Multiple bifurcation trees of period-1 motions to chaos in a periodically forced,time-delayed,hardening Duffing oscillator

In this paper, bifurcation trees of periodic motions in a periodically forced, time-delayed, hardening Duffing oscillator are analytically predicted by a semi-analytical method. Such a semi-analytical method is based on the differential equation discretization of the time-delayed, nonlinear dynamical system. Bifurcation trees for the stable and unstable solutions of periodic motions to chaos in such a time-delayed, Duffing oscillator are achieved analytically. From the finite discrete Fourier series, harmonic frequency-amplitude curves for stable and unstable solutions of period-1 to period-4 motions are developed for a better understanding of quantity levels, singularity and catastrophes of harmonic amplitudes in the frequency domain. From the analytical prediction, numerical results of periodic motions in the time-delayed, hardening Duffing oscillator are completed. Through the numerical illustrations, the complexity and asymmetry of period-1 motions to chaos in nonlinear dynamical systems are strongly dependent on the distributions and quantity levels of harmonic amplitudes. With the quantity level increases of specific harmonic amplitudes, effects of the corresponding harmonics on the periodic motions become strong, and the certain complexity and asymmetry of periodic motion and chaos can be identified through harmonic amplitudes with higher quantity levels.     

Dynamics of an impact-progressive system

An impact oscillator with a frictional slider is considered. The basic function of the investigated system is to overcome the frictional force and move downwards. Based on the analysis of the oscillatory and progressive motions of the system, we introduce an impact Poincaré map with dynamical variables defined at the impact instants. The nonlinear dynamics of the impact system with a frictional slider is analyzed by using the impact Poincaré map. The stability and bifurcations of single-impact periodic motions are analyzed, and some information about the existence of other types of periodic-impact motions is provided. Since the system equilibrium is moving downwards, one way to monitor the progression rate is to calculate its progression in a finite time. The simulation results show that in a finite time, the largest progression of the system is found to occur for period-1 multi-impact motions existing in the regions of low forcing frequencies. Secondly, the progression of the period-1 single-impact motion with peak-impact velocity is also distinct enough. However, it is important to note, that the largest progression for period-1 multi-impact motion existing at a low forcing frequency is not an optimal choice for practical engineering applications. The greater the number of the impacts in an excitation period, the more distinct the adverse effects such as high noise levels and wear and tear caused by impacts. As a result, the progression of the period-1 single-impact motion with the peak-impact velocity is still optimal for practical applications. The influence of parameter variations on the oscillatory and progressive motions of the impact-progressive system are elucidated accordingly, and feasible parameter regions are provided.     

Synchronization of a periodically forced Duffing oscillator with a periodically excited pendulum

In this paper, the analytical conditions for a periodically forced Duffing oscillator synchronized with a chaotic pendulum are developed through the theory of discontinuous dynamical systems. From the analytical conditions, the synchronization invariant domains are developed. For a better understanding of synchronization of two different dynamical systems, the partial and full synchronizations of the Duffing oscillator with the chaotic pendulum are presented for illustrations. The control parameter map is developed from the analytical conditions. Under special parameters, the two systems can be fully and partially synchronized. Since the forced pendulum has librational and rotational chaotic motions, the periodically forced Duffing oscillator can be synchronized only with the librational chaotic motions of the pendulum. It is impossible for the forced Duffing oscillator to be synchronized with the rotational chaotic motions.     

Bifurcation sequences of vibroimpact systems near a 1:2 strong resonance point

Two vibroimpact systems are considered, which can exhibit symmetrical double-impact periodic motions under suitable system parameter conditions. Dynamics of such systems are studied by use of maps derived from the equations of motion, between impacts, supplemented by transition conditions at the instants of impacts. Two-parameter bifurcations of fixed points in the vibroimpact systems, associated with 1:2 strong resonance, are analyzed. Interesting features like Neimark–Sacker bifurcation of period-1 double-impact symmetrical motion, tangent bifurcation of period-2 four-impact motion, period-doubling bifurcation of period-2 four-impact motion and Neimark–Sacker bifurcation of period-4 eight-impact motion, etc., are found to occur near 1:2 resonance point of a vibroimpact system. The quasi-periodic attractor, associated with the fixed point of period-1 double-impact symmetrical motion, is destroyed as a tangent bifurcation of fixed points of period-2 four-impact motion occurs. However, for the other vibroimpact system the quasi-periodic attractor is restored via the collision of stable and unstable fixed points of period-2 four-impact motion. The results mean that there exist possibly more complicated bifurcation sequences of period-two cycle near 1:2 resonance points of non-linear dynamical systems.     

