Routes to chaos in continuous mechanical systems. Part 3: The Lyapunov exponents, hyper, hyper-hyper and spatial–temporal chaos

Awrejcewicz, Jan; Krysko, Anton V.; Папкова, И. В.; Krysko, Vadim A.

doi:10.1016/j.chaos.2012.02.002

Cited by 45 publications

(12 citation statements)

References 10 publications

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“…Case 6. Figures 17,18,19,20,21,22,23,24,and 25 show that the frequency response curves for practical case stability, where 1 ̸ = 0, 2 ̸ = 0, and 3 ̸ = 0. Figure 17 shows that the effects of the detuning parameter 1 on the amplitudes of the three modes.…”

Section: Response Curve Of Casementioning

confidence: 99%

“…Stability of the system is studied using frequency response equations and the phaseplane method. Awrejcewicz et al [23][24][25] studied the chaotic dynamics of continuous mechanical systems such as flexible plates and shallow shells. The considered problems are solved by the Bubnov-Galerkin, Ritz method with higher approximations, and finite difference method.…”

Section: Introductionmentioning

confidence: 99%

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Vibration, Stability, and Resonance of Angle-Ply Composite Laminated Rectangular Thin Plate under Multiexcitations

Sayed

Mousa

2013

Mathematical Problems in Engineering

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An analytical investigation of the nonlinear vibration of a symmetric cross-ply composite laminated piezoelectric rectangular plate under parametric and external excitations is presented. The method of multiple time scale perturbation is applied to solve the nonlinear differential equations describing the system up to and including the second-order approximation. All possible resonance cases are extracted at this approximation order. The case of 1 : 1 : 3 primary and internal resonance, whereΩ3≅ω1,ω2≅ω1, andω3≅3ω1, is considered. The stability of the system is investigated using both phase-plane method and frequency response curves. The influences of the cubic terms on nonlinear dynamic characteristics of the composite laminated piezoelectric rectangular plate are studied. The analytical results given by the method of multiple time scale is verified by comparison with results from numerical integration of the modal equations. Reliability of the obtained results is verified by comparison between the finite difference method (FDM) and Runge-Kutta method (RKM). It is quite clear that some of the simultaneous resonance cases are undesirable in the design of such system. Such cases should be avoided as working conditions for the system. Variation of the parametersμ1,μ2,α7,β8,ω1,ω2,f1,f2leads to multivalued amplitudes and hence to jump phenomena. Some recommendations regarding the different parameters of the system are reported. Comparison with the available published work is reported.

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Section: Response Curve Of Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Vibration, Stability, and Resonance of Angle-Ply Composite Laminated Rectangular Thin Plate under Multiexcitations

Sayed

Mousa

2013

Mathematical Problems in Engineering

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“…A few selected examples of such applications are mentioned in [12][13][14]. In a series of three articles, Krysko et al used the FFT analysis to study (i) dynamics of continuous dynamical systems such as flexible plate and shallow shells [15], (ii) classical and novel scenarios of transition from periodic to chaotic solutions of dissipative continuous mechanical systems [16], (iii) dynamic loss of stability and different routes of transition to chaos of flexible curvilinear beam using Lyapunov exponents [17]. One can also mention a few examples of the FFT application in cases similar to the system analyzed in this paper.…”

Section: Introductionmentioning

confidence: 99%

FFT Bifurcation Analysis of Routes to Chaos via Quasiperiodic Solutions

Borkowski

Stefański

2015

Mathematical Problems in Engineering

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The dynamics of a ring of seven unidirectionally coupled nonlinear Duffing oscillators is studied. We show that the FFT analysis presented in form of a bifurcation graph, that is, frequency distribution versus a control parameter, can provide a valuable and helpful complement to the corresponding typical bifurcation diagram and the course of Lyapunov exponents, especially in context of detailed identification of the observed attractors. As an example, bifurcation analysis of routes to chaos via 2-frequency and 3-frequency quasiperiodicity is demonstrated.

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“…It was found that the supercritical solution branch takes a quasiperiodicity and phase locking route to chaos while the subcritical branch follows the Ruelle-Takens-Newhouse scenario. The transitions from regular to chaotic dynamics and analysis of the hyper, hyper-hyper, and spatial-temporal chaos using the Lyapunov exponents of continuous mechanical systems have been studied in [21][22][23][24]; they found the Sharkovskii windows of periodicity in the systems investigated.…”

Section: Introductionmentioning

confidence: 99%

On a Five-Dimensional Chaotic System Arising from Double-Diffusive Convection in a Fluid Layer

Idris

Siri

Hashim

2013

Abstract and Applied Analysis

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A chaotic system arising from double-diffusive convection in a fluid layer is investigated in this paper based on the theory of dynamical systems. A five-dimensional model of chaotic system is obtained using the Galerkin truncated approximation. The results showed that the transition from steady convection to chaos via a Hopf bifurcation produced a limit cycle which may be associated with a homoclinic explosion at a slightly subcritical value of the Rayleigh number.

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Routes to chaos in continuous mechanical systems. Part 3: The Lyapunov exponents, hyper, hyper-hyper and spatial–temporal chaos

Cited by 45 publications

References 10 publications

Vibration, Stability, and Resonance of Angle-Ply Composite Laminated Rectangular Thin Plate under Multiexcitations

Vibration, Stability, and Resonance of Angle-Ply Composite Laminated Rectangular Thin Plate under Multiexcitations

FFT Bifurcation Analysis of Routes to Chaos via Quasiperiodic Solutions

On a Five-Dimensional Chaotic System Arising from Double-Diffusive Convection in a Fluid Layer

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