Showing papers in "Nonlinearity in 1997"

Pattern formation in the one-dimensional Gray - Scott model

Abstract: In this work, we analyse a pair of one-dimensional coupled reaction-diffusion equations known as the Gray-Scott model, in which self-replicating patterns have been observed. We focus on stationary and travelling patterns, and begin by deriving the asymptotic scaling of the parameters and variables necessary for the analysis of these patterns. Single-pulse and multi- pulse stationary waves are shown to exist in the appropriately scaled equations on the infinite line. A (single) pulse is a narrow interval in which the concentration U of one chemical is small, while that of the second, V , is large, and outside of which the concentration U tends (slowly) to the homogeneous steady state U 1, while V is everywhere close to V 0. In addition, we establish the existence of a plethora of periodic steady states consisting of periodic arrays of pulses interspersed by intervals in which the concentration V is exponentially small and U varies slowly. These periodic states are spatially inhomogeneous steady patterns whose length scales are determined exclusively by the reactions of the chemicals and their diffusions, and not by other mechanisms such as boundary conditions. A complete bifurcation study of these solutions is presented. We also establish the non-existence of travelling solitary pulses in this system. This non-existence result reflects the system's degeneracy and indicates that some event, for example pulse splitting, 'must' occur when two pulses move apart from each other (as has been observed in simulations): these pulses evolve towards the non-existent travelling solitary pulses. The main mathematical techniques employed in this analysis of the stationary and travelling patterns are geometric singular perturbation theory and adiabatic Melnikov theory. Finally, the theoretical results are compared to those obtained from direct numerical simulation of the coupled partial differential equations on a 'very large' domain, using a moving grid code. It has been checked that the boundaries do not influence the dynamics. A subset of the family of stationary single pulses appears to be stable. This subset determines the boundary of a region in parameter space in which the self-replicating process takes place. In that region, we observe that the core of a time-dependent self-replicating pattern turns out to be precisely a stationary periodic pulse pattern of the type that we construct. Moreover, the simulations reveal some other essential components of the pulse-splitting process and provide an important guide to further analysis.

Modulational instability: first step towards energy localization in nonlinear lattices

Abstract: We study the modulational instability in discrete lattices and we show how the discreteness drastically modifies the stability condition. Analytical and numerical results are in very good agreement. We predict also the evolution of a linear wave in the presence of noise and we show that modulational instability is the first step towards energy localization.

Spatiotemporal chaos in terms of unstable recurrent patterns

Abstract: Spatiotemporally chaotic dynamics of a Kuramoto - Sivashinsky system is described by means of an infinite hierarchy of its unstable spatiotemporally periodic solutions. An intrinsic parametrization of the corresponding invariant set serves as an accurate guide to the high-dimensional dynamics, and the periodic orbit theory yields several global averages characterizing the chaotic dynamics.

Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators

Abstract: We address the issue of spatially localized periodic oscillations in coupled networks - so-called discrete breathers - in a general context. This context is concerned with general conditions which allow continuation of periodic solutions of vector fields. One advantage of our approach is to encompass in the same mathematical framework the cases of conservative and dissipative systems. An essential feature is that we allow the period to vary. In particular, we deduce existence of discrete breathers in networks where each site has an equilibrium and some sites have a limit cycle, and in Hamiltonian networks without requiring local anharmonicity. The latter case is dealt with by considering the persistence of families of periodic solutions in the more general context of systems with an integral, not just Hamiltonian ones.

Computer assisted proof of chaos in the Rössler equations and in the Hénon map

Abstract: We introduce horseshoe-type mappings which are geometrically similar to Smale's horseshoes. For such mappings we prove by means of the fixed point index the existence of chaotic dynamics - the semi-conjugacy to the shift on a finite number of symbols. Our theorem does not require any assumptions concerning derivatives, it is a purely topological result. The assumptions of our theorem are then rigorously verified by computer assisted computations for the classical Henon map and for classical Rossler equations.

Finite-time aggregation into a single point in a reaction - diffusion system

Abstract: We consider the following system: which has been used as a model for various phenomena, including motion of species by chemotaxis and equilibrium of self-attracting clusters. We show that, in space dimension N = 3, (S) possess radial solutions that blow-up in a finite time. The asymptotic behaviour of such solutions is analysed in detail. In particular, we obtain that the profile of any such solution consists of an imploding, smoothed-out shock wave that collapses into a Dirac mass when the singularity is formed. The differences between this type of behaviour and that known to occur for blowing-up solutions of (S) in the case N = 2 are also discussed.

