Neimark Bifurcation Research Articles

SYMMETRY, GENERIC BIFURCATIONS, AND MODE INTERACTION IN NONLINEAR RAILWAY DYNAMICS

We investigate Cooperrider's complex bogie, a mathematical model of a railway bogie running on an ideal straight track. The speed of the bogie v is the control parameter. Taking symmetry into account, we find that the generic bifurcations from a symmetric periodic solution of the model are Hopf bifurcations for maps (or Neimark bifurcations), saddle-node bifurcations, and pitchfork bifurcations. The last ones are symmetry-breaking bifurcations. By variation of an additional parameter, bifurcations of higher degeneracy are possible. In particular, we consider mode interactions near a degenerate bifurcation. The bifurcation analysis and path-finding are done numerically.

A NECESSARY AND SUFFICIENT CONDITION TO HAVE r PAIRS OF COMPLEX MULTIPLIERS WITH MODULUS EQUAL TO UNITY IN THE CASE OF AN n-DIMENSIONAL MAP

Considering a cycle (or periodic point) of an n-dimensional map, a bifurcation occurs when at least one of its multipliers crosses the unit circle in the complex.plane. This paper presents a necessary and sufficient condition for the bifurcations occurring when r pairs of complex multipliers cross the unit circle. Among this kind of bifurcations, the most known is the Neimark's bifurcation corresponding to the case when r is equal to one. From the established condition, an algorithm to trace out these bifurcation curves in a parameter plane of the considered map is displayed.

Neimark bifurcation, almost-periodicity and chaos in the forced van der Pol-Duffing system in the neighbourhood of the principal resonance

The occurrence of chaotic motion in the forced van der Pol-Duffing oscillator in close neighbourhood of the principal resonance is considered and interpreted in connection with subcritical Neimark bifurcation. Computer simulations and approximate analytical analysis confirm that the chaotic motion region forms a transition zone between two regular motions: the T-periodic and almost-periodic oscillations.