Finite-dimensional Dynamical Systems Research Articles

Comparison of Lagrangian approach and method of moments for reducing dimensionality of soliton dynamical systems

For equations that cannot be solved exactly, the trial function approach to modelling soliton solutions represents a useful approximate technique. It has to be supplemented with the Lagrangian technique or the method of moments to obtain a finite dimensional dynamical system which can be analyzed more easily than the original partial differential equation. We compare these two approaches. Using the cubic-quintic complex Ginzburg-Landau equation as an example, we show that, for a wide class of plausible trial functions, the same system of equations will be obtained. We also explain where the two methods differ.

Constrained controllability of semilinear systems with delays

In the paper, finite-dimensional dynamical control systems described by semilinear ordinary differential state equations with multiple point time-variable delays in control are considered. Using a generalized open mapping theorem, sufficient conditions for constrained local relative controllability are formulated and proved. It is generally assumed that the values of admissible controls are in a convex and closed cone with the vertex at zero. The special case of constant multiple point delays is also discussed. Moreover, some remarks and comments on the existing results for controllability of nonlinear and semilinear dynamical systems are also presented.

KAM tori: persistence and smoothness

Ten open problems in Kolmogorov–Arnold–Moser theory for finite-dimensional dynamical systems are presented and discussed. These problems concern the preservation of frequencies and Floquet exponents of invariant tori of partially integrable systems under small perturbations in the presence of external parameters.

Model-Free Adaptive Control for a Class of Nonlinear Discrete-Time Systems Based on the Partial Form Linearization

The NARMA model is an exact representation of the input-output behavior of finite-dimensional nonlinear discrete-time dynamical systems in the neighborhood of the equilibrium state. However, it is inconvenient for purposes of adaptive control due to its nonlinear dependence on the control input, even by using the neural network method. In this paper, we introduce a so called model-free adaptive control (MFAC) method, which is based on some new dynamical linearization model and concept, the partial form linearization (PFL) and the pseudo-partial derivative (PPD) of a SISO nonlinear discrete-time system. The model-free means that the controller design is only based on the I/O data of the controlled plant, no training process, no structure information and no model are needed. Rigorous analysis and extensive simulations have shown that it has BIBO stability and performs very well.

Necessary and sufficient conditions for the existence of self-oscillations in a finite-dimensional continuous dynamical system

Necessary and sufficient conditions for the existence of self-oscillations in a finite-dimensional continuous dynamical system

Symmetries and integrability

This is a survey on finite-dimensional integrable dynamical systems related to Hamiltonian G-actions. Within a framework of noncommutative integrability we study integrability of G-invariant systems, collective motions and reduced integrability. We also consider reductions of the Hamiltonian flows restricted to their invariant submanifolds generalizing classical Hess-Appel'rot case of a heavy rigid body motion.

Constrained Controllability of Second Order Semilinear Systems

Abstract In the paper finite-dimensional dynamical control systems described by second order semilinear stationary ordinary differential state equations are considered. Using a generalized open mapping theorem, sufficient conditions for constrained local controllability in a given time interval are formulated and proved. These conditions require verification of constrained global controllability of the associated linear first-order dynamical control system. It is generally assumed, that the values of admissible controls are in a convex and closed cone with vertex at zero. Moreover, several remarks and comments on the existing results for controllability of semilinear dynamical control systems are also presented. Finally, simple numerical example, which illustrates theoretical considerations is also given. It should be pointed out, that the results given in the paper extend for the case of semilinear second-order dynamical systems constrained controllability conditions, which were previously known only for linear second-order systems.

Controlling a time-delay system using multiple delay feedback control

In this paper multiple delay feedback control (MDFC) with different and independent delay times is shown to be an efficient method for stabilizing fixed points in finite-dimensional dynamical systems. Whether MDFC can be applied to infinite-dimensional systems has been an open question. In this paper we find that for infinite-dimensional systems modelled by delay differential equations, MDFC works well for stabilizing (unstable) steady states in long-, moderateand short-time delay regions, in particular for the hyperchaotic case.

