arXiv subjects

Gregory Berkolaiko

Publications and source records attributed to Gregory Berkolaiko.

58 records · Page 4Linked to original sources

arXiv2006

Almost Sure Convergence of Solutions to Non-Homogeneous Stochastic Difference Equation

We consider a non-homogeneous nonlinear stochastic difference equation X_{n+1} = X_n (1 + f(X_n)ξ_{n+1}) + S_n, and its important special case X_{n+1} = X_n (1 + ξ_{n+1}) + S_n, both with initial value X_0, non-random decaying free coefficient S_n and independent random variables ξ_n. We establish results on \as convergence of solutions X_n to zero. The necessary conditions we find tie together certain moments of the noise ξ_n and the rate of decay of S_n. To ascertain sharpness of our conditions we discuss some situations when X_n diverges. We also establish a result concerning the rate of decay of X_n to zero.

math.PR↗

arXiv2004

Multiplicity of periodic solutions in bistable equations

We study the number of periodic solutions in two first order non-autonomous differential equations both of which have been used to describe, among other things, the mean magnetization of an Ising magnet in the time-varying external magnetic field. When the strength of the external field is varied, the set of periodic solutions undergoes a bifurcation in both equations. We prove that despite profound similarities between the equations, the character of the bifurcation can be very different. This results in a different number of coexisting stable periodic solutions in the vicinity of the bifurcation. As a consequence, in one of the models, the Suzuki-Kubo equation, one can effect a discontinuous change in magnetization by adiabatically varying the strength of the magnetic field.

cond-mat.stat-mech↗

arXiv2003

Form factor for a family of quantum graphs: An expansion to third order

For certain types of quantum graphs we show that the random-matrix form factor can be recovered to at least third order in the scaled time $τ$ from periodic-orbit theory. We consider the contributions from pairs of periodic orbits represented by diagrams with up to two self-intersections connected by up to four arcs and explain why all other diagrams are expected to give higher-order corrections only. For a large family of graphs with ergodic classical dynamics the diagrams that exist in the absence of time-reversal symmetry sum to zero. The mechanism for this cancellation is rather general which suggests that it may also apply at higher-orders in the expansion. This expectation is in full agreement with the fact that in this case the linear-$τ$ contribution, the diagonal approximation, already reproduces the random-matrix form factor for $τ<1$. For systems with time-reversal symmetry there are more diagrams which contribute at third order. We sum these contributions for quantum graphs with uniformly hyperbolic dynamics, obtaining $+2τ^{3}$, in agreement with random-matrix theory. As in the previous calculation of the leading-order correction to the diagonal approximation we find that the third order contribution can be attributed to exceptional orbits representing the intersection of diagram classes.

nlin.CD↗

arXiv2002

The Leading Off-Diagonal Correction to the Form Factor of Large Graphs

Using periodic-orbit theory beyond the diagonal approximation we investigate the form factor, $K(τ)$, of a generic quantum graph with mixing classical dynamics and time-reversal symmetry. We calculate the contribution from pairs of self-intersecting orbits that differ from each other only in the orientation of a single loop. In the limit of large graphs, these pairs produce a contribution $-2τ^2$ to the form factor which agrees with random-matrix theory.

nlin.CD↗