Computing Lyapunov spectra with continuous Gram - Schmidt orthonormalization

Abstract: We present a straightforward and reliable continuous method for computing the full or partial Lyapunov spectrum associated with a dynamical system specified by a set of differential equations. We do this by introducing a stability parameter and augmenting the dynamical system with an orthonormal k-dimensional frame and a Lyapunov vector such that the frame is continuously Gram - Schmidt orthonormalized and at most linear growth of the dynamical variables is involved. We prove that the method is strongly stable when where is the kth Lyapunov exponent in descending order and we show through examples how the method is implemented. It extends many previous results.

How projections affect the dimension spectrum of fractal measures

Abstract: We introduce a new potential-theoretic definition of the dimension spectrum of a probability measure for q > 1 and explain its relation to prior definitions. We apply this definition to prove that if and is a Borel probability measure with compact support in , then under almost every linear transformation from to , the q-dimension of the image of is ; in particular, the q-dimension of is preserved provided . We also present results on the preservation of information dimension and pointwise dimension. Finally, for and q > 2 we give examples for which is not preserved by any linear transformation into . All results for typical linear transformations are also proved for typical (in the sense of prevalence) continuously differentiable functions.

Hele-Shaw flow and pattern formation in a time-dependent gap

Abstract: We consider flow in a Hele - Shaw cell for which the upper plate is being lifted uniformly at a specified rate. This lifting puts the fluid under a lateral straining flow, sucking in the interface and causing it to buckle. The resulting short-lived patterns can resemble a network of connections with triple junctions. The basic instability - a variant of the Saffman - Taylor instability - is found in a version of the two-dimensional Darcy's law, where the divergence condition is modified to account for the lifting of the plate. For analytic data, we establish the existence, uniqueness and regularity of solutions when the surface tension is zero. We also construct some exact analytic solutions, both with and without surface tension. These solutions illustrate some of the possible behaviours of the system, such as cusp formation and bubble fission. Further, we present the results of numerical simulations of the bubble motion, examining in particular the distinctive pattern formation resulting from the Saffman - Taylor instability, and the effect of surface tension on a bubble evolution that in the absence of surface tension would fission into two bubbles. AMS classification scheme numbers: 76E30, 76D45

On the normal behaviour of partially elliptic lower-dimensional tori of Hamiltonian systems

Abstract: The purpose of this paper is to study the dynamics near a reducible lower-dimensional invariant tori of an autonomous analytic Hamiltonian system with degrees of freedom. We will focus in the case in which the torus has (some) elliptic directions. First, let us assume that the torus is totally elliptic, with Diophantine frequencies. In this case, it is shown that the diffusion time (the time to move away from the torus) is exponentially big with the initial distance to the torus. The result is valid, in particular, when the torus is of maximal dimension and when it is of dimension 0 (elliptic point). In the maximal-dimension case, our results coincide with previous ones. In the zero-dimension case, our results improve the existing bounds in the literature. Let us assume now that the torus (of dimension r, ) is partially elliptic (let us call to the number of these directions), and satisfying some generic conditions of nondegeneracy and nonresonance. In this case we show that, given a fixed number of elliptic directions (let us call to this number), there exist a Cantor family of invariant tori of dimension , that generalize the linear oscillations corresponding to these elliptic directions. Moreover, the Lebesgue measure of the complementary of this Cantor set (in the frequency space ) is proven to be exponentially small with the distance to the initial torus. This is a sort of `Cantorian central manifold' theorem, in which the central manifold is completely filled up by invariant tori and it is uniquely defined.

The complex Ginzburg - Landau equation on large and unbounded domains: sharper bounds and attractors

Abstract: Using weighted -norms we derive new bounds on the long-time behaviour of the solutions improving on the known results of the polynomial growth with respect to the instability parameter. These estimates are valid for quite arbitrary, possibly unbounded domains. We establish precise estimates on the maximal influence of the boundaries on the dynamics in the interior. For instance, the attractor for the domain with periodic boundary conditions is upper semicontinuous to .

Breathers on a diatomic FPU chain

Abstract: Existence of self-localized time-periodic vibrations (discrete breathers) is proved for alternating mass chains with anharmonic coupling and no external potential, provided the mass ratio is large enough. This result is significant because, except for special cases, previous proofs of existence of discrete breathers require the phonon spectrum to be a narrow band bounded away from zero (optic phonons), whereas this problem has also an acoustic band (frequencies arbitrarily close to zero). The method can be adapted to many other cases with both acoustic and optic phonons.

