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Extra info for Cylinders
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Indeed a NNM-based approach seems to be natural for the study of TET and the associated strongly nonlinear phenomena discussed in this work. , 2005a). In an additional series of works (King and Vakakis, 1993, 1994, 1995a; Vakakis and King, 1995; Andrianov, 2008) methodologies for analysing the NNMs (and their bifurcations) of nonlinear elastic and continuous systems have been developed. In King and Vakakis (1994) stationary and traveling solitary waves (breathers) in a class of nonlinear partial differential equations are regarded as localized NNMs over domains of infinite spatial extent and are studied analytically.
1996). 6). , 1997) has been investigated. A review of mode localization in systems governed by nonlinear partial differential equations was given in Vakakis (1996). We will demonstrate some aspects of nonlinear mode localization in coupled oscillators by considering two examples, one involving a low-dimensional cyclic system, and a second one with a nonlinear medium of infinite spatial extent. The latter example will underline the theoretical link between NNMs and solitary waves. , 1993): y¨i + yi + εµyi3 + εk(yi − yi+1 ) + εk(yi − yi−1 ) = 0, y0 ≡ yN , yN+1 ≡ y1 i = 0, .
The ranges of system parameters, external excitations and initial conditions) for the realization of TET in the dissipative system. The topological structure and bifurcations of the NNMs of the underlying non-dissipative systems will be carefully studied in this work – especially in the frequency-energy domain, since the energy dependencies of NNMs (and damped NNM invariant manifolds) play a key role regarding TET; this holds especially for NNMs whose spatial distributions change from non-localized to localized with decreasing energy.