A study of nonlinear periodic systems via the point mapping method

Flashner, Henryk; Hsu, C. S.

doi:10.1002/nme.1620190204

Cited by 28 publications

(4 citation statements)

References 11 publications

(7 reference statements)

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“…4, while the (A 1 , A 2 ) solution becomes unstable. This agrees with the results found by Flashner and Hsu [1983] and is also confirmed by the direct numerical integration of the new stable 2T solutions shown in Fig. 10 at β = 2.26.…”

Section: Methods Of Analysissupporting

confidence: 92%

“…This period 2T solution is plotted in the state space in Fig. 3 for four different values of β past the primary bifurcation point along with the Poincaré points att = 0 and T. It is seen that the amplitude of the periodic solution grows with an increase in the periodic parameter, a result also found by Flashner and Hsu [1983].…”

Section: Methods Of Analysissupporting

confidence: 60%

“…This is accomplished by using the information about the spectral density of the new solution and by assuming a parabolic-type bifurcation of the Poincaré points. Bifurcation diagrams are then constructed and the results are found to agree with those obtained by Flashner and Hsu [1983] using the numerical point mapping method.…”

Section: Introductionmentioning

confidence: 68%

“…It is assumed that the support of the pendulum is subjected to a sinusoidal motion in the vertical direction with frequency ω. The dimensionless equation of motion for the system is given by Flashner and Hsu [1983] as…”

Section: Methods Of Analysismentioning

confidence: 99%

See 3 more Smart Citations

Symbolic Computation of Secondary Bifurcations in a Parametrically Excited Simple Pendulum

Butcher

Sinha

1998

Int. J. Bifurcation Chaos

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A symbolic computational technique is used to study the secondary bifurcations of a parametrically excited simple pendulum as an explicit function of the periodic parameter. This is made possible by the recent development of an algorithm which approximates the fundamental solution matrix of linear time-periodic systems in terms of system parameters in symbolic form. By evaluating this matrix at the end of the principal period, the parameter-dependent Floquet transition matrix (FTM), or the linear part of the Poincaré map, is obtained. The subsequent use of well-known criteria for the local stability and bifurcation conditions of equilibria and periodic solutions enables one to obtain the equations for the bifurcation boundaries in the parameter space. Since this method is not based on expansion in terms of a small parameter, it can successfully be applied to periodic systems whose internal excitation is strong. By repeating the linearization and computational procedure after each bifurcation of an equilibrium or periodic solution, it is shown how the bifurcation locations as well as the new linearized equations may be obtained in closed form as a function of the periodic parameter. Bifurcation diagrams are constructed and the results are compared with those obtained elsewhere using the point mapping method.

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Section: Methods Of Analysissupporting

confidence: 92%

Section: Methods Of Analysissupporting

confidence: 60%

Section: Introductionmentioning

confidence: 68%

Section: Methods Of Analysismentioning

confidence: 99%

See 2 more Smart Citations

Symbolic Computation of Secondary Bifurcations in a Parametrically Excited Simple Pendulum

Butcher

Sinha

1998

Int. J. Bifurcation Chaos

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A numerical method for computing domains of attraction for dynamical systems

Guttalu

Flashner

1988

Numerical Meth Engineering

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SUMMARYThe backward mapping approach for computation of global domains of attraction of asymptotically stable non-critical equilibrium points of dynamical systems is presented. A basis for the proposed approach is an extension of Lyapunov's direct method due to LaSalle and Lefschetz. An iterative process that converges to the global domain of attraction of an asymptotically stable equilibrium point is formulated. The method applies to both continuous time and discrete time multidimensional systems. It is shown that the backward mapping approach proposed by C. S. Hsu for spiral equilibrium points of second order discrete time systems is a particular case of the algorithm presented here. The proposed method can be used for autonomous systems as well as for systems with periodic coefficients. When applied to discrete time formulation of dynamical systems, the method can be used to determine the regions of stability of periodic solutions. The paper concludes with a number of illustrative examples that demonstrate the usefulness of the proposed approach.

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Analysis of Dynamical Systems by Truncated Point Mappings and Cell Mapping

Guttalu

Flashner

1990

Nonlinear Dynamics in Engineering Systems

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A study of nonlinear periodic systems via the point mapping method

Cited by 28 publications

References 11 publications

Symbolic Computation of Secondary Bifurcations in a Parametrically Excited Simple Pendulum

Symbolic Computation of Secondary Bifurcations in a Parametrically Excited Simple Pendulum

A numerical method for computing domains of attraction for dynamical systems

Analysis of Dynamical Systems by Truncated Point Mappings and Cell Mapping

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