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This paper is devoted to the design of new chaotic Pseudo Random Number Generator (CPRNG). Exploring several topologies of network of 1-D coupled chaotic mapping, we focus first on two dimensional networks. Two coupled maps are studied: TTL^{RC} non-alternate, and TTL^{SC} alternate. The primary idea of the novel maps has been based on an original coupling of the tent and logistic maps to achieve excellent random properties and homogeneous /uniform/ density in the phase plane, thus guaranteeing maximum security when used for chaos base cryptography. In this aim a new nonlinear CPRNG: MTTL_{2}^{SC} is proposed. In addition, we explore higher dimension and the proposed ring coupling with injection mechanism enables us to achieve the strongest security requirements.

The motion of any mechanical system composed of rigid bodies or discretized elastic bodies is always described by a set of nonlinear ordinary differential equations of second order of the form (Chapter 1)

A neural network with weights described by the position of interacting Brownian particles is considered. Each weight is taken to correspond to a Gaussian particle. Neural learning aims at leading a set of M weights (Brownian particles) with different initial values on the 2-D phase plane, to the desirable final position. A Lyapunov function describes the evolution of the phase diagram towards the equilibrium Convergence to the goal state is assured for each particle through the negative definiteness of the associated Lyapunov function. The update of the weight (trajectory in the phase plane) is affected by (1) a drift force due to the harmonic potential, and (2) the interaction with neighboring weights (particles). It is finally shown that the mean of the particles will converge to the equilibrium while using LaSalle’s theorem it is shown that the individual particles will remain within a small area encircling the equilibrium.

Figure 5.1 displays the rotor of an automotive turbocharger in the center housing and rotating assembly (CHRA) including the rotor shaft, compressor and turbine wheels, rotating floating ring bearings (RFRBs), seal rings, and thrust rings. All rotor components must be taken into account in the rotor dynamic computation to study the rotor vibration response, such as the frequency components in the Waterfall plot, rotor orbits in the phase plane, and response amplitudes in the time domain diagram. Open image in new window Fig. 5.1 Automotive turbocharger rotor setup in the CHRA (Courtesy BMTS)

This study investigates joint kinematics, joint angular positions, and orbital dynamic stability of human walking with different prosthetic feet by using graphical methods of phase plane portraits, Poincare maps and Floquet multipliers, respectively. The Flex foot, SACH foot, Seattle foot and one non-specific optimized foot are taken as the research objects. Numerical experiments are performed to compare and evaluate human locomotion traits on several aspects by focusing on the concerned four kinds of prosthetic feet.

Braitenberg vehicles are a widely used model of animal behaviour in robotics and Artificial Life. This paper presents the first comprehensive and formal analysis of the behaviour of Braitenberg vehicles 2 and 3. After a review of their intuitive behaviour we present their models as dynamical systems, that under circularly symmetric stimuli can be simplified and analysed using the phase plot technique. We prove that intuitive understanding is not enough to determine the potential behaviour of vehicles 2b and 3a and, under certain circumstances, they could behave as vehicles 2a and 3b respectively. This work paves the way to a proper understanding of Braitenberg vehicles, as it provides theoretical support for their many existing empirically grounded works.

Syllogistic reasoning is modeled in analog hardware and some hardware models, i.e. syllogisms Baroco and Darii are presented. Chaining of syllogisms is modeled by using original min-max entities (circuits), “to see” whether the two rules, modeled in dedicated hardware, i.e., IF A THEN B and IF B THEN C imply the “hardware” rule IF A THEN C. The preliminaries include original min-max circuits based on operational amplifiers (Op-Amp), straight lines Op-Amp generators and different test circuits designed in Electronics WorkBench simulation environment and in real hardware. The “stage” to perform modeling is the phase plane.

The chapter begins with a short introduction to solitions and solitary waves in nonlinear dynamics. Thereupon the hierarchical equation and the coupled system are treated separately in the nonlinear case. In both models necessary and sufficient conditions for the existence of solitary waves are established and physical and geometrical properties of the solitary waves are studied. The waves are bell-shaped but asymmetric in the case of presence of the nonlinearity in the micro-scale. In the case of the hierarchical equation, a series axpansion of the wave function is deduced, too. The proofs are based on the canonization of the equations and analysis of trajectories in the phase plane. In some cases nonlinear transforms are applied in order to simplify the treatment of phase curves.

The discussions of deterministic evolution in Chap. 7 were focused on the modeling of the evolution of one variable (as heat, mass, the position of any body, or a population density). The consideration of such relatively simple problems is helpful for a basic understanding of the structure and the range of applicability of equations for typical problems. However, only a narrow range of problems can be described in this way: the analysis of most real problems requires the consideration of the multivariate evolution of several variables. The latter is required, for example, regarding the interaction of biological species and motions of bodies or fluids in three-dimensional space. To deal with such cases we extend here the concepts used for the modeling of mechanical and population ecology processes in Chap. 7 to the modeling of the joint evolution of several variables. We will continue with the consideration of global properties that change in time but not in space, i.e., partial differential equations that describe the evolution of processes in space will be not considered. The mathematics of models for the evolution of such processes can be formulated in terms of linear and nonlinear systems of coupled ordinary differential equations.

