Dynamical systems with time-dependent coupling: Clustering and critical behaviour

We study the collective behaviour of an ensemble of coupled motile elements whose interactions depend on time and are alternatively attractive or repulsive. The evolution of interactions is driven by individual internal variables with autonomous dynamics. The system exhibits different dynamical regimes, with various forms of collective organization, controlled by the range of interactions and the dispersion of time scales in the evolution of the internal variables. In the limit of large interaction ranges, it reduces to an ensemble of coupled identical phase oscillators and, to some extent, admits to be treated analytically. We find and characterize a transition between ordered and disordered states, mediated by a regime of dynamical clustering.Comment: to appear in Physica

Scaling Invariance in a Time-Dependent Elliptical Billiard

We study some dynamical properties of a classical time-dependent elliptical billiard. We consider periodically moving boundary and collisions between the particle and the boundary are assumed to be elastic. Our results confirm that although the static elliptical billiard is an integrable system, after to introduce time-dependent perturbation on the boundary the unlimited energy growth is observed. The behaviour of the average velocity is described using scaling arguments

Particle systems with locally dependent branching : long-time behaviour, genealogy and critical parameters

We consider the long-time behaviour of spatially extended random populations with locally dependent branching. We treat two classes of models: 1) Systems of continuous-time random walks on the d-dimensional grid with state dependent branching rate. While there are k particles at a given site, a branching event occurs there at rate s(k), and one of the particles is replaced by a random number of offspring (according to a fixed distribution with mean 1 and finite variance). 2) Discrete-time systems of branching random walks in random environment. Given a space-time i.i.d. field of random offspring distributions, all particles act independently, the offspring law of a given particle depending on its position and generation. The mean number of children per individual, averaged over the random environment, equals one The long-time behaviour is determined by the interplay of the motion and the branching mechanism: In the case of recurrent symmetrised individual motion, systems of the second type become locally extinct. We prove a comparison theorem for convex functionals of systems of type one which implies that these systems also become locally extinct in this case, provided that the branching rate function grows at least linearly. Furthermore, the analysis of a caricature model leads to the conjecture that local extinction prevails generically in this case. In the case of transient symmetrised individual motion the picture is more complex: Branching random walks with state dependent branching rate converge towards a non-trivial equilibrium, which preserves the initial intensity, whenever the branching rate function grows subquadratically. Systems of type 1) and systems of type 2) with quadratic branching rate function show very similar behaviour. They converge towards a non-trivial equilibrium if a conditional exponential moment of the collision time of two random walks of an order that reflects the variability in the branching mechanism is finite almost surely. The equilibrium population has finite variance of the local particle number if the corresponding unconditional exponential moment is finite. These results are proved by means of genealogical representations of the locally size-biased population. Furthermore, we compute the threshold values for existence of conditional exponential moments of the collision time of two random walks in terms of the entropy of the transition functions, using tools from large deviations theory. Our results prove in particular that - in contrast to the classical case of independent branching - there is a regime of equilibria with variance of the local number of particles.Wir betrachten das Langzeitverhalten von zufälligen, räumlich ausgebreiteten Populationen mit lokal abhängiger Verzweigung, speziell werden zwei Klassen von Modellen untersucht: 1) Zeitkontinuierliche Systeme von Irrfahrten auf dem d-dimensionalen Gitter mit zustandsabhängiger Verzweigungsrate. Wenn an einem Ort gerade k Teilchen sind, findet dort ein Verzweigungsereignis mit Rate s(k) statt, und eines der Teilchen wird durch eine zufällige Anzahl Nachkommen (gemäß einer vorgegebenen Verteilung mit Mittelwert 1 und endlicher Varianz) ersetzt. 2) Zeitdiskrete Systeme von verzweigenden Irrfahrten in zufälliger Umgebung. Wir betrachten ein Raum-Zeit-Feld von unabhängigen Kinderzahlverteilungen, gegeben dieses Feld verhalten sich alle Teilchen unabhängig, die Verteilung der Anzahl Nachkommen eines Teilchens hängt von seiner Position und seiner Generation ab. Die mittlere Anzahl Nachkommen pro Individuum, gemittelt über die zufällige Umgebung, ist exakt eins. Das Langzeitverhalten wird durch das Zusammenspiel von Bewegungs- und Verzweigungsmechanismus bestimmt: Bei rekurrenter symmetrisierter Individualbewegung sterben Systeme vom zweiten Typ stets aus. Wir beweisen ein Vergleichresultat für konvexe Funktionale von Systemen vom Typ 1, das impliziert, dass auch dort im rekurrenten Fall lokales Aussterben vorherrscht, sofern die Verzweigungsratenfunktion mindestens linear wächst. Darüberhinaus erhärten wir anhand eines Karikaturmodells die Vermutung, dass lokales Aussterben generisch vorliegt. Im Fall transienter symmetrisierter Individualbewegung bietet sich ein reichhaltigeres Bild: Verzweigende Irrfahrten mit zustandsabhängiger Verzweigung konvergieren gegen ein nicht-triviales Gleichgewicht, das die Anfangsintensität erhält, sofern die Verzweigungsratenfunktion subquadratisch wächst. Wir zeigen eine Parallele zwischen Systemen vom Typ 2 und Systemen vom Typ 1 mit quadratischer Verzweigungsratenfunktion. Wenn ein bedingtes exponentielles Moment der Kollisionszeit zweier unabhängiger Irrfahrten von einer Ordnung, die von der Variabilität im Verzweigungsmechanismus abhängt, fast sicher endlich bleibt, so konvergieren die Systeme gegen nichttriviale Gleichgewichte. Die zweiten Momente der lokalen Teilchenanzahlen im Gleichgewicht sind genau dann endlich, wenn auch das entsprechende unbedingte Moment endlich ist. Wir erzielen diese Resultate mittels genealogischer Darstellungen der lokal größenverzerrten Populationen. Darüberhinaus berechnen wir unter Verwendung von Hilfsmitteln aus der Theorie der großen Abweichungen die Schwellwerte für die Existenz der bedingten exponentiellen Momente der Kollisionszeit in Termen der Entropie der Übergangsmatrizen der Irrfahrt. Dies zeigt insbesondere, dass - im Gegensatz zum klassischen Fall unabhängiger Verzweigung - ein Regime von Gleichgewichten mit unendlicher Varianz der lokalen Anzahl existiert

