General Periodic Systems Research Articles

Nonlinear periodic system with unilateral constraints

We consider a general periodic system driven by a nonlinear, nonhomogeneous differential operator, with a maximal monotone term which is not defined everywhere. Using a topological approach based on Leray-Schauder alternative principle, we show the existence of a periodic solution.

Stability analysis problems of periodic piecewise polynomial systems

This paper studies the stability analysis problems of periodic piecewise systems, in which subsystems are given in the time-varying polynomial forms. A Lyapunov function with continuous time-varying Lyapunov matrix is adopted, which relaxes constraints on the variation of Lyapunov function in each subsystem. Using the time interval information, the exponential stability and stabilizing controller synthesis are studied. The results provide a possible alternative method to study the general periodic time-varying systems, which may further support the analysis and synthesis of general periodic systems. The effectiveness of the proposed method is validated and illustrated via numerical examples.

Towards AC-induced optimum control of dynamical localization

It is shown that dynamical localization (quantum suppression of classical diffusion) in the context of ultracold atoms in periodically shaken optical lattices subjected to time-periodic modulations having equidistant zeros depends on the impulse transmitted by the external modulation over half-period rather than on the modulation amplitude. This result provides a useful principle for optimally controlling dynamical localization in general periodic systems, which is capable of experimental realization.

Dynamics of decaying two-dimensional magnetohydrodynamic turbulence

High-resolution numerical studies of decaying two-dimensional magnetohydrodynamic turbulence were performed with up to 10242 collocation points in general periodic systems using various initial states, but restricting consideration to weak velocity-magnetic field correlation ρ. The global evolution is self-similar with constant kinetic to magnetic energy ratio EV/EM, macro- and microscale Reynolds numbers, and correlation ρ, while the total energy decays as E(t)∝(t+t0)−1. As in three dimensions, dissipative small-scale turbulence adjusts in such a way as to make the energy dissipation rate ε independent of the collisional dissipation coefficients. Normalized energy spectra are also invariant. The spectral index in the inertial range is, in general, close to 3/2 in agreement with Kraichnan’s Alfvén wave argument Ek =DB1/2ε1/2k−3/2, B=(EM)1/2, D≂1.8±0.2, but may be close to 5/3 in transient states, in which turbulence is concentrated in regions of weak magnetic field. In the dissipation range, intermittency gives rise to a modified dissipation scale leff =(l2λ)1/3, with l=Kolmogorov scale and λ=Taylor microscale. This reflects the intermittency of the dissipation process, which is consistent with the picture of current microsheets of thickness l and width and spacing λ.

A Generalized Hill’s Method for the Stability Analysis of Parametrically Excited Dynamic Systems

A method is described for investigating the stability of the null solution for a general system of linear second-order differential equations with periodic coefficients. The method is based on a generalization of Hill’s analysis and leads to a generalized Hill’s infinite determinant. Following a proof of its absolute convergence, a closed-form expression for the characteristic infinite determinant is obtained. Methods for the stability analysis utilizing different forms of the characteristic determinant are discussed. For cases where the instabilities are of the simple parametric type, a truncated form of the determinant may be used directly to locate the boundaries of the resonance regions in terms of appropriate system parameters. The present generalized Hill’s method is applied to a multidegree-of-freedom discretized system describing pipes conveying pulsating fluid. It is demonstrated that the method is a flexible and efficient computational tool for the stability analysis of general periodic systems.

Parametric aspects of mode and stability diagrams for general periodic systems

Stability and natural modes of general linear periodically time-varying systems are investigated using the classical procedure of Hill infinite determinant analysis. It is shown that for a normally stable passive system, instability can be invoked most easily by varying any one of the system parameters at a frequency of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2 \beta/k, k</tex> integral, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\beta</tex> is a natural frequency of the system. This is a generalization of a result well known for systems of second order. In addition, a conjecture propounded by Keenan [5] in 1966 regarding natural modes, is extended to its most general form. Embodied in the analysis is a new choice of parameter for the vertical axis of made and stability diagrams. Whilst the horizontal axis is retained as the amplitude of the time-varying system parameter the vertical axis is redefined in terms of the period of the variation. Examples for an analytically tractable system are presented and exhibit features predicted by the theory.