Search

With both analytical and numerical methods, chaotic motions of the Rayleigh–Duffing oscillator with a non-smooth periodic perturbation and harmonic excitation are investigated in this paper. Chaos arising from homoclinic or heteroclinic intersections is studied with the Melnikov method. Chaos threshold is obtained and chaotic feature on the system parameters is investigated in detail. Some new interesting dynamic phenomena including “controllable frequency interval”, “chaotic band” and “uncontrollable parameters” are presented and proved rigorously. Numerical simulations are given to verify chaos threshold obtained by the analytical results.

In this manuscript, multiple patterns of slow-fast motion in an accelerating viscoelastic beam model are analyzed from both analytic and numerical view. Based on the modified averaging method, the static solutions of amplitude response equations are obtained. The stability analysis about the zero solution of the averaged equations shows that the zero solution loses its stability through pitchfork bifurcation and the small eigenvalue indicates the existence of strong transient. The numerical bifurcation diagrams and time histories of the fast sub-system verify the results obtained from the averaging process, especially strong transient effects in the vicinity of small amplitude branch. Based on these, two cases of low frequency excitation are considered. When the period of the slow excitation is much larger than the critical time for the fast sub-system to converge to the intermediate branch, the transitions related to pitchfork bifurcation are always periodic. While, if the period of the slow excitation possesses the same magnitude as that of the critical time, transitions between the small amplitude branch and intermediate amplitude can present both regular and irregular manners, which results in complex dynamic buckling regimes. Thus, multiple patterns of slow-fast motion including “periodic bursts with quasi-periodic spiking”, “torus/short transient” mixed mode oscillations, periodic “pitchfork/long transient” mixed mode oscillations, amplitude-modulated oscillations, as well as irregular mixed mode oscillations, are analyzed from numerical view.

In this manuscript, slow-fast motions in a two degrees of freedom model under slow parametric and fast external excitations are analyzed using analytical and numerical methods. By expressing the state variables as the sums of slowly varying average and fast oscillations, the averaged equations which govern the slow dynamics for different magnitudes of external amplitude are obtained. For the averaged equations, analyzation about the equilibria and the corresponding stability shows that, the averaged slow manifolds constitute a pitchfork structure. Meanwhile, the harmonic balancing method is used to compute the slowly varying envelopes of the fast oscillations. On the other hand, bifurcation diagrams of the fast sub-system show structures in accordance with the pitchfork obtained from averaged equations as well as the amplitude curves computed by harmonic balancing method. Two forms of relative location between the folds of cycle on the handle and pitchfork bifurcation point are presented. Based on these, concerning on the slow-fast flow, two patterns of transition behavior between the handle and the upper (lower) branch of pitchfork namely “pitchfork/pitchfork” and “fold/pitchfork” types are analyzed, which indicate the dynamic buckling behaviors. Particularly, discussion about the influence of external excitation shows that, for case of near-resonant external frequency, increasing the amplitude of external excitation will change the local structures as well as the quantitative properties of the pitchfork, but holistic form of pitchfork type is a topological invariant.

Chaotic dynamics and subharmonic bifurcations of a soft Duffing oscillator with a non-smooth periodic perturbation and a harmonic excitation are investigated analytically in this paper. With the Fourier series, the system is expanded to the equivalent smooth system, and chaos arising from heteroclinic intersections is studied with the Melnikov method. The chaotic feature on the system parameters is investigated in detail. Some new interesting dynamic phenomena, such as chaotic bands for some excitation frequencies, are presented. The relationship between the frequency range of chaotic bands and the amplitude of the excitation as well as the damping is obtained analytically. Particularly, for some system parameters satisfying a particular relationship, chaos cannot occur for any excitation amplitudes or frequencies. Subharmonic bifurcations are investigated with a subharmonic Melnikov method. It is analytically proved that the system may undergo chaotic motions through infinite or finite odd order subharmonic bifurcations. Numerical simulations are given to verify the chaos threshold and revolution from subharmonic bifurcations to chaos obtained by analytical methods.

In this paper, a dry friction oscillator with periodic excitation is investigated. Melnikov method is used to study chaos of this model. The parameter regions of chaos are obtained. Numerical simulations are also given, which verify the analytical results.

