We have studied an integrate-and-#re model neuron driven by a periodic sequence of Gaussianshaped current pulses of width 1=2N and amplitude . For large N the current is a sequence of delta pulses and for smaller N it approaches a sine. When N is large, the #ring rate f vs. curve is a staircase of entrainment steps yielding an Arnold tongue structure in the vs. plane. In the small N limit we #n...

An exact expression for the determinant of the splitting matrix is derived: it allows us to analyze the asymptotic behaviour needed to amend the large angles theorem proposed in Ann. Inst. H. Poincaré, B-60, 1, 1994. The asymtotic validity of Mel-nokov's formulae is proved for the class of models considered, which include polynomial perturbations. §1. Introduction. Recently V. Gelfreich noted t...

Ravishankar K. Iyer F the past four years, it has been my privilege and pleasure to serve as the Inaugural Editor-in-Chief of IEEE’s Transactions on Dependable and Secure Computing. During the relatively short time since its launch, TDSC has become the premier journal in our fi eld. One indicator of success is the impact factor, as measured by the IEEE. TDSC has had one of the highest impact fa...

The low-dimensional Gross–Neveu models are studied. For the systems, related to the Lie algebras so(4), so(5), sp(4), sl(3), we prove that they have Birkhoff-Gustavson normal forms which are integrable and non-degenerate in Kolmogorov–Arnold–Moser (KAM) theory sense. Unfortunately, this is not the case for systems with three degrees of freedom, related to the Lie algebras so(6) ∼ sl(4), so(7), ...

1. Dima Arnold in my life, by Dmitry Fuchs Unfortunately, I have never been Arnold’s student, although as a mathematician, I owe him a lot. He was just two years older than I, and according to the University records, the time distance between us was still less: when I was admitted to the Moscow State University as a freshman, he was a sophomore. We knew each other, but did not communicate much....

In this paper we study Arnold’s tongues in a Z2-symmetric electronic circuit. They appear in a rich bifurcation scenario organized by a degenerate codimension-three Hopf–pitchfork bifurcation. On the one hand, we describe the transition open-to-closed of the resonance zones, finding two different types of Takens–Bogdanov bifurcations (quadratic and cubic homoclinic-type) of periodic orbits. The...

We study n-degree-of-freedom, nearly integrable Hamiltonian systems near the intersection of a stronger and a weaker resonance. We show the existence of motions that cross the weaker resonance faster than Arnold diiusion. We also describe families of complicated homoclinic orbits which create observable chaotic dynamics near double resonances. Quite remarkably, these families undergo a universa...

It is difficult to do scientifically rigorous research on treatments that must be administered urgently or emergently. Therefore, such treatments are often provided without a strong evidence base. Research would be facilitated if it were permissible to waive the requirement for parental consent. However, that raises a different set of concerns. Federal regulations allow waiver of the requiremen...

Some scaling properties of the chaotic sea for a particle confined inside an infinitely deep potential box containing a time varying barrier are studied. The dynamics of the particle is described by a two-dimensional, nonlinear and area-preserving mapping for the variables energy of the particle and time. The phase space of the model exhibits a mixed structure with Kolmogorov– Arnold–Moser isla...

We analytically establish the role of a spectrum of Lyapunov exponents in the evolution of phase-space distributions r(p ,q). Of particular interest is l2 , an exponent that quantifies the rate at which chaotically evolving distributions acquire structure at increasingly smaller scales and is generally larger than the maximal Lyapunov exponent l for trajectories. The approach is trajectory inde...