Chaotic Systems with Hyperbolic Sine Nonlinearity

In recent years, exploring and investigating chaotic systems with hyperbolic sine nonlinearity has gained the interest of many researchers. With two back-to-back diodes to approximate the hyperbolic sine nonlinearity, these chaotic systems can achieve simplicity of the electrical circuit without any multiplier or sub-circuits. In this chapter, the genesis of chaotic systems with hyperbolic sine nonlinearity is introduced, followed by the general method of generating nth-order (n > 3) chaotic systems. Then some derived chaotic systems/torus-chaotic system with hyperbolic sine nonlinearity is discussed. Finally, the applications such as random number generator algorithm, spread spectrum communication and image encryption schemes are introduced. The contribution of this chapter is that it systematically summarizes the design methods, the dynamic behavior and typical engineering applications of chaotic systems with hyperbolic sine nonlinearity, which may widen the current knowledge of chaos theory and engineering applications based on chaotic systems

Constructing multiwing attractors from a robust chaotic system with non-hyperbolic equilibrium points

We investigate a three-dimensional (3D) robust chaotic system which only holds two nonhyperbolic equilibrium points, and finds the complex dynamical behaviour of position modulation beyond amplitude modulation. To extend the application of this chaotic system, we initiate a novel methodology to construct multiwing chaotic attractors by modifying the position and amplitude parameters. Moreover, the signal amplitude, range and distance of the generated multiwings can be easily adjusted by using the control parameters, which enable us to enhance the potential application in chaotic cryptography and secure communication. The effectiveness of the theoretical analyses is confirmed by numerical simulations. Particularly, the multiwing attractor is physically realized by using DSP (digital signal processor) chip

Discrete Breathers

Nonlinear classical Hamiltonian lattices exhibit generic solutions in the form of discrete breathers. These solutions are time-periodic and (typically exponentially) localized in space. The lattices exhibit discrete translational symmetry. Discrete breathers are not confined to certain lattice dimensions. Necessary ingredients for their occurence are the existence of upper bounds on the phonon spectrum (of small fluctuations around the groundstate) of the system as well as the nonlinearity in the differential equations. We will present existence proofs, formulate necessary existence conditions, and discuss structural stability of discrete breathers. The following results will be also discussed: the creation of breathers through tangent bifurcation of band edge plane waves; dynamical stability; details of the spatial decay; numerical methods of obtaining breathers; interaction of breathers with phonons and electrons; movability; influence of the lattice dimension on discrete breather properties; quantum lattices - quantum breathers. Finally we will formulate a new conceptual aproach capable of predicting whether discrete breather exist for a given system or not, without actually solving for the breather. We discuss potential applications in lattice dynamics of solids (especially molecular crystals), selective bond excitations in large molecules, dynamical properties of coupled arrays of Josephson junctions, and localization of electromagnetic waves in photonic crystals with nonlinear response.Comment: 62 pages, LaTeX, 14 ps figures. Physics Reports, to be published; see also at http://www.mpipks-dresden.mpg.de/~flach/html/preprints.htm

Dynamical Casimir effect in a periodically changing domain: A dynamical systems approach

We study the problem of the behavior of a quantum massless scalar field in the space between two parallel infinite perfectly conducting plates, one of them stationary, the other moving periodically. We reformulate the physical problem into a problem about the asymptotic behavior of the iterates of a map of the circle, and then apply results from theory of dynamical systems to study the properties of the map. Many of the general mathematical properties of maps of the circle translate into properties of the field in the cavity. For example, we give a complete classification of the possible resonances in the system, and show that small enough perturbations do not destroy the resonances. We use some mathematical identities to give transparent physical interpretation of the processes of creation and amplification of the quantum field due to the motion of the boundary and to elucidate the similarities and the differences between the classical and quantum fields in domains with moving boundaries.Comment: 19 pages, 7 figure