With analytical and numerical methods, dynamic response of a composite laminated buckled beam with a lumped mass and an axial excitation is investigated in this paper. The dynamic equation of the first mode for this model, which is a perturbation of a Hamilton system with both quadratic and cubic nonlinearities as well as external and parametric excitations, is obtained with the Galerkin method. The distribution of the equilibriums, homoclinic orbits and periodic orbits for the unperturbed system under different system parameters are obtained analytically. Chaotic vibrations arising from homoclinic intersections are studied with Melnikov method. Chaotic feature on the location of the lumped mass is investigated in detail. It is presented that there exist controllable frequencies decreasing monotonously with the location of the lumped mass for this system. Subharmonic bifurcation is also studied with the subharmonic Melnikov method. It is numerically proved that the system may undergo chaotic motions through finite subharmonic bifurcations. Numerical simulations including the phase portraits, time histories, Poincar e ´ sections, Lyapunov spectrums and bifurcation diagram are given, which confirm the analytical results.

Based on the idea of slow-fast dynamics, two tapping modes are investigated in the Atomic Force Microscope (AFM) model with Lennard-Jones force and slow-fast base motion. According to the approximation derived from Taylor expansion in the vicinity of static equilibrium of the linearized system, we give a prediction about harmonic responses of the fast sub-system (FSS). Then a truncated version of harmonic balancing method (HBM) is combined with the Floquet theory to give the distribution and stability of the harmonic responses, which indicates the contact and non-contact modes dominated by cycle manifolds. By directly integrating the FSS, the numerical bifurcation diagrams and the frequency spectrums corresponding to the periodic solutions are obtained to verify the results given by the half-analytic process. Moreover, distribution of basins of attraction shows that, when the tri-stability is destroyed by fold bifurcation of cycle, due to the highly fragmented fractal structures in basins, transition between different mode manifolds may present an weak irregular way. On the other hand, when the slow excitation is introduced, the regular and irregular “cycle-cycle” type mixed mode oscillations (MMOs) triggered by “fold of cycle/fold of cycle” hysteresis can be observed, which results in two tapping mode, .i.e., the slow tapping mode and the “compound of contact/snap tapping” mode. Besides that, our results show that, for the AFM under slow and near-resonant excitation, the beneficial effects of constant contact mode and snap tapping mode can be utilized.

Using both analytical and numerical methods on the global dynamics, including the existence and uniqueness of solutions, subharmonic bifurcations and dynamic responses, of an elliptically excited pendulum model are investigated in this paper. The heteroclinic orbits, as well as periodic orbits with [Formula: see text] and [Formula: see text] types of unperturbed systems are obtained analytically. Chaotic vibrations arising from heteroclinic intersections are studied by means of the Melnikov method. The critical curves separating the chaotic and nonchaotic regions are plotted for different system parameters. The chaotic feature on the system parameter [Formula: see text], named the ratio between the horizontal and the vertical diameter of the upright ellipse traced out by the pivot during each period, is discussed in detail. The conditions for subharmonic bifurcations with the [Formula: see text] type or the [Formula: see text] type are also presented with the subharmonic Melnikov method. It is proved rigorously that the system can undergo chaotic motions through finite subharmonic bifurcations with the [Formula: see text] type. In addition, chaotic motions can occur through infinite subharmonic bifurcations with the [Formula: see text] type. An interesting dynamical phenomenon, i.e. “controllable frequency”, which decreases monotonically with the system parameter [Formula: see text], is presented. A number of related numerical simulations are given to confirm the analytical results.

The dynamics equation of the compound pendulum system is studied with the Melnikov method. The critical condition of the chaotic motion is discussed and the critical curve of the parameter is drawn. With the Runge-Kutta method, Numerical simulations are given, which verified the analytical results. It is concluded that there exists Smale horseshoe chaos arising from transversal intersections of stable and unstable manifold of the homoclinic orbit. The critical values for chaos first increase and then decrease with the increasing of the driving torque.

Subharmonic bifurcations and chaotic dynamics are investigated both analytically and numerically for a class of ship power system. Chaos arising from heteroclinic intersections is studied with the Melnikov method. The critical curves separating the chaotic and nonchaotic regions are obtained. The chaotic feature on the system parameters is discussed in detail. It is shown that there exist chaotic bands for this system. The conditions for subharmonic bifurcations with O type or R type are also obtained. It is proved that the system can be chaotically excited through finite subharmonic bifurcations with O type, and it also can be chaotically excited through infinite subharmonic bifurcations with R type. Some new dynamical phenomena are presented. Numerical simulations are given, which verify the analytical results.

