A minimal model of pattern formation in a prey-predator system

The spatio-temporal dynamics of a prey-predator community is described by two reaction-diffusion equations. It is shown that for a class of initial conditions the spatio-temporal system dynamics resembles a “phase transition” between a regular and an irregular phase, separated by a moving boundary. A simple approach to specify spatio-temporal chaos is proposed.     

Minimizing transient times in a coupled solid‐state laser model

We consider a system of ordinary differential equations modelling the dynamics of two coupled solid‐state lasers. Under the dynamics, this system may execute transitions between in‐phase and out‐of‐phase states. For satellite communications and high‐speed data transfer the transition times should be reduced to their shortest possible duration. In this paper, we apply optimal control theory to find the values of various laser parameters (e.g. the amplitude of the injected field, detunings, and coupling constants) which minimize the transient times between out‐of‐phase and in‐phase states. The effect of each parameter is shown to be independent of the other two, and the transient time is shown to be a strictly increasing function of detuning and a strictly decreasing function of the coupling constant and amplitude of the injected field. The effect of initial conditions on transient times is also analysed. Published in 2005 by John Wiley & Sons, Ltd.     

On Singular Points of Equations of Mechanics

A system of ordinary differential equations whose right-hand side has no limit at some singular point is considered. This situation is typical of mechanical systems with Coulomb friction in a neighborhood of equilibrium. The existence and uniqueness of solutions to the Cauchy problem is analyzed. A key property is that the phase curve reaches the singular point in a finite time. It is shown that the subsequent dynamics depends on the extension of the vector field to the singular point according to the physical interpretation of the problem: systems coinciding at all point, except for the singular one, can have different solutions. Uniqueness conditions are discussed.     

Spiral chaos in the nonholonomic model of a Chaplygin top

This paper presents a numerical study of the chaotic dynamics of a dynamically asymmetric unbalanced ball (Chaplygin top) rolling on a plane. It is well known that the dynamics of such a system reduces to the investigation of a three-dimensional map, which in the general case has no smooth invariant measure. It is shown that homoclinic strange attractors of discrete spiral type (discrete Shilnikov type attractors) arise in this model for certain parameters. From the viewpoint of physical motions, the trace of the contact point of a Chaplygin top on a plane is studied for the case where the phase trajectory sweeps out a discrete spiral attractor. Using the analysis of the trajectory of this trace, a conclusion is drawn about the influence of “strangeness” of the attractor on the motion pattern of the top.     

Stability and optimal control of thermomechanical processes with non-convex free energies of Ginzburg–Landau type

In this paper we study a coupled non-linear system of partial differential equations that models the dynamics of structural phase transitions in a one-dimensional non-viscous and heat-conducting solid. The corresponding Helmholtz free energy density is assumed in Ginzburg–Landau form; to allow for phase transitions and hysteresis phenomena, it is not assumed convex in the order parameter. It is shown that the solution of the system depends continuously upon the data, and we prove an existence result for an associated optimal control problem.     

Quantum dynamics as a stochastic process in phase space

It is shown that quantum dynamics is equivalent to a stochastic process in phase space. The process is described by normalized but not necessarily positive probability distributions (“pseudoprobabilities”). Evolution of the simultaneous probability distribution of momentum and coordinate exactly conincides with the dynamics of the Wigner function of a quantum system. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 3, pp. 454–458, March, 1997.     

An evolution equation in phase space and the Weyl correspondence

The Weyl correspondence that associates a quantum-mechanical operator to a Hamiltonian function on phase space is defined for all tempered distributions on R². The resulting Weyl operators are shown to include most Schroedinger operators for a system with one degree of freedom. For each tempered distribution, an evolution equation in phase space is defined that is formally equivalent to the dynamics of the Heisenberg picture. The evolution equation is studied both through a separation of variables technique that expresses the evolution operator as the difference of two Weyl operators and through the geometric properties of the distribution. For real tempered distributions with compact support the evolution equation has a unique solution if and only if the Weyl equation does. The evolution operator has skew-adjoint extensions that solve the evolution equation if the distribution satisfies an orthogonal symmetry condition.     

Switched system modeling and robust steering control of the tail end phase in a hot strip mill

In this article, a robust steering control for the last phase of the rolling process in a hot strip mill is proposed. This phase, called tail end phase, may be modelled as a linear switched system. The switchings make the system unstable and the task of the tail end steering control consists in guaranteeing the safety of the industrial plant. The system involves a two time scales dynamics. Hence, the singular perturbation method is used in order to design the control law. The controller has to take into account the physical variations of the rolled products and an uncertainty in the switching time. Results concerning the ArcelorMittal hot strip mill of Eisenhüttenstadt are presented.     