Theoretical and numerical studies of chaotic mixing

Theoretical and numerical studies of chaotic mixing are performed to circumvent the difficulties of efficient mixing, which come from the lack of turbulence in microfluidic devices. In order to carry out efficient and accurate parametric studies and to identify a fully chaotic state, a spectral element algorithm for solution of the incompressible Navier-Stokes and species transport equations is developed. Using Taylor series expansions in time marching, the new algorithm employs an algebraic factorization scheme on multi-dimensional staggered spectral element grids, and extends classical conforming Galerkin formulations to nonconforming spectral elements. Lagrangian particle tracking methods are utilized to study particle dispersion in the mixing device using spectral element and fourth order Runge-Kutta discretizations in space and time, respectively. Comparative studies of five different techniques commonly employed to identify the chaotic strength and mixing efficiency in microfluidic systems are presented to demonstrate the competitive advantages and shortcomings of each method. These are the stirring index based on the box counting method, Poincare sections, finite time Lyapunov exponents, the probability density function of the stretching field, and mixing index inverse, based on the standard deviation of scalar species distribution. Series of numerical simulations are performed by varying the Peclet number (Pe) at fixed kinematic conditions. The mixing length (lm) is characterized as function of the Pe number, and lm ∝ ln(Pe) scaling is demonstrated for fully chaotic cases. Employing the aforementioned techniques, optimum kinematic conditions and the actuation frequency of the stirrer that result in the highest mixing/stirring efficiency are identified in a zeta potential patterned straight micro channel, where a continuous flow is generated by superposition of a steady pressure driven flow and time periodic electroosmotic flow induced by a stream-wise AC electric field. Finally, it is shown that the invariant manifold of hyperbolic periodic point determines the geometry of fast mixing zones in oscillatory flows in two-dimensional cavity

Performance analysis of filtering based chaotic synchronization and development of chaotic digital communication schemes

Ph.DDOCTOR OF PHILOSOPH

Semiclassical Theory for extreme events of quantum maps eigenstates

In this work, we explore the extreme events of the eigenstate intensities in the position basis of three parameter-dependent quantum maps: standard map, perturbed cat map, and kicked Harper map. In order to expand previous works, we move forward considering not only fully chaotic states but eigenstates from near-integrable and mixed regimes. Namely, we propose the kurtosis measure to quantify and assess the tail of the intensities distributions. For all addressed maps, a conspicuous result is sharp peaks in the kurtosis for specific parameters in the mixed regime. Therefore, a semiclassical expression of kurtosis is achieved through a doubled average by the position and the energy spectrum of the eigenstates, which enables us to discuss possible explanations for the phenomenon. From a semiclassical perspective, i.e., h → 0, we advocate for the stable island contribution to the peaks in the kurtosis but let other classical structures open to forward inspections. Ultimately, we discourse the quantum phases and show that they also important play a role.Neste trabalho examinamos os eventos extremos das intensidades dos autoestados na base de posição de três mapas quânticos dependentes de parâmetros: o mapa padrão, mapa do gato perturbado e o mapa quicado de Harper. A fim de ampliar trabalhos anteriores, consideramos não apenas estados totalmente caóticos mas também autoestados nos regimes quase-integráveis e mistos. Especificamente, propusemos medir a curtose para quantificar e avaliar a cauda das distribuições das intensidades. Para todos os mapas abordados, um resultado notável são picos na curtose para valores de parâmetro específicos na região mista. Portanto, uma expressão semiclássica da curtose é alcançada através de uma média dupla pela posição e pelo espectro de energia dos autoestados, o qual nos permite a discutir possíveis explicações para o fenômeno. Por uma perspectiva semiclássica, ou seja, h → 0, defendemos a contribuição das ilhas estáveis para os picos na curtose, mas deixamos outras estruturas clássicas abertas para investigações futuras. Por fim, abordamos as fases quânticas e mostramos que elas também desempenham um papel importante

Developing a flexible and expressive realtime polyphonic wave terrain synthesis instrument based on a visual and multidimensional methodology

The Jitter extended library for Max/MSP is distributed with a gamut of tools for the generation, processing, storage, and visual display of multidimensional data structures. With additional support for a wide range of media types, and the interaction between these mediums, the environment presents a perfect working ground for Wave Terrain Synthesis. This research details the practical development of a realtime Wave Terrain Synthesis instrument within the Max/MSP programming environment utilizing the Jitter extended library. Various graphical processing routines are explored in relation to their potential use for Wave Terrain Synthesis