Grazing phenomena in a periodically forced,friction-induced,linear oscillator

The criterion for grazing motions in a dry-friction oscillator is obtained from the local theory of non-smooth dynamical systems on the connectable and accessible domains. The generic mappings for such a dry-friction oscillator are also introduced. The sufficient and necessary conditions for grazing at the final states of mappings are expressed. The initial and final switching sets of grazing mapping, varying with system parameters, are illustrated for the grazing parametric characteristics. The initial and grazing, switching manifolds in the switching sets are defined through grazing mappings. Finally, numerical illustrations of grazing motions are very easily carried out with help of the analytical predictions. This paper provides a comprehensive investigation of grazing motions in the dry-friction oscillator for a better understanding of the grazing mechanism of such a discontinuous system. The investigation based on the local singularity theory is more intuitive and efficient than the discontinuous mapping techniques.     

Discontinuous dynamics of a non-linear, self-excited, friction-induced, periodically forced oscillator

The discontinuous dynamics of a non-linear, friction-induced, periodically forced oscillator is studied. The analytical conditions for motion switchability at the velocity boundary in such a nonlinear oscillator are developed to understand the motion switching mechanism. Using such analytical conditions of motion switching, numerical predictions of the switching scenarios varying with excitation frequency and amplitude are carried out, and the parameter maps for specific periodic motions are presented. Chaotic and periodic motions are illustrated through phase planes and switching sections for a better understanding of motion mechanism of the nonlinear friction oscillator. The periodic motions with switching in such a nonlinear, frictional oscillator cannot be obtained from the traditional analysis (e.g., perturbation and harmonic balance method).     

Sliding and transversal motions on an inclined boundary in a periodically forced discontinuous system

In this paper, sliding and transversal motions on the boundary in the periodically driven, discontinuous dynamical system is investigated. The simple inclined straight line boundary in phase space is considered as a control law for such a dynamical system to switch. The normal vector field for a flow switching on the separation boundary is adopted to develop the analytical conditions, and the corresponding transversality conditions of a flow to the boundary are obtained. The conditions of sliding and grazing flows to the separation boundary are presented as well. Using mapping structures, periodic motions of such a discontinuous system are predicted, and the corresponding local stability and bifurcation analysis of the periodic motion are carried out. Numerical illustrations of periodic motions with and without sliding on the boundary are given. The local stability analysis cannot provide the proper prediction of the sliding and grazing motions in discontinuous dynamical systems. Therefore, the normal vector fields of periodic flows are presented, and the normal vector fields on the switching boundary points give the analytical criteria for sliding and transversality of motions.     

Explanation of appearance and characteristics of intermittency chaos of the impact oscillator

Chaotic motion of an intermittency type of the impact oscillator appears near segments of saddle-node stability boundaries of subharmonic motions with two different impacts in motion period, which is n multiple (n3) of excitation period. Chaotic motion arises due to an additional impact, which interrupts the process of instability. It is proved and shown by numerical simulations of the system motion. More detail characteristics of the intermittency chaos are evaluated. Described phenomena present a non-usual example, when transition cross special segments of saddle-node stability boundaries of subharmonic impact motions is reversible.     

Grazing bifurcations of a harmonically excited oscillator moving on a time-varying translation belt

The criterion for grazing motions in a friction-induced oscillator with a time-varying transport belt is obtained. The three mappings for such a friction-induced oscillator are introduced for analytical prediction of the grazing motions. The sufficient and necessary conditions of grazing are expressed from mappings. With system parameter variations, the initial and final switching sets of grazing mapping are illustrated. Numerical illustrations of grazing motions are carried out from analytical predictions. This investigation provides a technique for how to determine the grazing on the time-varying boundary in discontinuous dynamical systems.     

