The disturbed damped linear oscillator does not explode

Simulations of models of epidemics, biochemical systems, and other bio-systems show that when deterministic models yield damped oscillations, stochastic counterparts show sustained oscillations at an amplitude well above the expected noise level. A characterization of damped oscillations in terms of the local linear structure of the associated dynamics is well known, but in general there remains the problem of identifying the stochastic process which is observed in stochastic simulations. Here we show that in a general limiting sense the stochastic path describes a circular motion modulated by a slowly varying Ornstein-Uhlenbeck process. Numerical examples are shown for the Volterra predator-prey model, Sel'kov's model for glycolysis, and a damped linear oscillator.

The author finds the general form of positive-energy Gaussian noise so that a linear damped (Bose or Fermi) oscillator obeying a quantum Langevin equation should remain canonical for all time. In a heat bath the system converges to a KMS state (see Kubo, 1957, Martin and Schwinger, 1959) as time t to infinity .

Making chaotic behavior in a damped linear harmonic oscillator

Using the modified Prelle-Singer approach, we point out that explicit time independent first integrals can be identified for the damped linear harmonic oscillator in different parameter regimes. Using these constants of motion, an appropriate Lagrangian and Hamiltonian formalism is developed and the resultant canonical equations are shown to lead to the standard dynamical description. Suitable canonical transformations to standard Hamiltonian forms are also obtained. It is also shown that a possible quantum mechanical description can be developed either in the coordinate or momentum representations using the Hamiltonian forms.

This paper presents the derivation of an exact solution for a damped nonlinear oscillator of arbitrary order (both integer and non-integer). A coefficient relationship was defined under which such a solution exists. The analytical procedure was developed based on the application of the Ateb (inverse beta) function. It has been shown that an exact solution exists for a specific relationship between the damping coefficient and the coefficient of the linear elastic term, and that this relationship depends on the order of nonlinearity. The exact amplitude of vibration was found to be a time-decreasing function, depending on the initial amplitude, damping coefficient, and the order of nonlinearity. The period of vibration was also shown to depend not only on the amplitude but also on both the nonlinearity coefficient and its order. For cases where the damping coefficient of the exact oscillator is slightly perturbed, an approximate solution based on the exact one was proposed. Three illustrative examples of oscillators with different orders of nonlinearity were considered: a nearly linear oscillator, a Duffing oscillator, and one with strong nonlinearity. For all cases, the high accuracy of the asymptotic solution was confirmed. Since no exact analytic solution exists for a purely nonlinear damped oscillator, an approximate solution was constructed using the solution of the corresponding undamped oscillator with a time-varying amplitude and phase. In the case of a purely cubic damped oscillator, the approximate solution was compared with numerical results, and good agreement was demonstrated.

Overdamping is a regime in which friction is sufficiently large that the motion either decays to its equilibrium position or it crosses the equilibrium position exactly once before returning monotonically towards the equilibrium position. The phenomena of overdamping has been studied classically and quantum mechanically only for the case of the linear damped harmonic oscillator. Here we study the classical and quantum dynamics of a family of over-damped non linear systems. The main objective of this paper is to find a Lagrangian and Hamiltonian framework to study over-damped non linear systems and to show that a quantum mechanical description can be developed in the momentum representation. Our results reduce to the well known solution of the linear damped harmonic oscillator when the non linear part is set to zero.

Let a(t), b(t) be continuous T‐periodic functions with . We establish one stability criterion for the linear damped oscillator x′′ + b(t)x′ + a(t)x = 0. Moreover, based on the computation of the corresponding Birkhoff normal forms, we present a sufficient condition for the stability of the equilibrium of the nonlinear damped oscillator x′′ + b(t)x′ + a(t)x + c(t)x2n−1 + e(t, x) = 0, where n ≥ 2, c(t) is a continuous T‐periodic function, e(t, x) is continuous T‐periodic in t and dominated by the power x2n in a neighborhood of x = 0.

In this work we consider the Riemannian geometry associated with the differential equations of one dimensional simple and damped linear harmonic oscillators. We show that the sectional curvatures are completely determined by the oscillation frequency and the friction coefficient and these physical constants can be thought as obstructions for the manifold to be flat. Moreover, equations of simple and damped harmonic oscillators describe nonisomorphic solvable Lie groups with nonpositive scalar curvature.