Stability results for steady, spatially periodic planforms

Abstract: We consider the symmetry-breaking steady state bifurcation of a spatially-uniform equilibrium solution of E(2)-equivariant PDEs. We restrict the space of solutions to those that are doubly-periodic with respect to a square or hexagonal lattice, and consider the bifurcation problem restricted to a finite-dimensional center manifold. For the square lattice we assume that the kernel of the linear operator, at the bifurcation point, consists of 4 complex Fourier modes, with wave vectors K_1=(a,b), K_2=(-b,a), K_3=(b,a), and K_4=(-a,b), where a>b>0 are integers. For the hexagonal lattice, we assume that the kernel of the linear operator consists of 6 complex Fourier modes, also parameterized by an integer pair (a,b). We derive normal forms for the bifurcation problems, which we use to compute the linear, orbital stability of those solution branches guaranteed to exist by the equivariant branching lemma. These solutions consist of rolls, squares, hexagons, a countable set of rhombs, and a countable set of planforms that are superpositions of all of the Fourier modes in the kernel. Since rolls and squares (hexagons) are common to all of the bifurcation problems posed on square (hexagonal) lattices, this framework can be used to determine their stability relative to a countable set of perturbations by varying a and b. For the hexagonal lattice, we analyze the degenerate bifurcation problem obtained by setting the coefficient of the quadratic term to zero. The unfolding of the degenerate bifurcation problem reveals a new class of secondary bifurcations on the hexagons and rhombs solution branches.

Persistence and stability of relative equilibria

Abstract: We consider relative equilibria in symmetric Hamiltonian systems, and their persistence or bifurcation as the momentum is varied. In particular, we extend a classical result about persistence of relative equilibria from values of the momentum map that are regular for the coadjoint action, to arbitrary values, provided that either (i) the relative equilibrium is at a local extremum of the reduced Hamiltonian or (ii) the action on the phase space is (locally) free. The first case uses just point-set topology, while in the second we rely on the local normal form for (free) symplectic group actions, and then apply the splitting lemma. We also consider the Lyapunov stability of extremal relative equilibria. The group of symmetries is assumed to be compact.

Complete transversals and the classification of singularities

Abstract: The classification of map-germs (up to a variety of equivalences) has many applications in differential geometry, to the study of wavefronts and caustics and to bifurcation theory. In a previous paper the first and last authors together with C T C Wall, gave some useful criteria for map-germs to be finitely determined with respect to a wide range of equivalence relations. The results presented in this paper, used in conjunction with these determinacy techniques, provide a very efficient classification procedure. Moreover, we show that the algebraic criteria involved in these calculations may be reduced to finite-dimensional symbolic problems which may be performed by a computer.

Existence and stability of quasiperiodic breathers in the discrete nonlinear Schrödinger equation

Abstract: We show that the discrete nonlinear Schrodinger (DNLS) equation exhibits exact solutions which are quasiperiodic in time and localized in space if the ratio between the nonlinearity and the linear hopping constant is large enough. These quasiperiodic breather solutions, which also exist for a generalized DNLS equation with on-site nonlinearities of arbitrary positive power, can be constructed by continuation from the anticontinuous limit (i.e. the limit of zero hopping) of solutions where two (or more) sites are oscillating with two incommensurate frequencies. By numerical continuation from the anticontinuous limit, some quasiperiodic breathers are explicitly calculated, and their domain of existence is determined. Using Floquet analysis, we also show that the simplest quasiperiodic breathers are linearly stable close to the anticontinuous limit, and we determine numerically the stability boundaries. The nature of the bifurcations occurring at the boundaries of the stability and existence regions, respectively, is investigated by analysing the band structure of the corresponding Newton operator. We find that the way in which the breather stability and existence is lost depends qualitatively on the ratio between its frequencies. In some cases the two-site breather becomes unstable with respect to a pinning mode, so that applying a small perturbation results in a splitting of the breather into one pinned and one moving part. In other cases, the breather develops an extended tail as some harmonic of its frequencies enters the linear phonon band and becomes a `phonobreather', which was found to be linearly stable in some domain of parameters.

