comparison theorem
the asymptotic behavior of solution for the initial boundary value problems are studied. 相似文献
19.
I. I. Klevchuk 《Ukrainian Mathematical Journal》1995,47(8):1169-1177
We consider a system of singularly perturbed differential-difference equations with periodic right-hand sides. A representation of the integral manifold of this system is obtained. The bifurcation of an invariant torus from an equilibrium state and subfurcation of periodic solutions are studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 8, pp. 1022–1028, August, 1995. 相似文献
20.
V. F. Butuzov 《Computational Mathematics and Mathematical Physics》2011,51(1):40-50
For singularly perturbed parabolic problems, asymptotic expansions of time-periodic solutions with boundary layers in a neighborhood
of interval’s endpoints are constructed and justified in the case where the degenerate equation has a double or a triple root. 相似文献