Gauge theory for finite-dimensional dynamical systems

Gauge theory is a well-established concept in quantum physics, electrodynamics, and cosmology. This concept has recently proliferated into new areas, such as mechanics and astrodynamics. In this paper, we discuss a few applications of gauge theory in finite-dimensional dynamical systems. We focus on the concept of rescriptive gauge symmetry, which is, in essence, rescaling of an independent variable. We show that a simple gauge transformation of multiple harmonic oscillators driven by chaotic processes can render an apparently "disordered" flow into a regular dynamical process, and that there exists a strong connection between gauge transformations and reduction theory of ordinary differential equations. Throughout the discussion, we demonstrate the main ideas by considering examples from diverse fields, including quantum mechanics, chemistry, rigid-body dynamics, and information theory.

The pole dynamics of rational solutions of the viscous Burgers equation

Rational solutions of the viscous Burgers equation are examined using the dynamics of their poles in the complex x-plane. The dynamical system for the motion of these poles is finite dimensional and not Hamiltonian. Nevertheless, we show that this finite-dimensional system is completely integrable, by explicit construction of a sufficient number of conserved quantities. The dynamical system has a class of non-equilibrium similarity solutions for which all poles have equal real part for t sufficiently large. Within the context of the finite-dimensional dynamical system these solutions are shown to be asymptotically stable.

Stochastic Controllability of Linear Systems With State Delays

Stochastic Controllability of Linear Systems With State DelaysA class of finite-dimensional stationary dynamic control systems described by linear stochastic ordinary differential state equations with a single point delay in the state variables is considered. Using a theorem and methods adopted directly from deterministic controllability problems, necessary and sufficient conditions for various kinds of stochastic relative controllability are formulated and proved. It will be demonstrated that under suitable assumptions the relative controllability of an associated deterministic linear dynamic system is equivalent to the stochastic relative exact controllability and the stochastic relative approximate controllability of the original linear stochastic dynamic system. Some remarks and comments on the existing results for the controllability of linear dynamic systems with delays are also presented. Finally, a minimum energy control problem for a stochastic dynamic system is formulated and solved.

Solving Normal Cone Inclusion Problems in Contact Mechanics by Iterative Methods

In this paper we show how inclusions originating from set-valued force laws can be written as non-linear equations for finite-dimensional second order dynamical systems. We express the set-valued force laws as subdifferentials of the indicator function to a given convex set C, apply the augmented Lagrangian and arrive at a proximal point problem which is solved by Jacobi or Gauss Seidel like iterative schemes. This proceed provides a simple unified access to frictional contact problems in dynamics and can be used either in event-driven or time-stepping simulation approaches.

CONSTRAINED CONTROLLABILITY OF SEMILINEAR SYSTEMS WITH MULTIPLE DELAYS IN CONTROL

In the present paper finite-dimensional dynamical control systems described by semilinear ordinary differential state equations with multiple point delays in control are considered. It is generally assumed, that the values of admissible controls are in a convex and closed cone with vertex at zero. Using generalized open mapping theorem, sufficient conditions for constrained local relative controllability near the origin are formulated and proved. Roughly speaking, it will be proved that under suitable assumptions constrained global relative controllability of a linear associated approximated dynamical system implies constrained local relative controllability near the origin of the original semilinear dynamical system. This is a generalization to constrained controllability case some previous results concerning controllability of linear systems with multiple point delays in the control and with unconstrained controls. Moreover, necessary and sufficient conditions for constrained global relative controllability of an associated linear dynamical system with multiple point delays in control are discussed. Simple numerical example which illustrates theoretical considerations is also given. Finally, some remarks and comments on the existing results for controllability of nonlinear dynamical systems are also presented.