Vorticity alignment results for the three-dimensional Euler and Navier - Stokes equations

Abstract: We address the problem in Navier - Stokes isotropic turbulence of why the vorticity accumulates on thin sets such as quasi-one-dimensional tubes and quasi-two-dimensional sheets. Taking our motivation from the work of Ashurst, Kerstein, Kerr and Gibbon, who observed that the vorticity vector aligns with the intermediate eigenvector of the strain-matrix S, we study this problem in the context of both the three-dimensional Euler and Navier - Stokes equations using the variables and where . This introduces the dynamic angle , which lies between and . For the Euler equations a closed set of differential equations for and is derived in terms of the Hessian matrix of the pressure . For the Navier - Stokes equations, the Burgers vortex and shear-layer solutions turn out to be the Lagrangian fixed-point solutions of the equivalent equations with a corresponding angle . Under certain assumptions for more general flows it is shown that there is an attracting fixed point of the equations which corresponds to positive vortex stretching and for which the cosine of the corresponding angle is close to unity. This indicates that near alignment is an attracting state of the system and is consistent with the formation of Burgers-like structures. PACS Numbers: 4727J, 4732C, 4715K

kinks - gradient flow and dynamics

Abstract: The symmetric dynamics of two kinks and one antikink in classical (1 + 1)-dimensional theory is investigated. Gradient flow is used to construct a collective coordinate model of the system. The relationship between the discrete vibrational mode of a single kink, and the process of kink - antikink pair production is explored.

Non-chaotic behaviour in three-dimensional quadratic systems

Abstract: It is shown that three-dimensional dissipative quadratic systems of ordinary differential equations with a total of four terms on the right-hand side of the equations do not exhibit chaos. This complements recent work of Sprott who has given many examples of chaotic quadratic systems with as few as five terms on the right-hand side of the equations. PACS Number: 0545

Complexity in the bifurcation structure of homoclinic loops to a saddle-focus

Abstract: We report on the study of bifurcations of multi-circuit homoclinic loops in two-parameter families of vector fields in the neighbourhood of a main homoclinic tangency to a saddle-focus with characteristic exponents satisfying the Shil'nikov condition . We prove that one-parameter subfamilies of vector fields transverse to the main homoclinic tangency (1) may be tangent to subfamilies with a triple-circuit homoclinic loop; (2) may have a tangency of an arbitrarily high order to subfamilies with a multi-circuit homoclinic loop. These theorems show the high structural instability of one-parameter subfamilies of vector fields in the neighbourhood of a homoclinic tangency to a Shil'nikov-type saddle-focus. Implications for nonlinear partial differential equations modelling waves in spatially extended systems are briefly discussed.

Low-energy chaos in the Fermi-Pasta-Ulam problem

Abstract: A possibility that in the FPU problem the critical energy for chaos goes to zero when the number of particles in the chain increases is discussed. The distribution for long linear waves in this regime is found and an estimate for the new border of the transition to energy equipartition is given.

Noncompact drift for relative equilibria and relative periodic orbits

Abstract: In the context of equivariant dynamical systems with a compact Lie group, Gamma, of symmetries, Field and Krupa have given sharp upper bounds on the drifts associated with relative equilibria and relative periodic orbits. For relative equilibria consisting of points of trivial isotropy, the drifts correspond to tori in Gamma. Generically, these are maximal tori. Analogous results hold when there is a nontrivial isotropy subgroup Sigma, with Gamma replaced by N(Sigma)/Sigma. In this paper, we generalize the results of Field and Krupa to noncompact Lie groups. The drifts now correspond to tori or lines (unbounded copies of R) in Gamma and generically these are maximal tori or lines. Which of these drifts is preferred, compact or unbounded, depends on Gamma: there are examples where compact drift is preferred (Euclidean group in the plane), where unbounded drift is preferred (Euclidean group in three-dimensional space) and where neither is preferred (Lorentz group). Our results partially explain the quasiperiodic (Winfree) and linear (Barkley) meandering of spirals in the plane, as well as the drifting behaviour of spiral bound pairs (Ermakova et al). In addition, we obtain predictions for the drifting of the scroll solutions (scroll waves and scroll rings, twisted and linked) considered by Winfree and Strogatz.

Bifurcations of cuspidal loops

Abstract: A cuspidal loop for a planar vector field X consists of a homoclinic orbit through a singular point p, at which X has a nilpotent cusp. This is the simplest non-elementary singular cycle (or graphic) in the sense that its singularities are not elementary (i.e. hyperbolic or semihyperbolic). Cuspidal loops appear persistently in three-parameter families of planar vector fields. The bifurcation diagrams of unfoldings of cuspidal loops are studied here under mild genericity hypotheses: the singular point p is of Bogdanov - Takens type and the derivative of the first return map along the orbit is different from 1. An analytic and geometric method based on the blowing up for unfoldings is proposed here to justify the two essentially different models for generic bifurcation diagrams presented in this work. This method can be applied for the study of a large class of complex multiparametric bifurcation problems involving non-elementary singularities, of which the cuspidal loop is the simplest representative. The proofs are complete in a large part of parameter space and can be extended to the complete parameter space modulo a conjecture on the time function of certain quadratic planar vector fields. In one of the cases we can prove that the generic cuspidal loop bifurcates into four limit cycles that are close to it in the Hausdorff sense.

