Complex dynamics in the 1:3 spatial resonance
Introduction
The analysis of wave–wave interactions plays a vital role in understanding the behavior of continuous systems [1]. In Hamiltonian systems such interactions are responsible for mode conversion as well as various types of instabilities. In driven dissipative systems related processes lead to wave number selection [2]. Even in systems, such as convection, that do not support wave-like disturbances, mode–mode interactions are often responsible for the presence of unexpected dynamics. The basic mechanism is described most clearly by Dangelmayr [3] in the context of the interaction between two spatial modes with wave numbers in the ratio m:, assumed to be coprime. For problems on the real line with periodic boundary conditions and symmetry under spatial reflection (x→−x) the resulting problem is equivariant under the standard action on of the group O(2) of rotations and reflections of a circle. If m>1 pure modes bifurcate from the trivial state followed by secondary bifurcations to mixed modes which in turn can undergo either a Hopf bifurcation leading to standing oscillations, or a parity-breaking bifurcation leading to drifting mixed modes, i.e., to traveling waves. The analysis also applies to problems with Neumann (and in appropriate cases, Dirichlet) boundary conditions since such problems have hidden O(2) symmetry [4], [5], but the traveling waves are then absent. On the real line the competing wave numbers may be close to one another and the wave number selection process is then described by the Ginzburg–Landau equation, and is the consequence of the so-called Eckhaus or sideband instability. As discussed by Tuckerman and Barkley [6], on a large but finite domain the Ginzburg–Landau equation gives rise to a number of mode interactions between adjacent wave numbers that are responsible for the presence of multiple stable steady states at finite amplitude, familiar from studies of finite domains [7], [8]. However, as shown by Mizushima and Fujimura [9] the familiar Eckhaus boundary for convection rolls is substantially modified by the presence of strong resonances, particularly those involving the wave number 1. Of these the 1:2 steady-state interaction has been extensively studied and is known, under appropriate circumstances, to lead to attracting structurally stable heteroclinic cycles [10], [11]. Mizushima and Fujimura pointed out, however, that in Rayleigh–Bénard convection with identical boundary conditions at the top and bottom of the layer (and no non-Boussinesq effects) it is the 1:3 resonance that plays an important role. This is because the resulting midplane reflection symmetry pushes the resonant terms for the 1:2 resonance to fifth order [12], while the resonant terms for the 1:3 resonance remain at third order. Prat et al. [13] show by direct integration of the two-dimensional equations describing Rayleigh–Bénard convection with periodic boundary conditions that the 1:3 resonance does indeed organize the wave number selection process in large regions of parameter space, despite the fact that it is always shielded by instabilities to other modes. We use these results to motivate our interest in the 1:3 resonance. Related issues arise in the theory of unstable electrostatic waves when the equilibrium electron velocity distribution function is reflection-symmetric [14].
Explicit computation of the normal form coefficients for the 1:3 resonance in Rayleigh–Bénard convection [9] indicates that interesting time-dependence may be present for sufficiently low Prandtl numbers. Although this is so even with Neumann boundary conditions [8], with periodic boundary conditions Mizushima and Fujimura noted the presence of both periodic and quasi-periodic traveling waves. We show below that the solutions of the equations describing the 1:3 mode interaction are in fact much richer. In particular we describe, via careful numerical studies, the fate of the quasi-periodic oscillations, both in the case studied by Mizushima and Fujimura, and in others. We uncover a number of new global bifurcations involving circles of steady states and of standing waves, as well as the trivial state. Continuous deformation from one heteroclinic cycle to another is found to explain a number of features of the bifurcation diagrams determined numerically. This explanation is supported by analytically constructed return maps which yield, as a byproduct, conditions on the eigenvalues and Floquet multipliers of the various solution types for the presence of stable complex dynamics. In contrast to the behavior identified in [10], [11] for the 1:2 resonance much of the behavior of interest in the present problem is associated with structurally unstable heteroclinic cycles, although structurally stable heteroclinic cycles can also be present.
This paper is organized as follows. In Section 2 we construct the truncated normal form equations studied in this paper, and rewrite these in several equivalent but useful forms. In 3 Principal solutions, 4 Local bifurcations we summarize the properties of the simplest solutions and of the bifurcations leading to them. Section 5 looks at the geometry of the various representations of the vector field. Section 6 summarizes our numerical results for five choices of the normal form coefficients, chosen to produce nontrivial dynamics. In Section 7 we describe our interpretation of these results, and support it with a detailed discussion of the properties of the various return maps associated with the presence of the global connections identified in Section 6. These results are extended to other 1:n resonances in Section 8 and the results are summarized in Section 9.
