399 research outputs found
Localization-delocalization transition on a separatrix system of nonlinear Schrodinger equation with disorder
Localization-delocalization transition in a discrete Anderson nonlinear
Schr\"odinger equation with disorder is shown to be a critical phenomenon
similar to a percolation transition on a disordered lattice, with the
nonlinearity parameter thought as the control parameter. In vicinity of the
critical point the spreading of the wave field is subdiffusive in the limit
. The second moment grows with time as a powerlaw , with exactly 1/3. This critical spreading finds its
significance in some connection with the general problem of transport along
separatrices of dynamical systems with many degrees of freedom and is
mathematically related with a description in terms fractional derivative
equations. Above the delocalization point, with the criticality effects
stepping aside, we find that the transport is subdiffusive with
consistently with the results from previous investigations. A threshold for
unlimited spreading is calculated exactly by mapping the transport problem on a
Cayley tree.Comment: 6 pages, 1 figur
A topological approximation of the nonlinear Anderson model
We study the phenomena of Anderson localization in the presence of nonlinear
interaction on a lattice. A class of nonlinear Schrodinger models with
arbitrary power nonlinearity is analyzed. We conceive the various regimes of
behavior, depending on the topology of resonance-overlap in phase space,
ranging from a fully developed chaos involving Levy flights to pseudochaotic
dynamics at the onset of delocalization. It is demonstrated that quadratic
nonlinearity plays a dynamically very distinguished role in that it is the only
type of power nonlinearity permitting an abrupt localization-delocalization
transition with unlimited spreading already at the delocalization border. We
describe this localization-delocalization transition as a percolation
transition on a Cayley tree. It is found in vicinity of the criticality that
the spreading of the wave field is subdiffusive in the limit
t\rightarrow+\infty. The second moment grows with time as a powerlaw t^\alpha,
with \alpha = 1/3. Also we find for superquadratic nonlinearity that the analog
pseudochaotic regime at the edge of chaos is self-controlling in that it has
feedback on the topology of the structure on which the transport processes
concentrate. Then the system automatically (without tuning of parameters)
develops its percolation point. We classify this type of behavior in terms of
self-organized criticality dynamics in Hilbert space. For subquadratic
nonlinearities, the behavior is shown to be sensitive to details of definition
of the nonlinear term. A transport model is proposed based on modified
nonlinearity, using the idea of stripes propagating the wave process to large
distances. Theoretical investigations, presented here, are the basis for
consistency analysis of the different localization-delocalization patterns in
systems with many coupled degrees of freedom in association with the asymptotic
properties of the transport.Comment: 20 pages, 2 figures; improved text with revisions; accepted for
publication in Physical Review
Theoretical Analysis and Circuit Verification for Fractional-Order Chaotic Behavior in a New Hyperchaotic System
A novel nonlinear four-dimensional hyperchaotic system and its fractional-order form are presented. Some dynamical behaviors of this system are further investigated, including Poincaré mapping, parameter phase portraits, equilibrium points, bifurcations, and calculated Lyapunov exponents. A simple fourth-channel block circuit diagram is designed for generating
strange attractors of this dynamical system. Specifically, a novel network module fractance is introduced to achieve fractional-order
circuit diagram for hardware implementation of the fractional attractors of this nonlinear hyperchaotic system with order as low
as 0.9. Observation results have been observed by using oscilloscope which demonstrate that the fractional-order nonlinear hyperchaotic attractors exist indeed in this new system
Symmetry in Chaotic Systems and Circuits
Symmetry can play an important role in the field of nonlinear systems and especially in the design of nonlinear circuits that produce chaos. Therefore, this Special Issue, titled “Symmetry in Chaotic Systems and Circuits”, presents the latest scientific advances in nonlinear chaotic systems and circuits that introduce various kinds of symmetries. Applications of chaotic systems and circuits with symmetries, or with a deliberate lack of symmetry, are also presented in this Special Issue. The volume contains 14 published papers from authors around the world. This reflects the high impact of this Special Issue
Physics and Applications of Laser Diode Chaos
An overview of chaos in laser diodes is provided which surveys experimental
achievements in the area and explains the theory behind the phenomenon. The
fundamental physics underpinning this behaviour and also the opportunities for
harnessing laser diode chaos for potential applications are discussed. The
availability and ease of operation of laser diodes, in a wide range of
configurations, make them a convenient test-bed for exploring basic aspects of
nonlinear and chaotic dynamics. It also makes them attractive for practical
tasks, such as chaos-based secure communications and random number generation.