The local dynamics including stability and Hopf bifurcation of autonomoius van der Pol-Duffing circuit system containing parallel resistor are investigated both analytically and numerically in this paper. All the equilibrium points and their stability conditions are presented. The parametric conditions for Hopf bifurcation are also obtained. Numerical simulations verify the analytical results.

Employed both analytical and numerical methods, chaotic motions for the parametrically excited roll motion of a ship in regular longitudinal waves are investigated in this paper. It is presented that the mechanism for chaos is the intersection of the stable and unstable manifolds of the homoclinic orbits or heteroclinic cycle. The parameter conditions for chaos are obtained rigorously with the Melnikov method. The chaotic regions of the system parameters are illustrated. The chaotic feature on the system parameters is discussed in detail. It is obtained that the critical values of chaos for homoclinic orbits increase monotonously as the increasing of the nonlinear roll attenuation coefficient, while the critical values of chaos for the heteroclinic cycle decrease monotonously as the increasing of the nonlinear roll attenuation coefficient. On the other hand, critical values of chaos for both homoclinic orbits and heteroclinic cycle increase monotonously as the increasing of the excitation amplitude. It is also demonstrated that there exist chaotic bands for this model. Numerical simulations verify the analytical results.

In this paper, by employing both analytical and numerical methods, global dynamic responses including subharmonic bifurcations and chaos are investigated for a carbon nanotube supported by a Winkler and Pasternak foundation. The criteria of chaos arising from transverse intersections for stable and unstable manifolds of homoclinic orbits are proposed with the Melnikov method. The critical curves separating the chaotic and nonchaotic regions are plotted in the parameter plane. The parameter conditions for subharmonic bifurcations are also obtained by the subharmonic Melnikov method. It is proved rigorously that the route to chaos for this model is infinite subharmonic bifurcations. The stability of subharmonic bifurcations is also studied by the characteristic multipliers. Numerical simulations are given to confirm the analytical results.

The stability and chaotic motions of a damped satellite partially filled with liquid which is subjected to external disturbance are investigated in this paper. With linearization analysis, the stability of the two non-trivial equilibrium points is studied. The homoclinic and heteroclinic orbits are found by using the undetermined coefficient method, and the convergence of the series expansions of these two types of orbits is proved. It analytically demonstrates that there exist homoclinic orbits of the Si’lnikov type that join the two non-trivial equilibrium points to themselves, and therefore smale horseshoes and the horseshoe chaos occur for this system via the Si’lnikov criterion. In addition, there also exists a heteroclinic orbit connecting the two non-trivial equilibrium points. Numerical simulations are also given, which verify the analytical results. The system can be chaotic through period-doubling bifurcations as the amplitude of the external disturbance varies, and backward period-doubling bifurcations as the angular momentum of the rotor varies.

The chaotic dynamics of the transport equation for the L-mode to H-mode near the plasma in a tokamak is studied in detail with the Melnikov method. The transport equations represent a system with external and parametric excitation. The critical curves separating the chaotic regions and nonchaotic regions are presented for the system with periodically external excitation and linear parametric excitation, or cubic parametric excitation, respectively. The results obtained here show that there exist uncontrollable regions in which chaos always take place via heteroclinic bifurcation for the system with linear or cubic parametric excitation. Especially, there exists a controllable frequency, excited at which chaos does not occur via homoclinic bifurcation no matter how large the excitation amplitude is for the system with cubic parametric excitation. Some complicated dynamical behaviors are obtained for this class of systems.

Bifurcations and chaotic motions of a class of mechanical system subjected to a superharmonic parametric excitation or a nonlinear periodic parametric excitation are studied, respectively, in this paper. Chaos arising from the transverse intersections of the stable and unstable manifolds of the homoclinic and heteroclincic orbits is analyzed by Melnikov's method. The critical curves separating the chaotic and nonchaotic regions are plotted. Chaotic dynamics are compared for these systems with a periodic parametric excitation or a superharmonic parametric excitation, or a nonlinear periodic parametric excitation. Especially, some new dynamical phenomena are presented for the system with a nonlinear periodic parametric excitation.