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Chaotic and Regular Motion in Nonlinear Vibrating Systems
Local Techniques in Bifurcation Theory and Nonlinear Dynamics
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2T-subharmonic amplitude equation appear approximate criterion approximate solution assume axis Bifurcation diagram buckled beam center manifold chaotic behaviour chaotic motion chaotic vibrations chaotic zone close coefficients computer simulation correlation dimension criterion for chaotic critical value damping differential equation dimensional Duffing equation Duffing's equation eigenvalues eigenvector example experimental forcing parameter fractal dimension frequency spectrum function harmonic components hence Holmes homoclinic orbits Hopf bifurcation initial conditions invariant jump phenomena Large Orbit motion linear operator Lyapunov exponents mathematical Moon neighborhood nonlinear oscillator nonlinear vibrations normal form observed obtain period doubling bifurcations Periodic forcing periodic motion periodic solution phase plane phase portrait phase space pitchfork bifurcation Poincare map pointwise dimension portraits and Poincare problem resonance curves route to chaos saddle-node bifurcation sampling shown in Figure signal stability limit strange attractor subharmonic symmetric systems T-periodic Taylor expansion theoretical theory of nonlinear trajectories two-well potential Ueda unsymmetric solution variable variational equation x(nT x(tn