Solutions Of Nonlinear Dynamical Systems Research Articles

Stability and approximate solution of nonlinear dynamic system of a cylinder with two end faces in relative rotation

For the nonlinear dynamics system of a cylinder with two end faces in relative rotation, a qualitative analysis of the autonomous system dynamics equation is performed. The stability of the solution of the equation is established, and approximate solution of the equation under special conditions is obtained by averaging method.

Multiple positive solutions for nonlinear dynamical systems on a measure chain

In this paper, we consider the following dynamical system on a measure chain: u 1 ΔΔ(t)+f 1(t,u 1(σ(t)),u 2(σ(t)))=0, t∈[a,b], u 2 ΔΔ(t)+f 2(t,u 1(σ(t)),u 2(σ(t)))=0, t∈[a,b], with the Sturm–Liouville boundary value conditions αu i(a)−βu i Δ(a)=0, γu i(σ(b))+δu i Δ(σ(b))=0 for i=1,2. Some results are obtained for the existence of three positive solutions of the above problem by using Leggett–Williams fixed point theorem.

High-order variational calculation for the frequency of time-periodic solutions

We develop a convergent variational perturbation theory for the frequency of time-periodic solutions of nonlinear dynamical systems. The power of the theory is illustrated by applying it to the Duffing oscillator.

Nonlinear vibrations of dynamical systems with a general form of piecewise-linear viscous damping by incremental harmonic balance method

Incremental harmonic balance (IHB) method for computation of periodic solutions of nonlinear dynamical systems is extended here for analysis of a class of periodically excited systems with a general form of piecewise-linear viscous damping characteristics, with an explicit formulation being derived , which is in many respects distinctively advantageous over classical approaches, and especially excels in performing parametric studies as frequency response property. Numerical simulation of a specific periodically excited oscillator of the considered type is effectively carried out by the IHB scheme and the results compare very well with direct numerical integration. The formulation derived here can readily be combined with the existing IHB scheme designed for treating systems with only piecewise-linear stiffness in analyzing complex dynamical behavior as bifurcation and chaos of more general piecewise-linear systems.

Periodic waveform relaxation solutions of nonlinear dynamic equations

We present a simple theorem to safeguard the convergence of waveform relaxation (WR) solutions of a dynamic system described by nonlinear ordinary differential equations (ODEs) with a periodic constraint. Namely, if a basic expression of certain constants issued from the system is less than one, the proposed WR algorithm is convergent to the exact solution. It is the first time that WR is used to treat periodic solutions of nonlinear dynamic systems. A numerical example is provided to confirm the theoretic work of the paper.

Normal form methods for symbolic creation of approximate solutions of nonlinear dynamical systems

We present a scheme for the computation (and further reduction) of normal forms of Hamiltonian systems, and we describe the scheme’s computer algebra application to some two-degrees-of-freedom mechanical models to build symbolic approximations for local families of their periodic solutions near the origin. More precisely, we consider one-parameter families of periodic solutions of real Hamiltonian systems, the parameter being the energy. We deal with some interesting cases in the generalized Hénon-Heiles system and another Hamiltonian considered by Churchill et al. The families of periodic solutions are represented as truncated Fourier series in the approximate frequencies and power series in the mechanical energy. It is also shown that some periods of the found families can be given closed expressions in terms of the energy constant in the form of very simple generalized hypergeometric functions.

RESEARCH ON THE PERIODIC ORBIT OF NON-LINEAR DYNAMIC SYSTEMS USING CHEBYSHEV POLYNOMIALS

In this paper, a new analysis method is presented to study the steady periodic solution of non-linear dynamical systems over one period. By using the good properties of Chebyshev polynomials, the state vectors appearing in the equations can be expanded in terms of Chebyshev polynomials over the principal period such that the original non-linear differential problem is simplified to a set of non-linear algebraic equations. Furthermore, all systems, including linear, weak non-linear and strong non-linear can be analyzed in the same way for no limitation of small parameter any more. It is also very efficient to get the asymptotic solution of periodical orbit even for high-dimensional dynamical systems. The numerical accuracy of the proposed technique is compared with that of the standard numerical Runge–Kutta method.

On the search for unstable periodic solutions of nonlinear dynamical systems

On the search for unstable periodic solutions of nonlinear dynamical systems

Nondispersive wave packets as solitonic solutions of level dynamics

We discuss the analogy between nondispersive wave packets in periodically driven hydrogen atoms and solitons as solitary wave solutions of nonlinear dynamical systems. Whereas the wave packet eigenstates should not be considered as solitons propagating in real space, they are shown to display the essential features of solitonic solutions as they propagate through the irregular lattice of quasienergy levels evolving with a fictitious time like the strength or the frequency of the driving force.