Shape correction factor for drying shrinkage in a concrete cross-section

A concrete member is subjected to loads for a long period of time, during which creep and shrinkage of concrete develop gradually. The prediction of this time-dependent behaviour is important as it may cause serious serviceability problems in concrete structures. A time-dependent analysis is commonly based on empirical equations according to design codes where the function describing the time dependent increment of shrinkage and creep is commonly, among others, defined based on the notional size of the element. In case of imbedded steel or insulated boundaries the moisture transport can be partially affected or prevented. Also, the geometry and size of the cross-section have an important effect on the shrinkage behaviour of a concrete member. Hence, the performance of commonly used empirical formulas may be improved by applying a correction factor on the notional size. In order to investigate the impact of these various factors on the net macroscopic shrinkage used in analysis and design, a discretized 2D physical model was developed. The model was used to simulate drying of a concrete cross-section by determining the moisture distribution in the cross-section as function of time

Cross-sectional associations between sleep duration, sedentary time, physical activity, and adiposity indicators among Canadian preschool-aged children using compositional analyses

Abstract Background Sleep duration, sedentary behaviour, and physical activity are three co-dependent behaviours that fall on the movement/non-movement intensity continuum. Compositional data analyses provide an appropriate method for analyzing the association between co-dependent movement behaviour data and health indicators. The objectives of this study were to examine: (1) the combined associations of the composition of time spent in sleep, sedentary behaviour, light-intensity physical activity (LPA), and moderate- to vigorous-intensity physical activity (MVPA) with adiposity indicators; and (2) the association of the time spent in sleep, sedentary behaviour, LPA, or MVPA with adiposity indicators relative to the time spent in the other behaviours in a representative sample of Canadian preschool-aged children. Methods Participants were 552 children aged 3 to 4 years from cycles 2 and 3 of the Canadian Health Measures Survey. Sedentary time, LPA, and MVPA were measured with Actical accelerometers (Philips Respironics, Bend, OR USA), and sleep duration was parental reported. Adiposity indicators included waist circumference (WC) and body mass index (BMI) z-scores based on World Health Organization growth standards. Compositional data analyses were used to examine the cross-sectional associations. Results The composition of movement behaviours was significantly associated with BMI z-scores (p = 0.006) but not with WC (p = 0.718). Further, the time spent in sleep (BMI z-score: γ sleep  = −0.72; p = 0.138; WC: γ sleep  = −1.95; p = 0.285), sedentary behaviour (BMI z-score: γ SB  = 0.19; p = 0.624; WC: γ SB  = 0.87; p = 0.614), LPA (BMI z-score: γ LPA  = 0.62; p = 0.213, WC: γ LPA  = 0.23; p = 0.902), or MVPA (BMI z-score: γ MVPA  = −0.09; p = 0.733, WC: γ MVPA  = 0.08; p = 0.288) relative to the other behaviours was not significantly associated with the adiposity indicators. Conclusions This study is the first to use compositional analyses when examining associations of co-dependent sleep duration, sedentary time, and physical activity behaviours with adiposity indicators in preschool-aged children. The overall composition of movement behaviours appears important for healthy BMI z-scores in preschool-aged children. Future research is needed to determine the optimal movement behaviour composition that should be promoted in this age group

Fermi acceleration and suppression of Fermi acceleration in a time-dependent Lorentz Gas

We study some dynamical properties of a Lorentz gas. We have considered both the static and time dependent boundary. For the static case we have shown that the system has a chaotic component characterized with a positive Lyapunov Exponent. For the time-dependent perturbation we describe the model using a four-dimensional nonlinear map. The behaviour of the average velocity is considered in two situations (i) non-dissipative and (ii) dissipative. Our results show that the unlimited energy growth is observed for the non-dissipative case. However, when dissipation, via damping coefficients, is introduced the senary changes and the unlimited engergy growth is suppressed. The behaviour of the average velocity is described using scaling approach