Section snippets
The amplitude equations
We consider a two-parameter family of vector fields on that are equivariant under the following representation of the symmetry group O(2):resulting from the effect of horizontal translations , and reflection R:x→−x on scalar fields of the formHere ‘⋯’ indicates higher harmonics whose amplitudes can be expressed in terms of the amplitudes of the primary modes. The equivariance requirement
Principal solutions
We now summarize the main types of simple equilibrium states. These solutions are described by Mizushima and Fujimura [9] and are also realized as special cases of the more general discussion given by Dangelmayr [3]. These fall into four catagories.
- 1.
Trivial state (z1,z3)=0. The presence of this solution is forced by the symmetry O(2). Its stability is specified by the two eigenvalues μ1 and μ3, each of which occurs with multiplicity 2.
- 2.
Pure modes (Pφ). The pure modes are given by
Local bifurcations
The nature of the interactions between the different solutions as μ1 and μ3 are varied depends on the values of the coefficients . In the following we discuss several of the more interesting choices in some detail. It is possible, nonetheless, to make some general statements about the possible bifurcation diagrams which result from traversing a given path in (μ1,μ3) space. For this purpose it is convenient to identify all the P and MM states with a representative solution,
Geometric properties
We now briefly describe the geometry and physical interpretation of the phase space associated with the four-dimensional system (8) and the three-dimensional system (12).
In the four-dimensional system the most significant features are a result of the O(2) symmetry of these equations. The real subspace (y1=y3=0) is fixed by R and the action of Tφ on this invariant two-dimensional subspace generates an S1 family of conjugate subspaces which are also fixed by reflection (about an appropriate
Numerical results
In this section we present detailed results for five choices of the coefficients dij, assuming always that σ=−1, i.e., that TW are possible. The most direct physical significance belongs to the set of coefficients referred to below as case B. These values correspond to two-dimensional Rayleigh–Bénard convection with no-slip fixed temperature boundary conditions at the top and bottom in the limit of small Prandtl number, and were obtained by Mizushima and Fujimura [9]. The remaining sets (A,
Interpretation and analysis
In this section we discuss in more detail the origin of the behavior described in Section 6. Our interpretation assumes that this behavior is in some sense typical. Support for this belief is provided in Fig. 13 which shows that the isola I5 for case B bears substantial similarity to the corresponding isola for case A (Fig. 5). In all the cases described the complex behavior occurs near the Hopf bifurcation to SW on the MM0 branch. In this parameter regime the origin is unstable to the pure
General 1:n resonance
In this section we briefly discuss the situation for a general 1:n spatial resonance and show that one can expect to find complex dynamics similar to that found for the 1:3 resonance.
The representation of the symmetry group O(2) appropriate for a general 1:n mode interaction isand the most general smooth vector field equivariant under (73) has the formwhere the invariants , and w are
Discussion
In this paper we described in some detail the unexpected complex dynamics that are present in the unfolding of a common type of mode interaction problem: the interaction of two steady modes with spatial wave numbers in the ratio 1:n in the presence of O(2) symmetry. Problems of this type have a variety of applications depending on the origin of the O(2) symmetry. This symmetry can come about through the imposition of periodic boundary conditions on a problem posed on the real line or from the
Acknowledgements
This work was supported in part by the National Science Foundation under Grant No. DMS-9703684.
References (31)
- et al.
Bifurcation analysis of the Eckhaus instability
Physica D
(1990) - et al.
Heteroclinic cycles and modulated travelling waves in systems with O(2) symmetry
Physica D
(1988) O(2)-symmetric bifurcation theory for convection rolls
Physica D
(1987)- et al.
Šil’nikov–saddle-node interaction near a codimension-2 bifurcation: laser with injected signal
Physica D
(1997) - et al.
The non-transverse Shil’nikov–Hopf bifurcation: uncoupling of homoclinic orbits and homoclinic tangencies
Physica D
(1999) - et al.
Period-doubling mode interactions with circular symmetry
Physica D
(1990) Normal forms for driven surface waves: boundary conditions, symmetry, and genericity
Physica D
(1991)- A.D.D. Craik, Wave Interactions and Fluid Flows, Cambridge University Press, Cambridge,...
- et al.
Symmetry and symmetry-breaking bifurcations in fluid mechanics
Ann. Rev. Fluid Mech.
(1991) Steady-state mode interactions in the presence of O(2)-symmetry
Dyn. Stability Syst.
(1986)