Visualization in mechanics: the dynamics of an unbalanced roller

It is well known that mechanical engineering students often find mechanics a difficult area to grasp. This article describes a system of equations describing the motion of a balanced and an unbalanced roller constrained by a pivot arm. A wide range of dynamics can be simulated with the model. The equations of motion are embedded in a graphical user interface for its numerical solution in MATLAB. This allows a student's focus to be on the influence of different parameters on the system dynamics. The simulation tool can be used as a dynamics demonstrator in a lecture or as an educational tool driven by the imagination of the student. By way of demonstration the simulation tool has been applied to a range of roller–pivot arm configurations. In addition, approximations to the equations of motion are explored and a second-order model is shown to be accurate for a limited range of parameters.     

Functional integral method for investigating the dynamics of a Dicke type model

Conclusions The approach to the investigation of nonequilibrium systems proposed in this paper has made it possible to treat fairly readily the dynamics of a Dicke type model in the thermodynamic limit. A number of other model systems can be investigated similarly. The results obtained from the expression for the generating functional of the temperature-time Green's functions generalize the corresponding result of [4]. Exact study of the dynamical behavior of the Dicke type model at large times has shown that it possesses different nonlinear dynamical regimes, the conditions for the existence of which depend strongly on the initial values of the parameters of the system and the statistics of the fields. The point of bifurcation of the dynamics is an analog of the critical temperature for an equilibrium phase transition.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 69, No. 2, pp. 279–286, November, 1987.     

Algoritmo compensador neuronal discreto de dinámica en robots móviles usando filtro de Kalman extendido

This paper presents the design of an algorithm based on neural networks in discrete time for its application in mobile robots. In addition, the system stability is analyzed and an evaluation of the experimental results is shown.The mobile robot has two controllers, one addressed for the kinematics and the other one designed for the dynamics. Both controllers are based on the feedback linearization. The controller of the dynamics only has information of the nominal dynamics (parameters). The neural algorithm of compensation adapts its behaviour to reduce the perturbations caused by the variations in the dynamics and the model uncertainties. Thus, the differences in the dynamics between the nominal model and the real one are learned by a neural network RBF (radial basis functions) where the output weights are set using the extended Kalman filter. The neural compensation algorithm is efficient, since the consumed processing time is lower than the one required to learning the totality of the dynamics. In addition, the proposed algorithm is robust with respect to failures of the dynamic controller. In this work, a stability analysis of the adaptable neural algorithm is shown and it is demonstrated that the control errors are bounded depending on the error of approximation of the neural network RBF. Finally, the results of experiments performed by using a mobile robot are shown to test the viability in practice and the performance for the control of robots.     

Near Optimization of Dynamic Systems by Decomposition and Aggregation

This paper is concerned with the reduction of a class of optimal control problems to simpler problems by using decomposition and aggregation. Decomposition is shown to provide a good approximation when the system dynamics involve nearly decomposable matrices or variables with strong and weak interactions. Aggregation provides a good approximation if each of the decomposed matrices has one or more dominant eigenvalues. It is shown how one can construct nearly-optimal controls for the given system from the optimal solutions of the simpler reduced problems.     

Computation of dynamical phase transitions in solids

This paper deals with the dynamics of phase boundaries in a nonlinear elastic two-phase material. We consider the elasticity system in 1D and the equations of anti-plane shear motion in 2D, where effects of viscosity and capillarity are neglected. These first-order conservation laws allow to represent phase boundaries as shock-like sharp interfaces. However, in contrast to what is known for homogeneous materials, the entropy inequality does not select a unique solution, and an additional criterion, the so-called kinetic relation, is required.Based on a scheme introduced by Hou, Rosakis and LeFloch [T. Hou, Ph. Rosakis, P.G. LeFloch, A level-set approach to the computation of twinning and phase-transition dynamics, J. Comput. Phys. 150 (1999) 302–331] we focus on the numerical solution of a specific model system. Using a level-set technique to enforce the kinetic relation on the discrete level leads to a reformulation of the original system in the form of a system of conservation laws coupled to a Hamilton–Jacobi equation for each phase boundary. The numerical method for the reformulated system is constructed for unstructured meshes (in 2D), and a self-adaptive algorithm is introduced.In the 1D-case we show that the reformulated system has a solution that corresponds to exact dynamical phase boundaries of the elasticity system which obey the kinetic relation. To validate the method in 2D, we present computations on the interaction of a plane wave with a phase boundary. The efficiency of the adaptation mechanism is demonstrated by an example showing the development of microstructures by twinning.     