The chaotic synchronization of a controlled pendulum with a periodically forced,damped Duffing oscillator

In this paper, the chaotic synchronization of the Duffing oscillator and controlled pendulum is investigated. From the analytical conditions developed in [1], the partial and full synchronizations of the controlled pendulum with chaotic motions in the Duffing oscillator are discussed. Compared with the periodic synchronization, in the chaotic synchronization, switching points for appearance and vanishing of the partial synchronization are chaotic. The control parameter map for the synchronization is developed from the analytical conditions, and the partial and full synchronizations are illustrated to show the analytical conditions. This synchronization is different from the controlled Duffing oscillator synchronizing with chaotic motion in the periodically excited pendulum. For a better understanding of synchronization characteristics between two different dynamical systems, effects with other parameters will be discussed later.     

On motions and switchability in a periodically forced,discontinuous system with a parabolic boundary

The analytical conditions for motion switchability on the switching boundary in a periodically forced, discontinuous system are developed through the G-function of the vector fields to the switching boundary. Periodic motions in such a discontinuous dynamical system are discussed by the use of mapping structures. Two periodic motions and the analytical conditions are presented for illustration. Further investigation should be carried out for a better understanding of the vanishing and stability of regular and chaotic motions.     

Flow switchability and periodic motions in a periodically forced,discontinuous dynamical system

This paper presents the switchability of a flow from one domain into another one in the periodically forced, discontinuous dynamical system. The inclined line boundary in phase space is used for the dynamical system to switch. The normal vector field product for flow switching on the separation boundary is introduced. The passability condition of a flow to the separation boundary is achieved through such a normal vector field product, and the sliding and grazing conditions to the separation boundary are presented as well. Using mapping structures, periodic motions in such a discontinuous system are predicted analytically, and the corresponding local stability and bifurcation analysis are carried out. With the analytical conditions of grazing and sliding motions, the parameter maps of specific motions are developed. Illustrations of periodic and chaotic motions are given, and the normal vector fields are presented to show the analytical criteria. This investigation may help one better understand the sliding mode control. The methodology presented in this paper can be applied to discontinuous, nonlinear systems.     

Periodic and chaotic synchronizations of two distinct dynamical systems under sinusoidal constraints

In this paper, periodic and chaotic synchronizations between two distinct dynamical systems under specific constraints are investigated from the theory of discontinuous dynamical systems. The analytical conditions for the sinusoidal synchronization of the pendulum and Duffing oscillator were obtained, and the invariant domain of sinusoidal synchronization is achieved. From analytical conditions, the control parameter map is developed. Numerical illustrations for partial and full sinusoidal synchronizations of chaotic and periodic motions of the controlled pendulum with the Duffing oscillator are carried out. This paper presents how to apply the theory of discontinuous dynamical systems to dynamical system synchronization with specific constraints. The function synchronization of two distinct dynamical systems with specific constraints should be identified only by G-functions. The significance of function synchronization of distinct dynamical systems is to make the synchronicity behaviors hidden, which is very useful for telecommunication synchronization and network security.     

Synchronization dynamics of two different dynamical systems

In this paper, synchronization dynamics of two different dynamical systems is investigated through the theory of discontinuous dynamical systems. The necessary and sufficient conditions for the synchronization, de-synchronization and instantaneous synchronization (penetration or grazing) are presented. Using such a synchronization theory, the synchronization of a controlled pendulum with the Duffing oscillator is systematically discussed as a sampled problem, and the corresponding analytical conditions for the synchronization are presented. The synchronization parameter study is carried out for a better understanding of synchronization characteristics of the controlled pendulum and the Duffing oscillator. Finally, the partial and full synchronizations of the controlled pendulum with periodic and chaotic motions are presented to illustrate the analytical conditions. The synchronization of the Duffing oscillator and pendulum are investigated in order to show the usefulness and efficiency of the methodology in this paper. The synchronization invariant domain is obtained. The technique presented in this paper should have a wide spectrum of applications in engineering. For example, this technique can be applied to the maneuvering target tracking, and the others.     

Controlling bifurcation and chaos of a plastic impact oscillator

A two-degree-of-freedom plastic impact oscillator is considered. Based on the analysis of sticking and non-sticking impact motions of the system, we introduce a three-dimensional impact Poincaré map with dynamical variables defined at the impact instants. The plastic impacts complicate the dynamic responses of the impact oscillator considerably. Consequently, the piecewise property and singularity are found to exist in the three-dimensional map. The piecewise property is caused by the transitions of free flight and sticking motions of two masses immediately after impact, and the singularity of the map is generated via the grazing contact of two masses and the instability of their corresponding periodic motions. The nonlinear dynamics of the plastic impact oscillator is analyzed by using the Poincaré map. The simulated results show that the dynamic behavior of this system is very complex under parameter variation, varying from different types of sticking or non-sticking periodic motions to chaos. Suppressing bifurcation and chaotic-impact motions is studied by using an external driving force, delay feedback and damping control law. The effectiveness of these methods is demonstrated by the presentation of examples of suppressing bifurcations and chaos for the plastic impact oscillator.     

