Asymmetric Oscillator Research Articles

Calculation of secular axial frequencies in a nonlinear ion trap with hexapole, octopole and decapole superpositions by a modified Lindstedt–Poincare method

In this paper the modified Lindstedt–Poincare method is used for calculation of axial secular frequencies of a nonlinear ion trap with hexapole, octopole and decapole superpositions. The motion of the ion in a rapidly oscillating field is transformed to the motion in an effective potential. The equations of ion motion in the effective potential are in the form of a Duffing-like equation. With only octopole superposition the resulted nonlinear equations are symmetric, however, in the presence of hexapole and decapole superpositions, they are asymmetric. For asymmetric oscillators, it has been pointed out that the angular frequency for positive amplitudes is different from the angular frequency for negative amplitudes. Considering this problem, the modified Lindstedt–Poincare method is used for solving the resulted nonlinear equations. As a result, the ion secular frequencies as a function of nonlinear field parameters are obtained. The calculated secular frequencies are compared with the results of some other methods and the exact results. There is an excellent agreement between the results of this paper and the exact results.

Nonlinear Resonance: a Comparison Between Landesman-Lazer and Ahmad-Lazer-Paul Conditions

Abstract We show that the Ahmad-Lazer-Paul condition for resonant problems is more general than the Landesman-Lazer one, discussing some relations with other existence conditions, as well. As a consequence, such a relation holds, for example, when considering resonant boundary value problems associated with linear elliptic operators, the p-Laplacian and, in the scalar case, with an asymmetric oscillator.

Unboundedness of solutions for perturbed asymmetric oscillators

In this paper, we consider the existence of unbounded solutions andperiodic solutions for the perturbed asymmetric oscillator with damping$x'' + f(x )x' + ax^+ - bx^-$ $+ g(x)=p(t), $ where $x^+ =\max\{x,0\}, x^-$ $=\max\{-x,0\}$, $a$ and $b$ are twopositive constants, $f(x)$ is a continuous function and $ p(t)$ isa $2\pi $-periodic continuous function, $g(x)$ is locallyLipschitz continuous and bounded. We discuss the existence ofperiodic solutions and unbounded solutions under two classes of conditions: the resonancecase $\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}}\in Q$ and thenonresonance case $\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}} \notinQ$. Unlike many existing results in the literature wherethe function $g(x)$ is required to have asymptotic limits atinfinity, our main results here allow $g(x)$ be oscillatorywithout asymptotic limits.

Homotopy perturbation for conservative Helmholtz–Duffing oscillators

The approximate periodic solutions of the Helmholtz–Duffing oscillator are obtained by homotopy perturbation. The Helmholtz–Duffing oscillator becomes a Duffing oscillator when the homotopy parameter degenerates to one and a Helmholtz oscillator when it is zero. Since the behaviors of the solutions in the positive and negative directions are quite different, the asymmetric equation is separated into two auxiliary equations. The auxiliary equations are solved by homotopy perturbation method. A new analytical period for the Helmholtz–Duffing equation is derived. The resulting second-order approximate periodic solutions are compared to the analytical solutions using numerical integration with improved accuracy over some existing methods. Thus, the homotopy perturbation is very effective for the asymmetric nonlinear oscillators.

On perturbation theory for an asymmetric anharmonic oscillator

A modified perturbation theory is formulated for an asymmetric anharmonic oscillator based on a choice of the main approximation constructed by analogy with the self-consistent field model. This perturbation theory has a much wider range of application in comparison with the standard approach and considers a finite number of energy levels in the potential well already in the main approximation. This approach is used for the construction of a two-atomic model of a quantum crystal. A quantum analog of the Lindeman criterion is obtained.

Exponential Growth Rates of Periodic Asymmetric Oscillators

Abstract In this paper we will study the dynamics of the periodic asymmetric oscillator xʺ + q+(t)x+ + q–(t)x– = 0, where q+; q– ∊ L1(ℝ / 2πℤ) and x+ = max(x; 0), x– = min(x; 0) for x ∊ ℝ. It will be proved that the exponential growth rate does exist for each non-zero solution x(t) of the oscillator. The properties of these rates, or the Lyapunov exponents, will be given using the induced circle di®eomorphism of the oscillator. The proof is extensively based on the Denjoy theorem in topological dynamics and the unique ergodicity theorem in ergodic theory.

On the resonance response of an asymmetric Duffing oscillator

The primary resonance response of an asymmetric Duffing oscillator with no linear stiffness term and with hardening characteristic is investigated in this paper. An approximate solution corresponding to the steady-state response is sought by applying the harmonic balance method. Its stability is also studied. It is found that different shapes of frequency–response curves can exist. Multiple-valued solutions, indicating the occurrence of jump phenomena, are observed analytically and confirmed numerically. The influence of the system parameters on the primary resonance response is also examined.

