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 solution of a vibration system with two mass elastic connecting rods

In this article, the dynamic characteristic of linkage bobbing machine with balanced double-mass bearing periodic excitation and damping is studied. Subharmonic Melnikov function of the oscillating periodic orbits is computed through Melnikov method. And, the relationship of parameters is given when the subharmonic bifurcation occurs. The periodic motion is simulated. Through analysis, as the excitation frequency varies, the periodic motion undergoes flip bifurcations and subharmonic bifurcation occurs, which finally leads to chaos. In addition, the ultra-subharmonic solution of the system is given.

Subharmonic Bifurcation for a Nonsmooth Oscillator

A nonsmooth pendulum model with multiple impulse effect is constructed to detect the bifurcation of a periodic orbit with multiple jump discontinuous points. Subharmonic Melnikov function of this kind of nonsmooth systems is studied. Differences of subharmonic Melnikov function between the nonsmooth system with multiple jump discontinuities and the smooth system are analyzed by using the Hamiltonian function and piecewise integral method. Applying the recursive method and perturbation principle, the effects of the jump discontinuous points on the subharmonic Melnikov function are converted to integral items which can be easily calculated. Hence, the subharmonic Melnikov function for the subharmonic orbit with multiple jump discontinuous points is obtained. Finally, the existence conditions for periodic motion of the subharmonic orbit are derived and the efficiency of the conclusions is verified via numerical simulations.

Homoclinic Orbit Bifurcation of a Rotating Truncated Conical Shell

This is a study on Homoclinic bifurcation and subharmonic bifurcation of a truncated conical shallow shell rotating around a single axle and excited by a transverse periodic load. A systematic numerical approach is used to study the nonlinear motion of the system. The conditions under which bifurcations occur are determined on the basis of the characteristics of the rotating shell. Hamilton’s singular distributions are also investigated in details.

Numerical Simulation of Multiplicity and Stability of Mixed Convection in Rotating Curved Ducts

A numerical study is made on the fully developed bifurcation structure and stability of the mixed convection in rotating curved ducts of square cross-section with the emphasis on the effect of buoyancy force. The rotation can be positive or negative. The fluid can be heated or cooled. The study reveals the rich solution and flow structures and complicated stability features. One symmetric and two symmetric/asymmetric solution branches are found with seventy five limit points and fourteen bifurcation points. The flows on these branches can be symmetric, asymmetric, 2-cell, and up to 14-cell structures. Dynamic responses of the multiple solutions to finite random disturbances are examined by the direct transient computation. It is found that possible physically realizable fully developed flows evolve, as the variation of buoyancy force, from a stable steady multicell state at a large buoyancy force of cooling to the coexistence of three stable steady multicell states, a temporal periodic oscillation state, the coexistence of periodic oscillation and chaotic oscillation, a chaotic temporal oscillation, a subharmonic-bifurcation-driven asymmetric oscillating state, and a stable steady 2-cell state at large buoyancy force of heating.