David K. Arrowsmith

Dynamical Systems: Differential Equations, Maps, and Chaotic Behaviour

Part 1 Introduction: preliminary ideas autonomous equations autonomous systems in the plane construction of phase portraits in the plane flows and evolution. Part 2 Linear systems: linear changes of variable similarity types for 2x2 real matrices phase portraits for canonical systems in the plane classification of simple linear phase portraits in the plane the evolution operator affine systems linear systems of dimension greater than two. Part 3 Non-linear systems in the plane: local and global behaviour linearization at a fixed point the linearization theorem non-simple fixed points stability of fixed points ordinary points and global behaviour first integrals limit points and limit cycles Poincare-Bendixson theory. Part 4 Flows on non-planar phase spaces: fixed points closed orbits attracting sets and attractors further integrals. Part 5 Applications I - planar phase spaces: linear models affine models non-linear models relaxation oscillations piecewise modelling. Part 6 Applications II - non-planar phase spaces, families of systems and bifurcations: the Zeeman models of heart beat and nerve impulse a model of animal conflict families of differential equations and bifurcations a mathematical model of tumor growth some bifurcations in families of one-dimensional maps some bifurcations in families of two-dimensional maps area-preserving maps, homoclinic tangles and strange attractors symbolic dynamics new directions.

Robustness of trans-European gas networks

Here, we uncover the load and fault-tolerant backbones of the trans-European gas pipeline network. Combining topological data with information on intercountry flows, we estimate the global load of the network and its tolerance to failures. To do this, we apply two complementary methods generalized from the betweenness centrality and the maximum flow. We find that the gas pipeline network has grown to satisfy a dual purpose. On one hand, the major pipelines are crossed by a large number of shortest paths thereby increasing the efficiency of the network; on the other hand, a nonoperational pipeline causes only a minimal impact on network capacity, implying that the network is error tolerant. These findings suggest that the trans-European gas pipeline network is robust, i.e., error tolerant to failures of high load links.

Geometry of p -adic Siegel discs

Abstract We survey recent advances in the study of regular motions over p-adic fields, show its varied connections with dynamics and number theory, and illustrate its significance to an important class of discrete dynamical systems. We also show that mappings supporting quasi-periodic motions can be naturally interpreted as flows with p-adic time.

Some p-adic representations of the Smale horseshoe

Abstract We extend one example of Verstegen on the p-adic representation of the chaotic doubling map to the Smale horseshoe map. We show that the horseshoe is topologically conjugate to a linear saddle-type diffeomorphism, defined on the product of two copies of the p-adic integers. We also find that there exist topological conjugacies of the horseshoe with homeomorphisms of a single copy of the p-adic integers, provided one relaxes the differentiability condition.

THE BOGDANOV MAP: BIFURCATIONS, MODE LOCKING, AND CHAOS IN A DISSIPATIVE SYSTEM

We investigate the bifurcations and basins of attraction in the Bogdanov map, a planar quadratic map which is conjugate to the Henon area-preserving map in its conservative limit. It undergoes a Hopf bifurcation as dissipation is added, and exhibits the panoply of mode locking, Arnold tongues, and chaos as an invariant circle grows out, finally to be destroyed in the homo-clinic tangency of the manifolds of a remote saddle point. The Bogdanov map is the Euler map of a two-dimensional system of ordinary differential equations first considered by Bogdanov and Arnold in their study of the versal unfolding of the double-zero-eigenvalue singularity, and equivalently of a vector field invariant under rotation of the plane by an angle 2π. It is a useful system in which to observe the effect of dissipative perturbations on Hamiltonian structure. In addition, we argue that the Bogdanov map provides a good approximation to the dynamics of the Poincare maps of periodically forced oscillators.

Resilience of Natural Gas Networks during Conflicts, Crises and Disruptions

Human conflict, geopolitical crises, terrorist attacks, and natural disasters can turn large parts of energy distribution networks offline. Europes current gas supply network is largely dependent on deliveries from Russia and North Africa, creating vulnerabilities to social and political instabilities. During crises, less delivery may mean greater congestion, as the pipeline network is used in ways it has not been designed for. Given the importance of the security of natural gas supply, we develop a model to handle network congestion on various geographical scales. We offer a resilient response strategy to energy shortages and quantify its effectiveness for a variety of relevant scenarios. In essence, Europes gas supply can be made robust even to major supply disruptions, if a fair distribution strategy is applied.

The emerging energy web

Abstract There is a general need of elaborating energy-effective solutions for managing our increasingly dense interconnected world. The problem should be tackled in multiple dimensions -technology, society, economics, law, regulations, and politics- at different temporal and spatial scales. Holistic approaches will enable technological solutions to be supported by socio-economic motivations, adequate incentive regulation to foster investment in green infrastructures coherently integrated with adequate energy provisioning schemes. In this article, an attempt is made to describe such multidisciplinary challenges with a coherent set of solutions to be identified to significantly impact the way our interconnected energy world is designed and operated. Graphical abstract

Effects of variations of load distribution on network performance

The paper is concerned with the characterization of the relationship between topology and traffic dynamics. We use a model of network generation that allows the transition from random to scale free networks. Specifically, we consider three different topological types of network: random; scale-free with /spl gamma/=3; scale-free with /spl gamma/=2. By using a novel LRD traffic generator, we observe best performance, in terms of transmission rates and delivered packets, in the case of random networks. We show that, even if scale-free networks are characterized by shorter characteristic-pathlength (the lower the exponent, the lower the pathlength), they show worst performances in terms of communication. We conjecture that this can be explained in terms of changes in the load distribution, defined as the number of shortest paths going through a given vertex. In fact, that distribution is characterized by (i) a decreasing mean and (ii) an increasing standard deviation, as the networks becomes scale-free (especially scale-free networks with low exponents). The use of a degree-independent server also discriminates against a scale-free structure. As a result, since the model is uncontrolled, most packets go through the same vertices, favoring the onset of congestion.

Chaotic maps for traffic modelling and queueing performance analysis

Abstract In this paper we present an overview of the progress made using chaotic maps to model individual and aggregated self-similar traffic streams and in particular their impact on queue performance. Our findings show that the asymptotic behaviour of the queue is a function only of the tail of the ON active periods, and that the Hurst parameter is not a good parameter to achieve traffic control: two different self-similar traffic traces can have the same Hurst parameter but have a very different effect on the queue statistics. These results are part of a framework for developing chaotic control of networks.

Mode-locking in coupled map lattices

Abstract We study propagation of pulses along one-way coupled map lattices, which originate from the transition between two superstable states of the local map. The velocity of the pulses exhibits a staircase-like behaviour as the coupling parameter is varied. For a piecewise linear local map, we prove that the velocity of the wave has a Devils staircase dependence on the coupling parameter. A wave travelling with rational velocity is found to be stable to parametric perturbations in a manner akin to rational mode-locking for circle maps. We provide evidence that mode-locking is also present for a broader range of maps and couplings.