Virial theorem and energy partitioning in systems with mixed power-law potentials

Clausius’ Virial Theorem is often invoked to predict the partitioning of kinetic and potential energies in either classical or quantum systems with simple power-law potentials. Here, the Virial Theorem, and related statistical mechanical identities, are used to investigate energy partitioning in classical systems with mixed power-law potentials, in either one or three dimensions, with either positive or negative mixed power-law exponents. Such classical systems are found to have temperature-dependent heat capacities reminiscent of quantum behaviour, as well as other unusual properties. Closed form analytical expressions are obtained for energy partitioning in a family of one-dimensional generalised asymmetric oscillators. Unlike the corresponding symmetric oscillators, these systems are predicted to undergo a thermally driven migration of energy between spatially separated regions. Analogous behaviour is obtained for three-dimensional systems with mixed inverse power-law potentials, as exemplified by non-ideal gases with Lennard–Jones n 0–n 1 interaction potentials. In this case the Virial Theorem implies that the ratio of the average energy associated with the negative (n 1) and positive (n 0) parts of the potential is exactly equal to −n 0/n 1 at the Boyle temperature (where the second virial coefficient vanishes), and approximately so at other temperatures.

Free and forced motion in an asymmetric liquid-column oscillator

The results of a combined theoretical, numerical and experimental study of liquid oscillations in an asymmetric U-tube are presented. The configuration under investigation is analogous to that of the tuned liquid-column damper used to suppress oscillatory motion in large semi-supported structures. The liquid motion is described by a second-order ordinary differential equation that is nonlinear when the widths of the two vertical columns are unequal. It is shown that this asymmetry can be used as a tuning parameter to determine the natural frequency of free oscillations in the system, in addition to the known tuning effect of the connecting chamber height. The effects of viscous damping and periodic forcing are considered, leading to a description of probable initial and long-term resonance behaviour in a practical asymmetric device.

Method of studying the synchronization of self-sustained oscillations using continuous wavelet analysis of univariant data

A new method for detecting the synchronization of a self-sustained oscillator under an external action with a linearly increasing frequency is proposed. It is based on the continuous wavelet transformation of univariant data (scalar time series). The efficiency of the method is demonstrated with a modified asymmetric van der Pol oscillator, Rossler oscillator, and experimental physiological data. In the last case, synchronization between rhythmic processes of different order in the cardiovascular and respiratory systems of the man is dem- onstrated using only the time series of the R - R intervals.

Detection of synchronization from univariate data using wavelet transform

A method is proposed for detecting from univariate data the presence of synchronization of a self-sustained oscillator by external driving with varying frequency. The method is based on the analysis of difference between the oscillator instantaneous phases calculated using continuous wavelet transform at time moments shifted by a certain constant value relative to each other. We apply our method to a driven asymmetric van der Pol oscillator, experimental data from a driven electronic oscillator with delayed feedback and human heartbeat time series. In the latest case, the analysis of the heart rate variability data reveals synchronous regimes between the respiration and slow oscillations in blood pressure.

Stability analysis of dynamic ratcheting in elastoplastic systems

This paper deals with the stability and the dynamics of a harmonically excited elastic perfectly plastic asymmetrical oscillator. The hysteretical system is written as a nonsmooth, forced autonomous system. The dimension of the phase space can be reduced using adapted variables. It is shown that asymmetry of boundary conditions (forcing term) and material asymmetry lead to an equivalent system for this simple structural case. The forced vibration of such an oscillator is treated by a numerical approach by using time locating techniques. Stability of the limit cycles is analytically investigated with a perturbation approach. The boundary between elastoplastic shakedown and dynamic ratcheting is given in closed form. It is shown that the divergence rate is strongly correlated to the internal asymmetry of the oscillator.

DisappearingQoperator

In the Schr\"odinger formulation of non-Hermitian quantum theories a positive-definite metric operator $\ensuremath{\eta}\ensuremath{\equiv}{e}^{\ensuremath{-}Q}$ must be introduced in order to ensure their probabilistic interpretation. This operator also gives an equivalent Hermitian theory, by means of a similarity transformation. If, however, quantum mechanics is formulated in terms of functional integrals, we show that the $Q$ operator makes only a subliminal appearance and is not needed for the calculation of expectation values. Instead, the relation to the Hermitian theory is encoded via the external source $j(t)$. These points are illustrated and amplified for two non-Hermitian quantum theories: the Swanson model, a non-Hermitian transform of the simple harmonic oscillator, and the wrong-sign quartic oscillator, which has been shown to be equivalent to a conventional asymmetric quartic oscillator.

THE ASYMPTOTIC ITERATION METHOD FOR THE EIGENENERGIES OF THE ASYMMETRICAL QUANTUM ANHARMONIC OSCILLATOR POTENTIALS $V(x)=\sum_{j=2}^{2\alpha}A_{j}x^{j}$

The asymptotic iteration method is used to calculate the eigenenergies for the asymmetrical quantum anharmonic oscillator potentials [Formula: see text], with (α = 2) for quartic, and (α = 3) for sextic asymmetrical quantum anharmonic oscillators. An adjustable parameter β is introduced in the method to improve its rate of convergence. Comparing the present results with the exact numerical values, and with the numerical results of the earlier works, it is found that asymptotically, this method gives accurate results over the full range of parameter values Aj.