Breathers in nonlinear lattices: numerical calculation from the anticontinuous limit

Breather solutions are time-periodic and space-localized solutions of nonlinear dynamical systems. We show that the concept of anticontinuous limit, which was used before for proving an existence theorem on breathers and multibreather solutions in arrays of coupled nonlinear oscillators, can be used constructively as a high-precision numerical method for finding these solutions. The method is based on the continuation of breather solutions which trivially exist at the anticontinuous limit. It is quite universal and applicable to a wide class of nonlinear models which can be of arbitrary dimension, periodic or random, with or without a driving force plus damping, etc. The main advantage of our method compared with other available methods is that we can distinguish unambiguously the different breather (or multibreather) solutions by their coding sequence. Another advantage is that we can obtain the corresponding solutions whether they are linearly stable or not. These solutions can be calculated in their full domain of existence. We illustrate the techniques with examples of breather calculations in several models. We mostly consider arrays of coupled anharmonic oscillators in one dimension, but we also test the method in two dimensions. Our method allows us to show that the breather solution can be continued while its frequency enters the phonon band (it then superposes to a band edge phonon with a finite amplitude). We also test that our method works when introducing an extra time-periodic driving force plus damping. Our method is applied for the calculation of breathers in so-called Fermi - Pasta - Ulam (FPU) chains, that is, one-dimensional chains of atoms with anharmonic nearest-neighbour coupling without on-site potential. The breather and multibreather solutions are then obtained by continuation from the anticontinuous limit of an extended model containing an extra parameter. Finally, we show that we can also calculate `rotobreathers' in arrays of coupled rotators, which correspond to solutions with one or several rotators rotating while the remaining rotators are only oscillating. The linear stability analysis of the obtained time periodic solutions (Floquet analysis) of all these models will be done in a forthcoming paper.

Exact stationary response solutions of nonlinear dynamic systems

Exact stationary response solutions of nonlinear dynamic systems

Exact stationary response solutions of nonlinear dynamic systems

Exact stationary solutions are constructed for nonlinear dynamic systems subjected to stochastic excitation. The results are then applied to both classical and nonclassical stochastic vibration systems.

Technical results for the study of robustness of Lagrange stability

We study the robustness of boundedness of solutions of nonlinear dynamical systems. A sufficient coordinate-free technical condition on the characterization of unmodelled effects is given. This characterization with some analogy with the input to state stability of [16] has the potential to encompass most classical uncertainties. In this context, we establish Lagrange stability results which look very much like a small gain theorem. We illustrate the use of these technical results in the robust stabilization problem by a class of interconnected systems.

Reduced basis alternatives to the solution of nonlinear dynamical systems

The research deals with predicting the geometric nonlinear dynamic response of a structure under impulsive loading by means of reduction methods. Techniques for reducing the number of equations to be solved, e.g., where such equations come from a finite element model, are referred to as reduction methods. The order of a dynamical system is reduced by a Rayleigh-Ritz technique using selective basis vectors. These vectors are Rayleigh-Ritz approximation functions which are not the exact eigenvectors of the system. Ritz vectors can be generated with less computational effort than needed to generate eigenvectors. Furthermore, the basis set is augmented by derivatives of Ritz vectors and updated Ritz vectors. These types of vectors are added to account for the nonlinearities of the dynamical system. The proposed reduction technique is enhanced by updating the stiffness matrix with a reasonable reassembly frequency. An error norm, which is a weighted Euclidean norm of the unbalanced force vector, is utilized to determine when updating basis vectors is necessary. Nomenclature C = damping matrix for the structure C = reduced damping matrix for the structure e = error norm defined by Eq. (44) /, = total unbalanced force vector at time t K = stiffness matrix for the structure K = reduced stiffness matrix for the structure K(d> = stiffness matrix with small change in 5th term of Z K* = effective stiffness matrix K* = reduced effective stiffness matrix M = mass matrix for the structure M = reduced mass matrix for the structure m = number of basis vectors Nf = frequency of updating stiffness Nu = number of basis vectors updating n = total number of DOF in the finite element model P = applied loads vector P* = effective loads vector P* = reduced effective loads vector td = time duration of impulsive load A? = time integration step U = displacement vector U = velocity vector U = acceleration vector U = reduced displacement vector with small change in 5th term zs = 5th components of reduced displacement vector I, = rth natural period of structure ^ = matrix of basis vectors fa = ith basis vector

Basic facts about homoclinic and heteroclinic orbits and their importance in the theory of analytical chaos

We begin the paper with short summaries of the geometry of the solutions of dynamical systems as well as of the results of symbolic dynamics. Based upon the summaries, it is shown how these well known mathematical concepts can be used to detect and describe chaotic behaviour of solutions of nonlinear dynamical systems.

Periodical solutions of nonlinear dynamical systems by decomposition method

Periodical solutions of nonlinear dynamical systems by decomposition method

Conditions for quadratic convergence of quick periodic steady-state methods

The periodic steady-state solution of nonlinear dynamic systems can be computed by the so-called quick steady-state methods, e.g., Newton iteration, optimization or extrapolation. These methods compute a vector z(t_{0}) of the periodic steady-state solution by requiring the solution to repeat itself periodically, z(t_{0} + T) = z(t_{0}) where T is the period. It is essential for the performance of the quick steady-state methods that the solution of the differential equations at t_{0} + T is a smooth function of the initial value at t_{0} . This paper gives sufficient conditions for the numerical approximation of the solution at t_{0} + T to be differentiable with Lipschitz continuous derivative with respect to the initial vector at t_{0} . The numerical approximation is obtained by the backward differentiation formulas, and two cases are considered. First the nonlinear algebraic corrector equations are solved exactly, and this case deals both with continuously differentiable problems and piecewise-linear problems. Secondly the algebraic equations are solved to an accuracy consistent with the discretization error of the backward differentiation formulas. Finally, the theorems are illustrated by examples.