Global Dynamics of an Elliptically Excited Pendulum Model

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.

Subharmonic bifurcation and chaos of a carbon nanotube supported by a Winkler and Pasternak foundation

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.

Subharmonic Bifurcations and Chaotic Dynamics for a Class of Ship Power System

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.

Subharmonic Bifurcations and Transition to Chaos in a Pipe Conveying Fluid under Harmonic Excitation

The subharmonic and chaotic behavior of a two end-fixed fluid conveying pipe whose base is subjected to a harmonic excitation are investigated. Melnikov method is applied for the system, and Melnikov criterions for subharmonic and homoclinic bifurcations are obtained analytically. The numerical simulations (including bifurcation diagrams, maximal Lyapunov exponents, phase portraits and Poincare map) confirm the analytical predictions and exhibit the complicated dynamical behaviors.