Horizontal fast excitation in delayed van der Pol oscillator

This paper investigates the interaction effect of horizontal fast harmonic parametric excitation and time delay on self-excited vibration in van der Pol oscillator. We apply the method of direct partition of motion to derive the main autonomous equation governing the slow dynamic of the oscillator. The method of averaging is then performed on the slow dynamic to obtain a slow flow which is analyzed for equilibria and periodic motion. This analysis provides analytical approximations of regions in parameter space where periodic self-excited vibrations can be eliminated. A numerical study is performed on the original oscillator and compared to analytical approximations. It was shown that in the delayed case, horizontal fast harmonic excitation can eliminate undesirable self-excited vibrations for moderate values of the excitation frequency. In contrast, the case without delay requires large excitation frequency to eliminate such motions. This work has application to regenerative behavior in high-speed milling.     

Periodic motions in a simplified brake system with a periodic excitation

In this paper, periodic motions for a simplified brake system under a periodical excitation are investigated, and the motion switchability on the discontinuous boundary is discussed through the theory of discontinuous dynamical systems. The onset and vanishing of periodic motions are discussed through the bifurcation and grazing analyses. Based on the discontinuous boundary, the switching planes and the basic mappings are introduced, and the mapping structures for periodic motions are developed. From the mapping structures, the periodic motions are analytically predicted and the corresponding local stability and bifurcation analysis is completed. Periodic motions will be illustrated for verification of analytical predictions. In addition, the relative force distributions along the displacement are illustrated for illustrations of the analytical conditions of motion switchability on the discontinuous boundary.     

Approximate solutions and chaotic motions of a piecewise nonlinear–linear oscillator

A global symmetric period-1 approximate solution is analytically constructed for the non-resonant periodic response of a periodically excited piecewise nonlinear–linear oscillator. The approximate solutions are found to be in good agreement with the exact solutions that are obtained from the numerical integration of the original equations. In addition, the dynamic behaviour of the oscillator is numerically investigated with the help of bifurcation diagrams, Lyapunov exponents, Poincare maps, phase portraits and basins of attraction. The existence of subharmonic and chaotic motions and the coexistence of four attractors are observed for some combinations of the system parameters.     

Nonlinear Dynamics of Marine Systems in Random Waves

The dynamics of ships or offshore structures is influenced by several different effects, some of which have a distinctly nonlinear characteristic. Even though in many situations the motion can sufficiently be described by linear models, nonlinear phenomena play a crucial role in the investigation of some more critical operating conditions: Large amplitude motions, sudden jumps in the dynamical behavior and sensitivity to the initial conditions are likely to occur under some circumstances. The response of floating systems such as moored buoys and barges in regular waves can be approximated by analytical or numerical techniques. These analyses reveal the characteristics of different periodic motions. In order to determine how these responses change under a more general forcing, the motion of floating structures under the influence of random disturbances is described by probability distributions. Different mathematical tools can efficiently be applied to models with few degrees of freedom. The localized statistical linearization used here is also promising for larger systems. Modelling aspects of offshore structures and random waves are discussed as well as the determination of probability distributions. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of period-doubling and chaos of a non-symmetric oscillator with piecewise-linearity

This paper presents an analysis of the dynamical behaviour of a non-symmetric oscillator with piecewise-linearity. The Chen–Langford (C–L) method is used to obtain the averaged system of the oscillator. Using this method, the local bifurcation and the stability of the steady-state solutions are studied. A Runge–Kutta method, Poincaré map and the largest Lyapunov’s exponent are used to detect the complex dynamical phenomena of the system. It is found that the system with piecewise-linearity exhibits periodic oscillations, period-doubling, period-3 solution and then chaos. When chaos is found, it is detected by examining the phase plane, bifurcation diagram and the largest Lyapunov’s exponent. The results obtained in this paper show that the vibration system with piecewise-linearity do exhibit quite similar dynamical behaviour to the discrete system given by the logistic map.