Avenues for future research and development of chaotic laser diodes are also
identified.Comment: Published in Nature Photonic
On the Application of PSpice for Localised Cloud Security
The work reported in this thesis commenced with a review of methods for creating random binary sequences for encoding data locally by the client before storing in the Cloud. The first method reviewed investigated evolutionary computing software which generated noise-producing functions from natural noise, a highly-speculative novel idea since noise is stochastic. Nevertheless, a function was created which generated noise to seed chaos oscillators which produced random binary sequences and this research led to a circuit-based one-time pad key chaos encoder for encrypting data. Circuit-based delay chaos oscillators, initialised with sampled electronic noise, were simulated in a linear circuit simulator called PSpice. Many simulation problems were encountered because of the nonlinear nature of chaos but were solved by creating new simulation parts, tools and simulation paradigms. Simulation data from a range of chaos sources was exported and analysed using Lyapunov analysis and identified two sources which produced one-time pad sequences with maximum entropy. This led to an encoding system which generated unlimited, infinitely-long period, unique random one-time pad encryption keys for plaintext data length matching. The keys were studied for maximum entropy and passed a suite of stringent internationally-accepted statistical tests for randomness. A prototype containing two delay chaos sources initialised by electronic noise was produced on a double-sided printed circuit board and produced more than 200 Mbits of OTPs. According to Vladimir Kotelnikov in 1941 and Claude Shannon in 1945, one-time pad sequences are theoretically-perfect and unbreakable, provided specific rules are adhered to. Two other techniques for generating random binary sequences were researched; a new circuit element, memristance was incorporated in a Chua chaos oscillator, and a fractional-order Lorenz chaos system with order less than three. Quantum computing will present many problems to cryptographic system security when existing systems are upgraded in the near future. The only existing encoding system that will resist cryptanalysis by this system is the unconditionally-secure one-time pad encryption
On the outer synchronization of complex dynamical networks
Complex network models have become a major tool in the modeling and analysis of many physical, biological and social phenomena. A complex network exhibits behaviors which emerge as a consequence of interactions between its constituent elements, that is, remarkably, not the same as individual components. One particular topic that has attracted the researchers' attention is the analysis of how synchronization occurs in this class of models, with the expectation of gaining new insights of the interactions taking place in real-world complex systems. Most of the work in the literature so far has been focused on the synchronization of a collection of interconnected nodes (forming one single network), where each node is a dynamical system governed by a set of nonlinear differential equations, possibly displaying chaotic dynamics. In this thesis, we study an extended version of this problem. In particular, we consider a setup consisting of two complex networks which are coupled unidirectionally, in such a way that a set of signals from the master network are injected into the response network, and then investigate how synchronization is attained. Our analysis is fairly general. We impose few conditions on the network structure and do not assume that the nodes in a single network are synchronized. This work can be divided into two main parts; outer synchronization in fractional-order networks, and outer synchronization in ordinary networks. In both cases the system parameters are perturbed by bounded, time varying and unknown perturbations. The synchronizer feedback matrix is possibly perturbed with the same type of perturbations as well. In both cases, of fractional-order and ordinary networks, we build up several theorems that ensure the attainment of synchronization in various scenarios, including, e.g., cases in which the coupling matrix of the networks is non-diffusive (hence we can avoid this assumption, which is almost invariably made in the literature). In all the cases of interest, we show that the scheme for coupling the networks is very simple, as it reduces to the computation of a single gain matrix whose dimension is independent of the number of network nodes. The structure of the designed synchronizer is also very simple, making it convenient for real-world applications. Although all of the proposed schemes are assessed analytically, numerical results (obtained by computer simulations) are also provided to illustrate the proposed methods. ---------------------------------------Las redes complejas se han convertido en una herramienta fundamental
en el análisis de muchos sistemas físicos, biológicos y sociales. Una red
compleja presenta comportamientos que "emergen" como consecuencia de
las interacciones entre sus elementos constituyentes pero que no se observan
de forma individual en estos elementos.
Un aspecto en concreto que ha atrapado la atención de muchos
investigadores es el análisis de cómo se producen fenómenos de sincronización
en esta clase de modelos, con la esperanza de alcanzar una mayor
comprensión de las interacciones que tienen lugar en sistemas complejos del
mundo real. La mayor parte del trabajo publicado hasta ahora ha estado
centrado en la sincronización de una colección de nodos interconectados (que
forman una única red con entidad propia), donde cada nodo es un sistema
dinámico gobernado por un conjunto de ecuaciones diferenciales no lineales,
posiblemente caóticas.
En esta tesis estudiamos una versión extendida de este problema.
En concreto, consideramos un sistema formado por dos redes complejas
acopladas unidireccionalmente, de manera que un conjunto de señales de la
red principal se inyectan en la red secundaria, e investigamos cómo se alcanza
un estado de sincronización. Este fenómeno se conoce como "sincronización
externa". Nuestro análisis es muy general. Se imponen pocas condiciones a
las estructura de las redes y no es necesario suponer que los nodos de cada
red estén sincronizados entre sí previamente.
Esta memoria se puede dividir en dos bloques: la sincronización externa
de redes descritas por ecuaciones diferenciales de orden fraccionario y
la sincronización externa de redes ordinarias (descritas por ecuaciones
diferenciales de orden entero). En ambos casos, se admite que los
parámetros del sistema puedan estar sujetos a perturbaciones desconocidas,
posiblemente variables con el tiempo, pero acotadas. La matriz de
realimentación del esquema de sincronización puede sufrir el mismo tipo
de perturbación. En ambos casos, con ecuaciones de orden fraccionario
o entero, construimos varios teoremas que aseguran que se alcance la
sincronización en escenarios diversos, incluyendo, por ejemplo, casos en los que la matriz de acoplamiento de las redes es no difusiva (por lo tanto,
podemos evitar esta hipótesis, que es ubicua en la literatura). En todos los
casos de interés, mostramos que el esquema necesario para interconectar las
redes es muy simple, puesto que se reduce al cálculo de una única matriz de
ganancia cuya dimensión es independiente de la dimensión total (número de nodos) de las redes. La estructura del sincronizados es también muy sencilla,
lo que la hace potencialmente adecuada para aplicaciones del mundo real.
Aunque todos los esquemas que se proponen se analizan de manera
rigurosa, también se muestran resultados numéricos (obtenidos mediante
simulación) para ilustrar los métodos propuestos.Programa de Doctorado en Multimedia y Comunicaciones por la Universidad Carlos III de Madrid y la Universidad Rey Juan CarlosPresidente: Ángel María Bravo Santos.- Secretario: David Luengo García.- Vocal: Irene Sendiña Nada
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