Analysis of Time-Periodic Nonlinear Dynamical Systems Undergoing Bifurcations

Pandiyan, R.; Sinha, Subhash C.

doi:10.1007/978-94-011-0367-1_2

Cited by 24 publications

(20 citation statements)

References 12 publications

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“…A close examination of Equation (11) reveals that resonance may only occur when the eigenvalues are purely imaginary, i.e., if there is no dissipation in the system. For example, in the case of cubic nonlinearity, Equation (14) takes the form (see Pandiyan and Sinha [12])…”

Section: Time-independent Resonancementioning

confidence: 99%

Analysis of periodic-quasiperiodic nonlinear systems via Lyapunov-Floquet transformation and normal forms

Wooden

Sinha

2006

Nonlinear Dyn

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In this paper a general technique for the analysis of nonlinear dynamical systems with periodicquasiperiodic coefficients is developed. For such systems the coefficients of the linear terms are periodic with frequency ω while the coefficients of the nonlinear terms contain frequencies that are incommensurate with ω. No restrictions are placed on the size of the periodic terms appearing in the linear part of system equation. Application of Lyapunov-Floquet transformation produces a dynamically equivalent system in which the linear part is time-invariant and the time varying coefficients of the nonlinear terms are quasiperiodic. Then a series of quasiperiodic near-identity transformations are applied to reduce the system equation to a normal form. In the process a quasiperiodic homological equation and the corresponding 'solvability condition' are obtained. Various resonance conditions are discussed and examples are included to show practical significance of the method. Results obtained from the quasiperiodic time-dependent normal form theory are compared with the numerical solutions. A close agreement is found.

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Section: Time-independent Resonancementioning

confidence: 99%

Analysis of periodic-quasiperiodic nonlinear systems via Lyapunov-Floquet transformation and normal forms

Wooden

Sinha

2006

Nonlinear Dyn

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“…1, also known as the Ziegler column [32,40,42]. The rigid and inextensible bars of length 2l have a mass m. The two bars are allowed to rotate at points O and B thanks to elastic hinges characterized by a rotational stiffness k. At rest, the biarticulated structure is lying in the horizontal direction (O, x).…”

Section: Nonlinear Equation Of Motionmentioning

confidence: 99%

“…A second technique consists in using the Lyapunov-Floquet transformation to recast a linear time-periodic system in a time-invariant one [37,38]. This transformation could allow to compute and analyze FFs but it has mostly been used as a step, which coupled with center manifold reduction techniques and normal form theories, enables the study of nonlinear time-periodic systems undergoing bifurcations [5,32,39].…”

Section: Introductionmentioning

confidence: 99%

“…The discrete dynamical system we consider is a classic 2D model consisting of two articulated rigid bars, connected by elastic hinges, that are submitted to an end periodic compressive load so that the elastic state of the Ziegler column is periodically modulated [32]. We focus on the transverse oscillatory modes of the structure around its trivial configuration that is the undeformed straight column in space, with periodic elasticity in time.…”

Section: Introductionmentioning

confidence: 99%

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Modal and stability analysis of structures in periodic elastic states: application to the Ziegler column

Bentvelsen

Lazarus

2017

Nonlinear Dyn

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We present a spectral method to compute the transverse vibrational modes, or Floquet Forms (FFs), of a 2D bi-articulated bar in periodic elastic state due to an end harmonic compressive force. By changing the directional nature of the applied load, the trivial straight Ziegler column exhibits the classic instabilities of stationary states of dynamical system. We use this simple structure as a numerical benchmark to compare the various spectral methods that consist in computing the FFs from the spectrum of a truncated Hill matrix. We show the necessity of sorting this spectrum and the benefit of computing the fundamental FFs that converge faster. Those FFs are almost-periodic entities that generalize the concept of harmonic modal analysis of structures in equilibria to structures in periodic states. Like their particular harmonic relatives, FFs allow to get physical insights in the bifurcations of periodic stationary states. Notably, the local loss of stability is due to the frequency lock-in of the FFs for certain modulation parameters. The presented results could apply to many structural problems in mechanics, from the vibrations of rotating machineries with shape imperfections to the stability of periodic limit cycles or of any slender structures with tensile or compressive periodic elastic stresses.

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“…However, when these matrices do not commute, one of many other methods to compute the transition matrix can be used to determine the eigenvalues of Φ(2π). For example, Pandiyan & Sinha [97] showed that there always exists some matrix C such that Φ(2π) = exp(2πC), reducing condition (4.18) to finding the eigenvalues of C. Utilising this property, a clearer condition than (4.18) could be formulated.…”

Section: Discussionmentioning

confidence: 99%

Spatially structured metapopulation models within static and dynamic environments

Smith¹

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Traditionally in population modelling, the mixing of individuals has been assumed to be homogeneous; that is, every individual can come into contact with every other individual. Within the last 40 years, however, a number of population models have been proposed that do not assume homogeneous mixing but rather assume populations are divided into disjoint habitable patches that are separated by uninhabitable space. Populations with this structure are known as metapopulations.When metapopulation modelling was first proposed, the habitable patches would be classified as either colonised or extinct and the dynamics of colonisation and extinction would be the only dynamics accounted for in the model. For example, the logistic model adapted to a metapopulation wouldwhere x(t) is the proportion of occupied patches at time t, λ is the rate an unoccupied patch becomes colonised when all patches are unoccupied and µ is the rate an occupied patch becomes extinct. Adding more detail to metapopulation models, the model examined in this thesis records the number of individuals in each location, thereby accounting for an individual's dynamics, such as the rates of births, deaths and migrations, within the model.The model is a continuous time Markov process that will be used to account for the demographic stochasticity within populations such as births, deaths and migration events. Results by Kurtz, and extended by Pollett, which can be applied to a family of Markov processes, termed asymptotically density dependent, will be used to determine an approximating system of ii differential equations. These differential equations are then analysed to determine conditions for persistence and extinction. Furthermore, an Allee effect, where the initial conditions of the population determine whether it persists or goes extinct, is confirmed to exist in a two patch system that has a large difference between the migration rate for the two patches.The model is extended in two ways. The first extension accounts for a deterministically changing environment. This is done by allowing the parameters of the system to depend on time. A new functional limit law is derived which can be applied to time inhomogeneous, asymptotically density dependent Markov processes. This functional limit law is used to derive a nonautonomous system of differential equations. This system is then analysed to provide conditions for persistence and extinction of the metapopulation.The second extension to the model accounts for a stochastically changing environment. Again, the parameters of the system are allowed to vary in time. However, the parameters vary stochastically according to an underlying Markov process, designed to model a stochastically changing environment. A functional limit law provides a way to approximate the process with a random dynamical system. The random dynamical system is then analysed to determine a sufficient condition for extinction. While not a complete description of the long term behaviour, such an approach facilitates more research ...

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Analysis of Time-Periodic Nonlinear Dynamical Systems Undergoing Bifurcations

Cited by 24 publications

References 12 publications

Analysis of periodic-quasiperiodic nonlinear systems via Lyapunov-Floquet transformation and normal forms

Analysis of periodic-quasiperiodic nonlinear systems via Lyapunov-Floquet transformation and normal forms

Modal and stability analysis of structures in periodic elastic states: application to the Ziegler column

Spatially structured metapopulation models within static and dynamic environments

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