A Poisson Integrator for Gaussian Wavepacket Dynamics

We consider the variational approximation~of the time-dependent Schrodinger equation by Gaussians wavepackets. The corresponding finite-dimensional dynamical system inherits a Poisson (or non-canonically symplectic) structure from the Schrodinger equation by its construction via the Dirac–Frenkel–McLachlan variational principle. The variational splitting between kinetic and potential energy turns out to yield an explicit, easily implemented numerical scheme. This method is a time-reversible Poisson integrator, which also preserves the L2 norm and linear and angular momentum. Using backward error analysis, we show long-time energy conservation for this splitting scheme. In the semi-classical limit the numerical approximations to position and momentum converge to those obtained by applying the Stormer–Verlet method to the classical limit system.

Dynamical models for dissipative localized waves of the complex Ginzburg-Landau equation

Finite-dimensional dynamical models for solitons of the cubic-quintic complex Ginzburg-Landau equation (CGLE) are derived. The models describe the evolution of the pulse parameters, such as the maximum amplitude, pulse width, and chirp. A clear correspondence between attractors of the finite-dimensional dynamical systems and localized waves of the continuous dissipative system is demonstrated. It is shown that stationary solitons of the CGLE correspond to fixed points, while pulsating solitons are associated with stable limit cycles. The models show that a transformation from a stationary soliton to a pulsating soliton is the result of a Hopf bifurcation in the reduced dynamical system. The appearance of moving fronts (kinks) in the CGLE is related to the loss of stability of the limit cycles. Bifurcation boundaries and pulse behavior in the regions between the boundaries, for a wide range of system parameters, are found from analysis of the reduced dynamical models. We also provide a comparison between various models and their correspondence to the exact results.

Exponential convergence and instability of solutions of uncertain dynamical systems

In the present paper we investigate exponential convergence and instability of finite-dimensional dynamical systems determined by uncertain ordinary differential equations. Our approach is based on the constructive use of matrix-valued Lyapunov functions. Examples are given which illustrate the improvement of the proposed conditions.

The discrete null space method for constrained mechanical systems in nonlinear structural and multibody dynamics

AbstractCorresponding to d'Alemberts principle in classical mechanics, the discrete null space method developed in [1] provides an energy‐momentum conserving time stepping scheme for conservative finite‐dimensional dynamical systems subject to constraints. The elimination of the Lagrange multipliers from the temporal discrete system leads to a reduced number of unknowns and to an improved condition number during the iterative solution of the nonlinear system. External and internal constraints are fulfilled likewise at the time nodes, wherefore the discrete null space method is particularly suited for the treatment of elastic multibody systems in structural dynamics. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Stabilization Mechanism for Two-Dimensional Solitons in Nonlinear Parametric Resonance

We consider a simple model system supporting stable solitons in two dimensions. The system is the parametrically driven damped nonlinear Schr\"odinger equation, and the soliton stabilises for sufficiently strong damping. The purpose of this note is to elucidate the stabilisation mechanism; we do this by reducing the partial differential equation to a finite-dimensional dynamical system. Our conclusion is that the negative feedback loop occurs via the enslaving of the soliton's phase, locked to the driver, to its amplitude and width.

Bifurcations from stationary to pulsating solitons in the cubic–quintic complex Ginzburg–Landau equation

Stationary to pulsating soliton bifurcation analysis of the complex Ginzburg–Landau equation (CGLE) is presented. The analysis is based on a reduction from an infinite-dimensional dynamical dissipative system to a finite-dimensional model. Stationary solitons, with constant amplitude and width, are associated with fixed points in the model. For the first time, pulsating solitons are shown to be stable limit cycles in the finite-dimensional dynamical system. The boundaries between the two types of solutions are obtained approximately from the reduced model. These boundaries are reasonably close to those predicted by direct numerical simulations of the CGLE.

Nonlinear Double-well Schrödinger Equations in the Semiclassical Limit

We consider time-dependent Schrodinger equations with a double well potential and an external nonlinear, both local and non-local, perturbation. In the semiclassical limit, the finite dimensional eigenspace associated to the lowest eigenvalues of the linear operator is almost invariant for times of the order of the beating period and the dominant term of the wavefunction is given by means of the solutions of a finite dimensional dynamical system. In the case of local nonlinear perturbation, we assume the spatial dimension d=1 or d=2.