Normal Modes and Localization in Nonlinear Systems by Alexander F. Vakakis

By Alexander F. Vakakis

This landmark booklet offers with nonlinear general modes (NNMs) and nonlinear mode localization. bargains an research which permits the examine of varied nonlinear phenomena having no counterpart in linear conception. On a extra theoretical point, the concept that of NNMs may be proven to supply a good framework for realizing quite a few distinctively nonlinear phenomena reminiscent of mode bifurcations and status or touring solitary waves.

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Hence, their results are only local in nature and cannot be applied to systems with relatively high energies. A nonlocal existence theorem for NNM can be formulated as follows (the notation used is defined below): Let s2 be a specified region containing the origin in the configuration space of an n DOF hamiltonian oscillator, for example, Q = { x ~ R n : V ( & ) i ) h ~ }or R = {q~Rn:llxllSA} where V(x) is the potential energy. Let dQ be the boundary of Q. Obtain criteria that guarantee the existence in s2 of a family of periodic solutions that exist in a neighborhood of q = Q and reach dQ.

1). ,pn) where diag(') denotes a diagonal matrix. 1 Bounds for the nonlinear restoring force. These special conditions imply that the nonlinear system is more or less stiff, respectively, than the linearized one. 2 1). , r In, the positive eigenvalues of the matrix M-", and assume that the quantities (Toi/27t ) are not integer numbers. 22) It follows that there exists an (n x n) Green matrix, E ( T , ~ , Twith ) , elements rik(T,u,T), by use of which the solution X(T) of the boundary value problem (2.

Find u finite value of the total etzergv, h *, such tlrat ,for any h Ih* there exists a solittion x(t) on the energy h that is qunlitatively the same with the solution xj*CosT (for the symmetric case) of the lineurized sysrem. 27), then the theorem guarantees the existence of such periodic solutions (NNMs) for any level of the total energy h. 14) does not guorantcc the monotonicity of the NNM solution x(t) for t E [O,T/Z]. 2 EXISTENCE THEOREM FOR NNMs 45 monotonicity takes place if the matrix [A(&)- N 3 is nonnegative for x > 0, X E Q .

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