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Centre manifold reduction for quasilinear discrete systems. (English) Zbl 1185.37158

Summary: We study the dynamics of quasilinear mappings in Hilbert spaces in the neighbourhood of a fixed point. The linearized map is a closed unbounded operator and thus the initial value problem is ill-posed. Under suitable spectral assumptions, we show that all solutions staying in some neighbourhood of the fixed point lie on an invariant centre manifold. We apply this result to the study of time-periodic oscillations of a class of infinite one-dimensional Hamiltonian lattices. In this context, our approach provides a mathematically justified and corrected version of the rotating-wave approximation method. The equations are viewed as recurrence relations in the discrete space coordinate, where the fixed point corresponds to the oscillators at rest. These problems yield finite-dimensional centre manifolds and thus can be locally reduced to the study of finite-dimensional mappings. In particular, we consider the Fermi-Pasta-Ulam (FPU) lattice, which describes a chain of nonlinearly coupled particles. When the frequency of solutions is close to the highest normal mode frequency, the reduction yields a two-dimensional reversible mapping. For interaction potentials satisfying a hardening condition, the reduced mapping admits homoclinic orbits to 0 which correspond to FPU “breathers” (time-periodic and spatially localized oscillations).

MSC:

37L10 Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems
37K60 Lattice dynamics; integrable lattice equations
34G20 Nonlinear differential equations in abstract spaces
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
47J99 Equations and inequalities involving nonlinear operators
82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
39A12 Discrete version of topics in analysis
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