Waggling conical pendulums and visualization in mechanics

Students often find mechanics a difficult area to grasp. This paper describes an equation of motion for a waggling conical pendulum. A wide range of pendulum dynamics can be simulated with this model. The equation of motion is embedded in a graphical user interface (GUI) for its numerical solution in MATLAB. This allows a student's focus to be on the influence of different parameters on the pendulums dynamics. The simulation tool can be used as a dynamics demonstrator in a lecture or as an educational tool driven by the imagination of the student. By way of demonstration, the simulation tool has been applied to two damped pendulums and an inverted damped pendulum. The model has also been used to simulate resonance and has shown that there is a wide range of behaviour possible depending on the type of forcing applied. Finally, a forced conical pendulum as a system for harnessing wave energy is considered.     

Reversible dynamics in discrete dissipative systems

The random walk model of Brownian motion is an example of a stochastic system which exhibits intrinsically irreversible behaviour. In spite of this, a simple discrete version of the model has been shown to harbour dynamics which are reversible and are described by a discrete form of Schrödinger's equation. The reversible dynamics appear as second order effects in this diffusive model, and the usual relationship between macroscopic irreversibility and microscopic reversibility is itself reversed. This will be discussed in the context of the `Brussels' school' on irreversibility.     

Bistability of rotational modes in a system of coupled pendulums

The main goal of this research is to examine any peculiarities and special modes observed in the dynamics of a system of two nonlinearly coupled pendulums. In addition to steady states, an in-phase rotation limit cycle is proved to exist in the system with both damping and constant external force. This rotation mode is numerically shown to become unstable for certain values of the coupling strength. We also present an asymptotic theory developed for an infinitely small dissipation, which explains why the in-phase rotation limit cycle loses its stability. Boundaries of the instability domain mentioned above are found analytically. As a result of numerical studies, a whole range of the coupling parameter values is found for the case where the system has more than one rotation limit cycle. There exist not only a stable in-phase cycle, but also two out-of phase ones: a stable rotation limit cycle and an unstable one. Bistability of the limit periodic mode is, therefore, established for the system of two nonlinearly coupled pendulums. Bifurcations that lead to the appearance and disappearance of the out-ofphase limit regimes are discussed as well.     

The rolling motion of a truncated ball without slipping and spinning on a plane

This paper is concerned with the dynamics of a top in the form of a truncated ball as it moves without slipping and spinning on a horizontal plane about a vertical. Such a system is described by differential equations with a discontinuous right-hand side. Equations describing the system dynamics are obtained and a reduction to quadratures is performed. A bifurcation analysis of the system is made and all possible types of the top’s motion depending on the system parameters and initial conditions are defined. The system dynamics in absolute space is examined. It is shown that, except for some special cases, the trajectories of motion are bounded.     

Existence of solutions for a mathematical model of structural phase transitions in shape memory alloys

In this paper a thermomechanical model for the dynamics of structural phase transitions in the so-called ‘shape memory alloys’ is developed. These materials exhibit rather spectacular hysteresis phenomena. The resulting mathematical model consists of a coupled and highly non-linear system of partial differential equations reflecting the balance laws of linear momentum and energy. For an appropriate weak formulation the local-in-time existence of weak solutions is shown.     

Chaos in delay-induced Leslie–Gower prey–predator–parasite model and its control through prey harvesting

In this paper we analyze a delay-induced predator–prey–parasite model with prey harvesting, where the predator–prey interaction is represented by Leslie–Gower type model with type II functional response. Infection is assumed to spread horizontally from one infected prey to another susceptible prey following mass action law. Spreading of disease is not instantaneous but mediated by a time lag to take into account the time required for incubation process. Both the susceptible and infected preys are subjected to linear harvesting. The analysis is accomplished in two phases. First we analyze the delay-induced predator–prey–parasite system in absence of harvesting and proved the local & global dynamics of different (six) equilibrium points. It is proved that the delay has no influence on the stability of different equilibrium points except the interior one. Delay may cause instability in an otherwise stable interior equilibrium point of the system and larger delay may even produce chaos if the infection rate is also high. In the second phase, we explored the dynamics of the delay-induced harvested system. It is shown that harvesting of prey population can suppress the abrupt fluctuations in the population densities and can stabilize the system when it exceeds some threshold value.