External Periodic Excitation Research Articles

Cascades of subharmonic stationary states in strongly non-linear driven planar systems

The dynamics of a one-degree-of-freedom oscillator with arbitrary polynomial non-linearity subjected to an external periodic excitation is studied. The sequences (cascades) of harmonic and subharmonic stationary solutions to the equation of motion are obtained by using the harmonic balance approximation adapted for arbitrary truncation numbers, powers of non-linearity, and orders of subharmonics. A scheme for investigating the stability of the harmonic balance stationary solutions of such a general form is developed on the basis of the Floquet theorem. Besides establishing the stable/unstable nature of a stationary solution, its stability analysis allows the regions of parameters to be obtained, where symmetry-breaking and period-doubling bifurcations occur. Thus, for period-doubling cascades, each unstable stationary solution is used as a base solution for finding a subsequent stationary state in a cascade. The procedure is repeated until this stationary state becomes stable provided that a stable solution can finally be achieved. The proposed technique is used to calculate the sequences of subharmonic stationary states in driven hardening Duffing's oscillator. The existence of stable subharmonic motions found is confirmed by solving the differential equation of motion numerically by means of a time-difference method, with initial conditions being supplied by the harmonic balance approximation.

Optimal Control of Homoclinic Bifurcation: Theoretical Treatment and Practical Reduction of Safe Basin Erosion in the Helmholtz Oscillator

A control method of the homoclinic bifurcation is developed and applied to the nonlinear dynamics of the Helmholtz oscillator. The method consists of choosing the shape of external and/or parametric periodic excitations, which permits us to avoid, in an optimal manner, the transverse intersection of the stable and unstable manifolds of the hilltop saddle. The homoclinic bifurcation is detected by the Melnikov method, and its dependence on the shape of the excitation is shown. We successively investigate the mathematical problem of optimization, which consists of determining the theoretical optimal excitation that maximizes the distance between stable and unstable manifolds for fixed excitation amplitude or, equivalently, the critical amplitude for homoclinic bifurcation. The optimal excitations in the reduced case with a finite number of superharmonic corrections are first determined, and then the optimization problem with infinite superharmonics is investigated and solved under a constraint on the relevant amplitudes, which is necessary to guarantee the physical admissibility of the mathematical solution. The mixed case of a finite number of constrained superharmonics is also considered. Some numerical simulations are then performed aimed at verifying the Melnikov's theoretical predictions of the homoclinic bifurcations and showing how the optimal excitations are indeed able to separate stable and unstable manifolds. Finally, we numerically investigate in detail the effectiveness of the control method with respect to the basin erosion and escape phenomena, which are the most important and dangerous practical aspects of the Helmholtz oscillator.

Optimal Control of Nonregular Dynamics in a Duffing Oscillator

A method for controlling nonlinear dynamics and chaos previouslydeveloped by the authors is applied to the classical Duffing oscillator.The method, which consists in choosing the best shape of externalperiodic excitations permitting to avoid the transverse intersection ofthe stable and unstable manifolds of the hilltop saddle, is firstillustrated and then applied by using the Melnikov method foranalytically detecting homoclinic bifurcations. Attention is focused onoptimal excitations with a finite number of superharmonics, because theyare theoretically performant and easy to reproduce. Extensive numericalinvestigations aimed at confirming the theoretical predictions andchecking the effectiveness of the method are performed. In particular,the elimination of the homoclinic tangency and the regularization offractal basins of attraction are numerically verified. The reduction ofthe erosion of the basins of attraction is also investigated in detail,and the paper ends with a study of the effects of control on delayingcross-well chaotic attractors.

