We consider decay of metastable states of forced vibrations of a quantum oscillator close to the bifurcation points where the states disappear. Decay occurs via quantum activation over a quasienergy barrier, a mechanism that differs from both tunneling and thermal activation. The decay probability W scales with the distance eta to the bifurcation point as /ln W/ proportional to eta(xi). The exp...

We consider the Reidemeister torsion associated with SL2(C)representations of a knot group. A bifurcation point in the SL2(C)-character variety of a knot group is a character which is given by both an abelian SL2(C)representation and a non-abelian one. We show that there exist limits of the non-acyclic Reidemeister torsion at bifurcation points and the limits are expressed by using the derivati...

Abstract: An algorithm is presented here, for discovering Hopf-Bifurcation varieties of polynomial dynamical systems. It is based on the expression of specific polynomials, as sums of products of first degree polynomials, with parametrical coefficients. By giving to these parameters certain values, we ensure the positiveness of some quantities, constructing thereby proper Lyapunov functions, wh...

This paper will introduce the topic of dynamical systems with both discrete and continuous time variables. Fixed points will be discussed, along with their properties such as stability or topological type. The paper will continue on to define the concept of hyperbolicity and its relevance in determining the structural stability of the system. It will conclude with a definition of a bifurcation ...

Border collision bifurcations refer to situations where a small parameter variation induces a change in the event sequence of a hybrid system. At such a bifurcation, the system trajectory makes tangential contact with an event triggering hypersurface. This bounding case separates regions of (generally) quite different dynamic behaviour. In the paper, the conditions governing bifurcation points ...