Quasi-geostrophic-type equations with initial data in Morrey spaces

Abstract: This paper studies the well posedness of the initial value problem for the quasi-geostrophic type equations where is a fixed parameter and is divergence free and determined from through the Riesz transform ( being a permutation of j, . The initial data is taken in certain Morrey spaces (see text for the definition). The local well posedness is proved for and the solution is global for sufficiently small data. Furthermore, the solution is shown to be smooth.

The Lyapunov exponent in the Sinai billiard in the small scatterer limit

Abstract: We show that the Lyapunov exponent for the Sinai billiard is with where R is the radius of the circular scatterer. We consider the disk-to-disk-map of the standard configuration where the disk is centred inside a unit square.

Isochronicity of plane polynomial Hamiltonian systems

Abstract: We study isochronous centres of plane polynomial Hamiltonian systems, and more generally, isochronous Morse critical points of complex polynomial Hamiltonian functions Our first result is that if the Hamiltonian function H is a non-degenerate semi-weighted homogeneous polynomial, then it cannot have an isochronous Morse critical point, unless the associate Hamiltonian system is linear, that is to say H is of degree two Our second result gives a topological obstruction for isochronicity Namely, let be a continuous family of one-cycles contained in the complex level set , and vanishing at an isochronous Morse critical point of H, as We prove that if H is a good polynomial with only simple isolated critical points and the level set contains a single critical point, then represents a zero homology cycle on the Riemann surface of the algebraic curve We give several examples of `non-trivial' complex Hamiltonians with isochronous Morse critical points and explain how their study is related to the famous Jacobian conjecture

A discrete $\phi^4$ system without a Peierls - Nabarro barrier

Abstract: A discrete system is proposed which preserves the topological lower bound on the kink energy. Existence of static kink solutions saturating this lower bound and occupying any position relative to the lattice is proved. Consequently, kinks of the model experience no Peierls - Nabarro barrier, and can move freely through the lattice without being pinned. Numerical simulations reveal that kink dynamics in this system is significantly less dissipative than that of the conventional discrete system, so that even on extremely coarse lattices the kink behaves much like its continuum counterpart. It is argued, therefore, that this is a natural discretization for the purpose of numerically studying soliton dynamics in the continuum model.

Reference systems for splitting of separatrices

Abstract: The investigation of exponentially small splitting of separatrices for high-frequency time-periodic perturbation of a Hamiltonian with one degree of freedom leads to a reference system in the complex phase space. The reference system is independent of a small parameter, e.g., the perturbation period of the original system. The splitting of the invariant manifolds is described for the system , which is a reference system for high-frequency perturbations of the pendulum.

Symmetry-breaking bifurcations on cubic lattices

Abstract: Steady-state symmetry-breaking bifurcations on the simple (SC), face-centred (FCC) and body-centred (BCC) cubic lattices are considered, corresponding to the 6-, 8- and 12-dimensional representations of the group . Methods of equivariant bifurcation theory are used to identify all primary solution branches and to determine their stability; branches with submaximal isotropy are generic for both the FCC and BCC lattices. Complete analysis of the local branching behaviour in the SC (three primary branches) and FCC (five primary branches) cases is given. The BCC case is substantially more complex, owing to the presence of a quadratic equivariant. A degeneracy that arises in the presence of an additional reflection symmetry is analysed first using a normal form truncated at third order. This problem, in which no quadratic equivariants are present, yields 10 primary branches with maximal isotropy and five with submaximal isotropy. The unfolding of the degeneracy is used to show that seven primary branches (six maximal and one submaximal) persist in the generic case, and to determine the form and properties of secondary branches. In certain cases higher-order terms are necessary. The study is motivated by the problem of pattern formation in three spatial dimensions, and extends earlier work by De Wit and co-workers.

Towards a kneading theory for Lozi mappings I: A solution of the pruning front conjecture and the first tangency problem

Abstract: We construct a kneading theory a la Milnor - Thurston for Lozi mappings (piecewise affine homeomorphisms of the plane). As a two-dimensional analogue of the kneading sequence, the pruning front and the primary pruned region are introduced, and the admissibility criterion for symbol sequences known as the pruning front conjecture is proven under a mild condition on the parameters. Using this result, we show that topological properties of the dynamics of the Lozi mapping are determined by its pruning front and primary pruned region only. This gives us a solution to the first tangency problem for the Lozi family, moreover the boundary of the set of all horseshoes in the parameter space is shown to be algebraic.