Class of kick-excited self-adaptive dynamical systems: “quantized” oscillation excitations

A class of kick-excited self-adaptive dynamical systems is formed and proposed. The class is characterized by a non-linear (inhomogeneous) external periodic excitation (as regards the coordinates of the excited system) and is remarkable for the occurrence of the following objective regularities: the phenomenon of “discrete” (“quantized”) oscillation excitation in macro-dynamical systems having multiple branch attractors and strong self-adaptive stability. The main features of the class of systems are studied both numerically and analytically on the basis of the general model of a pendulum under inhomogeneous action of a periodic force (referred to as a kicked pendulum). A diagram involving multiple bifurcations for the attractor set of the system under consideration is obtained and analyzed. The complex dynamics, evolution and the fractal boundaries of the multiple attractor basins in state space corresponding to energy and initial phase variables are obtained, traced and discussed. An analytic proof is presented showing the existence of “quantized” oscillations for the kick-excited pendulum. An analytic approach is given applicable to the cases of small and large amplitudes (small and large non-linearity). The spectrum of possible oscillation amplitudes for the pendulum is studied as well as its motion in a rotational regime under the influence of an external non-homogeneous periodic force. Generalized conditions for the excitation of pendulum oscillations under the influence of an external non-linear force are derived. A wide spectrum of applications of the formed class of systems is presented.

Solitons trapping for the nonlinear Klein–Gordon equation with an external excitation

The asymptotic perturbation (AP) method is applied to the study of the nonlinear Klein–Gordon equation and an external periodic excitation is supposed to be in primary resonance with the frequency of a generic mode. The AP method uses two different procedures for the solutions: introducing an asymptotic temporal rescaling and balancing of the harmonic terms with a simple iteration. Amplitude and phase modulation equations and external force–response and frequency–response curves are obtained and the presence of multistabilities and solitons trapping is demonstrated. The validity of the method is highlighted by comparing the approximate solutions with results of the numerical integration.

Dromions bound states

The asymptotic perturbation (AP) method is applied to the study of the nonlinear Klein–Gordon equation in 3+1 dimensions with harmonic potential and external periodic excitation supposed to be in primary resonance with the frequency of a generic mode. The AP method uses two different procedures for the solutions: introducing an asymptotic temporal rescaling and balancing of the harmonic terms with a simple iteration. Standard quantum mechanics can be used to derive the lowest order approximate solution and amplitude and phase modulation equations are obtained. External force–response and frequency–response curves are found and the existence of dromions trapped in bound states is demonstrated.

Onset of chaotic dynamics in a ball mill: Attractors merging and crisis induced intermittency.

In mechanical treatment carried out by ball milling, powder particles are subjected to repeated high-energy mechanical loads which induce heavy plastic deformations together with fracturing and cold-welding events. Owing to the continuous defect accumulation and interface renewal, both structural and chemical transformations occur. The nature and the rate of such transformations have been shown to depend on variables, such as impact velocity and collision frequency that depend, in turn, on the whole dynamics of the system. The characterization of the ball dynamics under different impact conditions is then to be considered a necessary step in order to gain a satisfactory control of the experimental set up. In this paper we investigate the motion of a ball in a milling device. Since the ball motion is governed by impulsive forces acting during each collision, no analytical expression for the complete ball trajectory can be obtained. In addition, mechanical systems exhibiting impacts are strongly nonlinear due to sudden changes of velocities at the instant of impact. Many different types of periodic and chaotic impact motions exist indeed even for simple systems with external periodic excitation forces. We present results of the analysis on the ball trajectory, obtained from a suitable numerical model, under growing degree of impact elasticity. A route to high dimensional chaos is obtained. Crisis and attractors merging are also found. (c) 2002 American Institute of Physics.

Turbulent Structure of a Gas-Liquid Impinging Jet

The turbulent structure of a submerged axisymmetric impinging jet containing small gas bubbles is studied experimentally under conditions of periodic external excitation. On the basis of measuring the surface-friction pulsatory component in the jet impinging on an obstacle, the effect of the suppression of large-scale eddies at large gas volume fractions is registered. The conditions of resonant growth of coherent structures and the suppression of wide-band turbulence are determined for both the single-phase and the two-phase impinging jet. An analysis of the development of different pulsatory friction components in the impinging-jet gradient region is presented.

Regulation of the Dynamics of a Carrying Body Using the Cycloidal Damper of Forced Oscillations with Dry Friction

We consider the forced vibrations of the vibro-protective system of rigid bodies "a heavy cylinder at the cycloidal depression of a carrying movable body", allowing for side forces of dry Coulomb friction, under the action of a periodic external excitation. The dynamic equations of joint motion of the cylinder at the cycloidal depression without slip and carrying body have been formed and numerically investigated. We obtained the graphic chart of the maximum amplitude of the carrying body upon the value of the dry friction coefficient

Regular and Chaotic Dynamic Analysis for a Vibratically Vibrating and Rotating Elliptic Tube Containing a Particle.

