Dynamics of Relative Phases: Generalised Multibreathers

For small Hamiltonian perturbation of a Hamiltonian systemof arbitrary number of degrees of freedom with anormally non-degenerate submanifold of periodic orbits we construct a nearbysubmanifold and an `effective Hamiltonian' on it such that the differencebetween the two Hamiltonian vector fields is small. The effective Hamiltonianis independent of one coordinate, the `overall phase', and hence thecorresponding action is preserved. Unlike standard averaging approaches,critical points of our effective Hamiltonian subject to given actioncorrespond to exact periodic solutions. We prove there has to be at least acertain number of these critical points given by global topological principles.The linearisation of the effective Hamiltonian about critical points is provedto give the linearised dynamics for the full system to leading order in theperturbation. Hence in the case of distinct eigenvalues which move at non-zerospeed with e,the linear stability type of the periodic orbit can be read offfrom the effective Hamiltonian. Our principal application is to networks ofoscillators or rotors where many such submanifolds of periodic orbits occurat the uncoupled limit – simply excite a number N ≥ 2 of the units inrational frequency ratio and put the others on equilibria, subject to anon-resonance condition. The resulting exact periodic solutions for weakcoupling are known as multibreathers. We call the approximate solutions givenby the effective Hamiltonian dynamics, `generalised multibreathers'. Theycorrespond to solutions which look periodic on a short time scale but therelative phases of the excited units may evolve slowly. Extensions aresketched to travelling breathers and energy exchange between degrees offreedom.

[1]  D. Bambusi,et al.  A property of exponential stability in nonlinear wave equations near the fundamental linear mode , 1998 .

[2]  B. Chirikov.  A universal instability of many-dimensional oscillator systems , 1979 .

[3]  Robert S. MacKay,et al.  Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators , 1997 .

[4]  Henri Poincaré,et al.  méthodes nouvelles de la mécanique céleste , 1892 .

[5]  D. Stuart.  Dynamics of Abelian Higgs vortices in the near Bogomolny regime , 1994 .

[6]  R. MacKay,et al.  Discrete breathers in disordered media , 1998 .

[7]  Dario Bambusi,et al.  Exponential stability of breathers in Hamiltonian networks of weakly coupled oscillators , 1996 .

[8]  A. Sievers,et al.  Intrinsic localized modes in anharmonic crystals. , 1988, Physical review letters.

[9]  S. Aubry,et al.  Breathers in nonlinear lattices: existence, linear stability and quantization , 1997 .

[10]  I. Ekeland,et al.  Symmetry breaking in Hamiltonian systems , 1987 .

[11]  R. MacKay,et al.  Stability of discrete breathers , 1998 .

[12]  G. Whitham,et al.  Linear and Nonlinear Waves , 1976 .

[13]  Taehoon Ahn.  Multisite oscillations in networks of weakly coupled autonomous oscillators , 1998 .

[14]  Kai Cieliebak,et al.  Symplectic Geometry , 1992, New Spaces in Physics.

[15]  Robert S. MacKay,et al.  Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators , 1994 .

[16]  R. MacKay,et al.  Breathers on a diatomic FPU chain , 1997 .

[17]  Jean-Luc Chabert,et al.  Chaos et déterminisme , 1992 .

[18]  Pouget,et al.  Breathing solitary waves in a Sine-Gordon two-dimensional lattice. , 1995, Physical review letters.

[19]  J. Moser,et al.  Regularization of kepler's problem and the averaging method on a manifold , 1970 .

[20]  A. Sievers,et al.  Intrinsic Localized Vibrational Modes in Anharmonic Crystals : Stationary Modes , 1988 .

[21]  A. Weinstein.  Bifurcations and Hamilton's principle , 1978 .

[22]  J. Schwartz.  Nonlinear Functional Analysis , 1969 .

[23]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[24]  S. M. Verduyn Lunel,et al.  Stochastic and spatial structures of dynamical systems , 1996 .

[25]  M. Morse.  The calculus of variation in the large , 1934 .

[26]  G. Iooss,et al.  Elementary stability and bifurcation theory , 1980 .

[27]  Peyrard,et al.  Energy localization in nonlinear lattices. , 1993, Physical review letters.

[28]  J. Guckenheimer,et al.  Three coupled oscillators: mode-locking, global bifurcations and toroidal chaos , 1991 .

[29]  Thierry Cretegny.  Dynamique collective et localisation de l'energie dans les reseaux non-lineaires , 1998 .

[30]  Ring laser configuration studied by collective coordinates , 1995 .

[31]  Movability of localized excitations in nonlinear discrete systems: A separatrix problem. , 1994, Physical review letters.

[32]  Mobility and reactivity of discrete breathers , 1997, cond-mat/9712046.

[33]  S. Aubry,et al.  Existence and stability of quasiperiodic breathers in the discrete nonlinear Schrödinger equation , 1997 .