Adam S. Landsberg

Direction-Reversing Traveling Waves

Abstract A simple mechanism for generating traveling waves which reverse their direction of propagation in a periodic manner is presented. This mechanism is generic in systems possessing O(2) symmetry, and corresponds to a codimension-one symmetry-breaking Hopf bifurcation from a circle of nontrivial steady states.

Hierarchical networks, power laws, and neuronal avalanches

We show that in networks with a hierarchical architecture, critical dynamical behaviors can emerge even when the underlying dynamical processes are not critical. This finding provides explicit insight into current studies of the brains neuronal network showing power-law avalanches in neural recordings, and provides a theoretical justification of recent numerical findings. Our analysis shows how the hierarchical organization of a network can itself lead to power-law distributions of avalanche sizes and durations, scaling laws between anomalous exponents, and universal functions-even in the absence of self-organized criticality or critical points. This hierarchy-induced phenomenon is independent of, though can potentially operate in conjunction with, standard dynamical mechanisms for generating power laws.

CHAOTIC DIRECTION-REVERSING WAVES

Symmetry-increasing bifurcations of strange attractors in systems with O 2 symmetry are shown to produce traveling waves that reverse their direction of propagation in a chaotic fashion. The resulting dynamics are illustrated using the normal form describing the triple zero instability. q 1999 Elsevier Science B.V. All rights reserved.

Short run dynamics of multi-class queues

We study the dynamics of general queueing systems with multiple classes of customers undergoing self-selection. We prove that if the capacity of a queue is sufficiently large, the equilibrium arrival rate will be globally stable. As the capacity is decreased, the arrival rate typically oscillates near the equilibrium.

New types of waves in systems with O(2) symmetry

Abstract A number of novel phenomena arising in systems with O(2) symmetry are described. These include pulsing traveling waves with either a periodically or quasiperiodically modulated phase velocity, and heteroclinic waves which alter their appearance with time, taking successively the form of pure traveling waves, standing waves and steady states. Chaotic waves can also be present. These states result from the codimension-two interaction between a reflection-breaking steady state bifurcation and a reflection-preserving Hopf bifurcation from a circle of nontrivial equilibria. The resulting waveforms are illustrated using a model of magnetoconvection with periodic boundary conditions.

Linewidth calculation for bare 2D Josephson arrays

Abstract We study how disorder affects the frequency-locking properties of a bare current-biased rectangular array of Josephson junctions. Our calculation is based on a simple physical picture wherein elements within each row lock by virtue of spontaneously induced shunt currents through the transverse junctions; no locking occurs between rows. Our analytic formula for the linewidth is in excellent agreement with numerical simulations of the nonlinear circuit equations.

Spectral Equivalence of Bosons and Fermions in One-Dimensional Harmonic Potentials

Recently, Schmidt and Schnack [Physica A 260, 479 (1998)], following earlier references, reiterate that the specific heat of N noninteracting bosons in a one-dimensional harmonic well equals that of N noninteracting fermions in the same potential. We show that this peculiar relationship between heat capacities results from a more dramatic equivalence between Bose and Fermi systems. Namely, we prove that the excitations of such Bose and Fermi systems are spectrally equivalent. Two complementary proofs of this equivalence are provided; one based on a combinatoric argument, the other from analysis of the underlying dynamical symmetry group.

Directed network motifs in Alzheimer's disease and mild cognitive impairment.

Directed network motifs are the building blocks of complex networks, such as human brain networks, and capture deep connectivity information that is not contained in standard network measures. In this paper we present the first application of directed network motifs in vivo to human brain networks, utilizing recently developed directed progression networks which are built upon rates of cortical thickness changes between brain regions. This is in contrast to previous studies which have relied on simulations and in vitro analysis of non-human brains. We show that frequencies of specific directed network motifs can be used to distinguish between patients with Alzheimer’s disease (AD) and normal control (NC) subjects. Especially interesting from a clinical standpoint, these motif frequencies can also distinguish between subjects with mild cognitive impairment who remained stable over three years (MCI) and those who converted to AD (CONV). Furthermore, we find that the entropy of the distribution of directed network motifs increased from MCI to CONV to AD, implying that the distribution of pathology is more structured in MCI but becomes less so as it progresses to CONV and further to AD. Thus, directed network motifs frequencies and distributional properties provide new insights into the progression of Alzheimer’s disease as well as new imaging markers for distinguishing between normal controls, stable mild cognitive impairment, MCI converters and Alzheimer’s disease.

Disorder and synchronization in a Josephson junction plaquette

We describe the effects of disorder on the coherence properties of a 2×2 array of Josephson junctions (a ‘‘plaquette’’). The disorder is introduced through variations in the junction characteristics. We show that the array will remain one‐to‐one frequency locked despite large amounts of the disorder, and determine analytically the maximum disorder that can be tolerated before a transition to a desynchronized state occurs. Connections with larger N×M arrays are also drawn.

SPATIAL SYMMETRIES AND GEOMETRICAL PHASES IN CLASSICAL DISSIPATIVE SYSTEMS

We describe the emergence of geometrical phases in dissipative systems with continuous spatial symmetries. The phase characterizes the spatial shift of a wave pattern that arises as the result of a cyclic adiabatic transport of control parameters of the system. Geometrical phases are calculated for both stationary and propagating wave patterns. Complementary formulations are provided for finite-dimensional and continuum systems. The theory is used to determine the phase shift for a traveling wave front in a standard reaction-diffusion model.