Linear delta expansion technique for the solution of anharmonic oscillations

The linear delta expansion technique has been developed for solving the differential equation of motion for symmetric and asymmetric anharmonic oscillators. We have also demonstrated the sophistication and simplicity of this new perturbation technique.

The stability of the equilibrium for a perturbed asymmetric oscillator

In this paper we will derive some stability criteria for the equilibrium of a perturbed asymmetric oscillator <p align="center"> $\ddot x +a^+ x^+ - a^-$ $x^-$ $+ b(t)x^2+r(t,x)=0,$ <p align="left" class="times"> where $a^+,a^-$ are two different positive numbers, $b(t)$ is a $2\pi$-periodic function, and the remaining term $r(t,x)$ is $2\pi$-periodic with respect to the time $t$ and dominated by the power $x^3$ in a neighborhood of the equilibrium $x=0$.

A variational approach to resonance for asymmetric oscillators

We consider in this note the equation <p align="center"> $x'' + \alpha x^+ - \beta x^- + g(x) =p(t),$ <p align="left" class="times"> where $x^+ =$ max{$x,0$} is the positive part of $x$, $x^- $ =max{$-x,0$} its negative part and $\alpha,\beta$ are positive parameters. We assume that $g :\mathbb R \to \mathbb R$ is continuous and bounded on $\mathbb R$, $p:\mathbb R \to \mathbb R$ is continuous and $2\pi$-periodic. We provide some sufficient conditions of Ahmad, Lazer and Paul type for the existence of $2\pi$-periodic solutions when $(\alpha,\beta)$ belongs to one of the curves of the Fučík spectrum corresponding to $2\pi$-periodic boundary conditions.

Discretely proceeding from cycle to chaos on Goodwin's path

Goodwin's [Goodwin, R.M., 1951. The nonlinear accelerator and the persistence of business cycles. Econometrica 19, 1–17; Goodwin, R.M., 1955. A model of cyclical growth. In: Lundberg, E. (Ed.), The Business Cycle in the Post-war World. Macmillan, London, pp. 203–221] nonlinear multiplier–accelerator model, worked out in continuous-time, is a recognised contribution to business cycle theory. It is rarely observed that its first version was a linear model formulated in discrete-time [Goodwin, R.M., 1946. Innovations and the irregularity of economic cycles. Review of Economics and Statistics 28, 95–104]. A few decades later, he restated the fully-fledged nonlinear version of the model in discrete-time showing that such a version may account better for the complex behaviour of empirical time series [e.g., Goodwin, R.M., 1985. An irregular, asymmetric oscillator, or The discrete charm of erraticism, Mimeo, Siena (reproduced, with the title The discrete charm of erraticism, in Goodwin, R.M., 1989. Essays in Nonlinear Economic Dynamics. Peter Lang, Frankfurt am Main, pp. 139–156); Goodwin, R.M., 1988. The multiplier/accelerator discretely revisited. In: Ricci, G., Velupillai, K. (Eds.), Growth Cycles and Multisectoral Economics: The Goodwin Tradition. Springer-Verlag, Berlin, pp. 19–29]. The article reconstructs the evolution of the multiplier–accelerator model in Goodwin's thought with special emphasis on the early and late discrete version. First, the genesis of the model is considered in some depth in order to clarify its foundations based on the constraints of a monetary economy. Second, the results of Goodwin's late contributions are amended and generalised. Finally, the path followed by Goodwin is reconstructed and appraised in the light of the dialectics between continuity and discontinuity, regularity and irregularity, stability and instability that steered its direction. The main conclusion is that Goodwin's path should be further pursued as an effective alternative to the equilibrium business cycle models.

A NONINTEGRABILITY CRITERION FOR ADIABATIC SYSTEMS

In the present paper we investigate the nonintegrability of adiabatic one degree of freedom Hamiltonian systems, with the additional assumption that the frozen system possesses an unstable fixed point with two asymmetric homoclinic loops. We prove a criterion for the nonexistence of an integral for such systems, and therefore we prove the nonexistence of a quantity which is conserved in an arbitrarily high order on ε. A specific application is given in the asymmetric quartic oscillator with adiabatic time dependence.

The stability of the equilibrium for a perturbed asymmetric oscillator

In this paper we will derive some stability criteria for theequilibrium of a perturbed asymmetric oscillator$\ddot x + a^+ x^+ - a^-$$ x^-$ $+ b(t)x^2+r(t,x)=0,$where $a^+,a^-$ are two different positive numbers, $b(t)$ is a$2\pi$-periodic function, and the remaining term $r(t,x)$ is$2\pi$-periodic with respect to the time $t$ and dominated by thepower $x^3$ in a neighborhood of the equilibrium $x=0$.