One of the important properties of feedback control systems is that under certain parameter conditions, their steady response is a periodic motion described by the limit cycles in the phase plane. As pointed out earlier, a stable limit cycle usually locates in the neighbourhood of an unstable equilibrium point. Actually, if the static characteristic of the nonlinear element has a hysteresis, the equilibrium point of the feedback control system is usually unstable provided the system parameters satisfy certain conditions under which the self-excited oscillation may occur. In addition, if the open-loop gain is higher than its critical value, the equilibrium position of some feedback control systems is also unstable. Here, the disturbed motion of the system diverging from the equilibrium position is restricted to the limited power of amplifiers and motors and will approach a periodic motion, namely, the self-excited oscillation.

This paper describes the Large-Eddy Simulation (LES) of a droplet laden mixing layer. The method uses an Eulerian description of the carrier flow with a Lagrangian formulation for the dispersed phase. Special treatment of inlet conditions for both phases is critical. Sub-grid scale dispersion effects are investigated, predictions of Sauter Mean Diameter and concentration agree closely with experimental data. The performance of a hybrid droplet/parcel approach for improved efficiency has also been investigated.

Self-Similar Blast Wave - The concept of self-similarity played a key role in the development of blast wave theory. More specifically, it has been formulated to explore the effects of the atom bomb explosion. For that purpose, the blast wave is considered as a gasdynamic flow field generated by the deposition of a finite amount of energy in zero time at a geometric point in an unbounded compressible medium – an event referred to as a point explosion: Ipso facto, the flow field is geometrically symmetric, while its extent is bounded by a gasdynamic front, whereas its structure is governed by conservation principles expressed in terms of time dependent, spatially one-dimensional partial differential equations. This chapter presents the salient features of linearly, cylindrically and spherically symmetric fields created by point explosions whose fronts propagate into a vacuum.

We study the generalised Chazy equation, \(\dddot x + x^q \ddot x + kx^{q - 1} \dot x^2 = 0\), which is characterised by the symmetries of time translation and rescaling. For a large class of initial conditions numerical computations reveal the asymptotic appearence of periodic solutions for k=q+1. These solutions are identical after rescaling and, in this sense, exhibit the property of a limit cycle in the three dimensional phase space. The periodic solutions are related to a conventional limit cycle of a class of second order ordinary differential equations which are connected to the existence of a first integral of the generalised Chazy equation.

The interactions between several types of intabilities lead to bifurcations of multiple codimension. The convectioe flow of a binary mixture in a porous medium is a good example of such an interaction of two instabilities. These instabilities depend on two parameters: the Rayleigh number Ra and the separation ratio ψ. The corresponding neutral curves intersect, in the (Ra, ψ) plane, at a polycritical point. We describe, in this paper, the influence, on this bifurcation, of a small low-frequency time variation of the thermal boundary conditions. The reduced nonlinear Mathieu equation depends on a parameter e, which characterizes the low frequency modulation. For e= 0 the Mathieu equation possesses, in the phase plane, heteroclinic and periodic solutions. For e ≠ 0, the system may exhibit a chaotic regime of Smale horseshoe type. The onset of the chaotic regime as well as the curve at which a saddle-node bifurcation occurs can be estimated by means of Melnikov's technique. Numerical simulations agree with the results of Melnikov's theory. This problem can be considered to be a model for other instability problems arising in other domains of the Physics, such as thermosolutal convection and convection in magnetic fields or under rotation.

A non-adiabatic thermal model of combustion is presented. The model admits ignition and extinction waves as exact solutions. The shape of these waves and their speed are determined numerically. Phase plane equations are used to understand the uniqueness of the wavespeeds and to determine watershed criteria for initial conditions.

A lumped-parameter model of a single-stage planetary gear train, which included the factors of the floating motion of the centroid of the sun gear, time-varying mesh stiffness, gear meshing transmission errors and gear backlashes, was established. The non-dimensional motion equations of system were solved by the method of numerical integration. The steady responses, namely period-1, period-2, period-4, period-8 and chaotic responses, were obtained under the different non-dimensional excitation frequency. The characteristic of system was described by time history, phase plane plot, Poincare section and FFT spectrum. The period doubling route to chaos was found by changing frequency of system.

The development of a general statistical physics of nonequilibrium systems was one of the main unfinished tasks of statistical physics of the 20th century. The aim of this work is the study of a special class of nonequilibrium systems where the formulation of an ensemble theory of some generality is possible. These are the so-called canonical-dissipative systems, where the driving terms are determined by invariants of motion. We construct canonical-dissipative systems which are ergodic on certain surfaces on the phase plane. These systems may be described by a nonequilibrium microcanocical ensemble, corresponding to an equal distribution on the target surface. Next we construct and solve Fokker-Planck equations; this leads to a kind of canonical-dissipative ensemble. In the last part we discuss the thoretical problem how to define bifurcations in the framework of nonequilibrium statistics and several possible applications.