The paper is to present the detailed dynamic analysis of a vertically vibrating and rotating elliptic tube containing a particle. By subjecting to an external periodic excitation, it has shown that the system exhibits both regular and chaotic motions. By using the Lyapunov direct method and Chetaev’s theorem, the stability and instability of the relative equilibrium position of the particle in the tube can be determined. The center manifold theorem is applied to verify the conditions of stability when system is under the critical case. The effects of the changes of parameters in the system can be found in the bifurcation and parametric diagrams. By applying various numerical results such as phase plane, Poincaré map and power spectrum analysis, a variety of the periodic solutions and the phenomena of the chaotic motion can be presented. Further, chaotic behavior can be verified by using Lyapunov exponents and Lyapunov dimensions.

Attractors of weakly damped forced KdV equations with external periodic excitation

Consider the KdV equations with the external periodic excitation ut+uux+uxxx+γu=f(t), with f(t+T)=f(t). Prove the existence of attractors of resulting discrete semigroup |S(Ʈ+mT)|m∈N.

“Quantized" Oscillations and Irregular Behavior of a Class of Kick-Excited Self-Adaptive Dynamical Systems

A class of kick-excited self-adaptive dynamical systems is formed and proposed. The class is characterized by nonlinear (inhomogeneous) external periodic excitation (with respect to the coordinates of the excited systems) and is remarkable for its objective regularities: the phenomenon of “quantized” oscillation excitation and strong self-adaptive stability. Multiple bifurcation diagram for the attractor set of the system under consideration is obtained and analysed.

Saddle-Node Bifurcations of Cycles in a Relief Valve

We extend the asymptotic perturbation (AP) method to the studyof a linear partial differential equation with nonlinear boundaryconditions. A relief valve under the combined effects of static anddynamic loadings is considered with the following resonances between thenth linear mode and the external periodic excitation: primaryresonance, subharmonic resonance of order one-half or one-third,superharmonic resonance of order-two or order-three and combinationresonance. The AP method uses two different procedures for thesolutions: introducing an asymptotic temporal rescaling and balancing ofthe harmonic terms with a simple iteration. We obtain amplitude andphase modulation equations and determine external force-response andfrequency-response curves. Stability of steady-state motions is alsoinvestigated. Saddle-node bifurcations of cycles are observed and underappropriate conditions the performance of the relief valve may beunsatisfactory due to the presence of jumps and hysteresis effects inthe system response. Global analysis is used in order to exclude theexistence of modulated motion. The validity of the method is highlightedby comparing approximate solutions with results of the numericalintegration.

“Discrete” Oscillations and multiple attractors in kick‐excited systems

A class of kick‐excited self‐adaptive dynamical systems is formed and proposed. The class is characterized by nonlinear (inhomogeneous) external periodic excitation (as regards to the coordinates of excited systems) and is remarkable for its objective regularities: the phenomenon of “discrete” (“quantized”) oscillation excitation and strong self‐adaptive stability. The main features of these systems are studied both numerically and analytically on the basis of a general model: a pendulum under inhomogeneous action of a periodic force which is referred to as a kicked pendulum. Multiple bifurcation diagram for the attractor set of the system under consideration is obtained and analyzed. The complex dynamics, evolution and the fractal boundaries of the multiple attractor basins in state space corresponding to energy and phase variables are obtained, traced and discussed. A two‐dimensional discrete map is derived for this case. A general treatment of the class of kick‐excited self‐adaptive dynamical systems is made by putting it in correspondence to a general class of dissipative twist maps and showing that the latter is an immanent tool for general description of its behavior.