Among the many methods available for modeling intraindividual time series, differential equation modeling has several advantages that make it promising for applications to psychological data. One interesting differential equation model is that of the damped linear oscillator (DLO), which can be used to model variables that have a tendency to fluctuate around some typical, or equilibrium, value. Methods available for fitting the damped linear oscillator model using differential equation modeling can yield biased parameter estimates when applied to univariate time series. The degree of this bias depends on a smoothing–like parameter, which balances the need for increasing smoothing to minimize error variance but not smoothing so much as to obscure change of interest. This article explores a technique that uses surrogate data analysis to select such a parameter, thereby producing approximately unbiased parameter estimates. Furthermore the smoothing parameter, which is usually researcher-selected, is produced in an automated manner so as to reduce the experience required by researchers to apply these methods. Focus is placed on the damped linear model; however, similar issues are expected with other differential equation models and other techniques in which parameter estimates depend on a smoothing parameter. An example using affect data from the Notre Dame Longitudinal Study of Aging (2004) is presented, which contrasts the use of a single smoothing parameter for all individuals versus use of a smoothing parameter for each individual.

Externally excited undamped and damped linear and nonlinear oscillators: Exact solutions and tuning to a desired exact form of the response

By introducing fundamental integrals of one-dimensional linear damped oscillators the other first integrals can be constructed, including time-irrelevant integrals. The above method is extended to multidimensional systems, in order to construct different integrals of two-dimensional and n-dimensional linear damped oscillators. It is proved that there are three independent time-irrelevant integrals for all kinds of two-dimensional linear damped oscillators, and 2n-1 independent time-irrelevant integrals for n-dimensional linear damped oscillators. Using the transformation of variables the first integrals of linear damped oscillator transform into ones of harmonic oscillator.

A mechanical system composed of two weakly coupled oscillators under harmonic excitation is considered. Its main part is a vibro-impact unit composed of a linear oscillator with an internally colliding small block. This block is coupled with the secondary part being a damped linear oscillator. The mathematical model of the system has been presented in a non-dimensional form. The analytical studies are restricted to the case of a periodic steady-state motion with two symmetric impacts per cycle near 1:1 resonance. The multiple scales method combined with the sawtooth-function-based modelling of the non-smooth dynamics is employed. A conception of the stability analysis of the periodic motions suited for this theoretical approach is presented. The frequency–response curves and force–response curves with stable and unstable branches are determined, and the interplay between various model parameters is investigated. The theoretical predictions related to the motion amplitude and the range of stability of the periodic steady-state response are verified via a series of numerical experiments and computation of Lyapunov exponents. Finally, the limitations and extensibility of the approach are discussed.

The stochastic resonance is studied for a damped linear oscillator subject to both parametric excitation of random noise and external excitation of periodically modulated random noise. By means of the Shapiro-Loginov formula, the expressions of the first-order and the second-order moments are obtained for the system response. It is found that there exist conventional stochastic resonance, bona fide stochastic resonance and stochastic resonance in a broad sense in the system. When the noise intensity ratio R≥1, the stochastic multi_resonance is found in the system. Moreover, the numerical results of power spectrum density of system response are presented to verify the analytic results.

People show stable differences in the way their affect fluctuates over time. Within the general framework of dynamical systems, the damped linear oscillator (DLO) model has been proposed as a useful approach to study affect dynamics. The DLO model can be applied to repeated measures provided by a single individual, and the resulting parameters can capture relevant features of the person's affect dynamics. Focusing on negative affect, we provide an accessible interpretation of the DLO model parameters in terms of emotional lability, resilience, and vulnerability. We conducted a Monte Carlo study to test the DLO model performance under different empirically relevant conditions in terms of individual characteristics and sampling scheme. We used state-space models in continuous time. The results show that, under certain conditions, the DLO model is able to accurately and efficiently recover the parameters underlying the affective dynamics of a single individual. We discuss the results and the theoretical and practical implications of using this model, illustrate how to use it for studying psychological phenomena at the individual level, and provide specific recommendations on how to collect data for this purpose. We also provide a tutorial website and computer code in R to implement this approach. (PsycInfo Database Record (c) 2025 APA, all rights reserved).

The aim of this work is to propose a mathematical model in terms of an exact analytical solution that may be used in numerical simulation and prediction of oscillatory dynamics of a one-dimensional viscoelastic system experiencing large deformations response. The model is represented with the use of a mechanical oscillator consisting of an inertial body attached to a nonlinear viscoelastic spring. As a result, a second-order first-degree Painlevé equation has been obtained as a law, governing the nonlinear oscillatory dynamics of the viscoelastic system. Analytical resolution of the evolution equation predicts the existence of three solutions and hence three damping modes of free vibration well known in dynamics of viscoelastically damped oscillating systems. Following the specific values of damping strength, over-damped, critically-damped and under-damped solutions have been obtained. It is observed that the rate of decay is not only governed by the damping degree but, also by the magnitude of the stiffness nonlinearity controlling parameter. Computational simulations demonstrated that numerical solutions match analytical results very well. It is found that the developed mathematical model includes a nonlinear extension of the classical damped linear harmonic oscillator and incorporates the Lambert nonlinear oscillatory equation with well-known solutions as special case. Finally, the three damped responses of the current mathematical model devoted for representing mechanical systems undergoing large deformations and viscoelastic behavior are found to be asymptotically stable.