Periodic and chaotic motions of an unsymmetrical oscillator in nonlinear structural dynamics

Rega, Giuseppe; Benedettini, F.; Salvatori, Antonello

doi:10.1016/0960-0779(91)90054-d

Cited by 33 publications

(13 citation statements)

References 25 publications

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“…Chaotic "islands" in Figure 2 for the classic Duffing system (a3 > 0) is, in fact, surrounded by a subaltraharmonic resonance of (he order of 5/2 and 7/3 and also by a 1/1 principal resonance (see Ueda [1] and Leung [8]). The other type of chaos region in the F -co plane for the hardening but unsymmetric oscillator (c~2 ¢ 0) was found by Benedettini and Rega [9,10]. It clearly shows that in this system chaos is related to the transition from the 1/2 subharmonic resonance to the 1/1 principal resonance.…”

Section: Introductionmentioning

confidence: 76%

The analytical predictive criteria for chaos and escape in nonlinear oscillators: A survey

Szemplińska-Stupnicka

1995

Nonlinear Dyn

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The purpose of this paper is to provide a brief summary of the various analytical predictive criteria in order for "strange phenomena" to occur in a class of softening nonlinear oscillators, oscillators which may exhibit escape from a potential well. Implications of Melnikov's criteria are discussed first and transient chaos in the twin-well potential oscillator is illustrated. Three different heuristic criteria for steady state chaos or escape solution, proposes by F. Moon, G. Schmidt and W. Szemplifiskia-Stupnicka, are then presented and compared to computer simulation results.

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Section: Introductionmentioning

confidence: 76%

The analytical predictive criteria for chaos and escape in nonlinear oscillators: A survey

Szemplińska-Stupnicka

1995

Nonlinear Dyn

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“…The associated cable belongs to the class of initially prestressed cables never undergoing compression, which is mechanically consistent with the circumstance that a cable can resist only tensile forces [2].…”

Section: Introductionmentioning

confidence: 86%

“…In this case, however, the occurrence of a sequence of PD bifurcations leading eventually to chaos within the unstable zone is already suggested by the response obtained via step-by-step long-time computer simulations of equation (1) with fixed i.c. [2]. In Figure 3 a detailed period chart of the response corresponding to zero i.c.…”

Section: Numerical and Geometrical Analysis Of Steady-state Responsementioning

confidence: 99%

“…A route to chaos with decreasing frequency actually characterized by a complete sequence of period doublings is observed, whiie the one with increasing frequency is seen to be discontinuous, with a sudden transition of the response from P -1 to chaotic. The actual chaotic nature of the response was validated through the calculation of positive values of the maximum Lyapunov exponent [2]. A magnification of the observed response curves in the frequency range of interest is shown in Figure 2b, where the various branches of stable or unstable fixed points are labelled using the notations mentioned in Section 2.…”

Section: Numerical and Geometrical Analysis Of Steady-state Responsementioning

confidence: 99%

“…The presence of both even and odd nonlinear terms in the oscillator equation (1) gives rise to a rich and complex set of responses [2][3][4]. Indeed, the model exhibits many fundamental characteristics of the behaviour of nonlinear dynamical systems: multiple coexisting periodic attractors, strange chaotic attractors occurring in various ranges of control parameter values, different routes to chaos from periodic solutions, and regular and fractal basin boundaries.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Numerical and geometrical analysis of bifurcation and chaos for an asymmetric elastic nonlinear oscillator

1995

Self Cite

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An asymmetric nonlinear oscillator representative of the finite forced dynamics of a structural system with initial curvature is used as a model system to show how the combined use of numerical and geometrical analysis allows deep insight into bifurcation phenomena and chaotic behaviour in the light of the system global dynamics. Numerical techniques are used to calculate fixed points of the response and bifurcation diagrams, to identify chaotic attractors, and to obtain basins of attraction of coexisting solutions. Geometrical analysis in control-phase portraits of the invariant manifolds of the direct and inverse saddles corresponding to unstable periodic motions is performed systematically in order to understand the global attractor structure and the attractor and basin bifurcations.

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Forced Harmonic Vibration of an Asymmetric Duffing Oscillator

Kovacic

Brennan

2011

The Duffing Equation

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Periodic and chaotic motions of an unsymmetrical oscillator in nonlinear structural dynamics

Cited by 33 publications

References 25 publications

The analytical predictive criteria for chaos and escape in nonlinear oscillators: A survey

The analytical predictive criteria for chaos and escape in nonlinear oscillators: A survey

Numerical and geometrical analysis of bifurcation and chaos for an asymmetric elastic nonlinear oscillator

Forced Harmonic Vibration of an Asymmetric Duffing Oscillator

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