The Response of an Oscillator Moving along a Parabola to an External Excitation

The behavior of a mass point moving along a parabola under theeffect of an external periodic excitation in resonance with the naturalfrequency of the oscillator is studied. The asymptotic perturbationmethod based on temporal rescaling and balancing of the harmonic termswith a simple iteration is used in order to determine the nonlinearmodulation equations for the amplitude and the phase of the oscillation.External force-response curves are shown and moreover jump phenomena arealso observed. In certain cases a second low frequency appears inaddition to the forcing frequency and then stable two-periodquasi-periodic motions are present with amplitudes depending on theinitial conditions. The value of the low frequency depends on theamplitude of the external excitation. A higher order perturbationanalysis is developed and the validity of the method is highlighted bycomparing the leading order and the higher order approximate analyticsolutions to numerical results.

BIFURCATION STRUCTURES GENERATED BY THE NONAUTONOMOUS DUFFING EQUATION

This paper deals with some properties of bifurcation structures in the parameter space related to the Duffing equation in the presence of an external periodical excitation B + B0 cos t. So global qualitative modifications of structures in the parameter plane (B, B0) are considered, when a third parameter ε (the damping term of the equation) varies. Complex sets of bifurcation curves are defined. They are based on the global bifurcation structure: crossroad area, saddle area, spring area, lip, quasi-lip, identified in the past with their "foliated representation", and their qualitative changes. For the Duffing model, special associations of the above areas, giving typical patterns called islands, are described with their qualitative modifications.

The Asymptotic Perturbation Method for Nonlinear Continuous Systems

We apply the asymptotic perturbation (AP) method to the study of the vibrations of Euler--Bernoulli beam resting on a nonlinear elastic foundation. An external periodic excitation is in primary resonance or in subharmonic resonance in the order of one-half with an nth mode frequency. The AP method uses two different procedures for the solutions: introducing an asymptotic temporal rescaling and balancing the harmonic terms with a simple iteration. We obtain amplitude and phase modulation equations and determine external force-response and frequency-response curves. The validity of the method is highlighted by comparing the approximate solutions with the results of the numerical integration and multiple-scale methods.

THE PASSAGE THROUGH RESONANCE IN A CATENARY–VERTICAL CABLE HOISTING SYSTEM WITH SLOWLY VARYING LENGTH

A passage through resonance in a catenary–vertical cable system with periodic external excitation is analyzed. Due to the time-varying length of the vertical cable the natural frequencies of the system vary slowly, and a transient resonance may occur when one of the frequencies coincides with the frequency of an external excitation at some critical time. A simplified model of the system with proportional damping is proposed. This model is analyzed by using a combined perturbation and numerical technique. The method of multiple scales is used to formulate a uniformly valid perturbation expansion for the response near the resonance, and a system of first order ordinary differential equations for the slowly varying amplitude and phase of the response results. This system is integrated numerically on a slow time scale. A model example is discussed, and the behavior of the essential dynamic properties of the system during the transition through resonance is examined.

Bifurcations and transition phenomena in an impact oscillator

The motion of mechanical systems with impacts is strongly nonlinear. Many different types of periodic and chaotic impact motions exist even for simple systems with external periodic excitation forces. The group of fundamental periodic motions is characterized by the different number of impacts in one motion period, which equals the excitation force period. Every motion has a region in the space of system parameters in which the solution can exist and is stable. There exist transition regions, so-named hysteresis regions and beat motion regions, which lie between the zones of neighbouring fundamental impact motions. Transition regions are determined by the boundaries of existence which correspond to grazing bifurcations and by the boundaries of stability corresponding to the period-doubling and saddle-node bifurcations. The transition between neighbouring periodic impact motions is never continuous, with the exception of singular points, where the existence boundaries and stability boundaries intersect. Jump phenomena appear on the hysteresis region boundaries and subharmonic and chaotic motions exist in the transition beat motion region. This paper presents results of theoretical analysis, analogue simulations and numerical solutions of a particular but typical impacting system.

Analysis of dynamics of monolithic passively mode-locked laser diodes under external periodic excitation

A distributed time-domain model is used for numerical analysis of the dynamics of a passively mode-locked laser diode under external modulation at a frequency close to the round-trip frequency of the laser. Dynamical regimes of the laser, such as synchronisation locking, quasilocking and RF frequency mixing, are studied. For the locked regime, steady-state parameters are defined, the crucial role of group-velocity dispersion in achieving locking is demonstrated and stages of the locking dynamics and corresponding time constants are identified.