Chaos control in passive walking dynamics of a compass-gait model

The compass-gait walker is a two-degree-of-freedom biped that can walk passively and steadily down an incline without any actuation. The mathematical model of the walking dynamics is represented by an impulsive hybrid nonlinear model. It is capable of displaying cyclic motions and chaos. In this paper, we propose a new approach to controlling chaos cropped up from the passive dynamic walking of the compass-gait model. The proposed technique is to linearize the nonlinear model around a desired passive hybrid limit cycle. Then, we show that the nonlinear model is transformed to an impulsive hybrid linear model with a controlled jump. Basing on the linearized model, we derive an analytical expression of a constrained controlled Poincaré map. We present a method for the numerical simulation of this constrained map where bifurcation diagrams are plotted. Relying on these diagrams, we show that the linear model is fairly close to the nonlinear one. Using the linearized controlled Poincaré map, we design a state feedback controller in order to stabilize the fixed point of the Poincaré map. We show that this controller is very efficient for the control of chaos for the original nonlinear model.     

An analysis of heart rhythm dynamics using a three-coupled oscillator model

Rhythmic phenomena represent one of the most striking manifestations of the dynamic behavior in biological systems. Understanding the mechanisms responsible for biological rhythms is crucial for the comprehension of the dynamics of life. Natural rhythms could be either regular or irregular over time and space. Each kind of dynamical behavior may be related to both normal and pathological physiological functioning. The cardiac conducting system can be treated as a network of self-excitatory elements and, since these elements exhibit oscillatory behavior, they can be modeled as nonlinear oscillators. This paper proposes a mathematical model to describe heart rhythms considering three modified Van der Pol oscillators connected with time delay couplings. Therefore, the heart dynamics is represented by a system of differential difference equations. Numerical simulations are carried out presenting qualitative agreement with the general heart rhythm behavior. Normal and pathological rhythms represented by the ECG signals are reproduced. Pathological rhythms are generated by either the coupling alterations that represents communications aspects in the heart electric system or forcing excitation representing external pacemaker excitation.     

Chaos control for the plates subjected to subsonic flow

The suppression of chaotic motion in viscoelastic plates driven by external subsonic air flow is studied. Nonlinear oscillation of the plate is modeled by the von-Kármán plate theory. The fluid-solid interaction is taken into account. Galerkin’s approach is employed to transform the partial differential equations of the system into the time domain. The corresponding homoclinic orbits of the unperturbed Hamiltonian system are obtained. In order to study the chaotic behavior of the plate, Melnikov’s integral is analytically applied and the threshold of the excitation amplitude and frequency for the occurrence of chaos is presented. It is found that adding a parametric perturbation to the system in terms of an excitation with the same frequency of the external force can lead to eliminate chaos. Variations of the Lyapunov exponent and bifurcation diagrams are provided to analyze the chaotic and periodic responses. Two perturbation-based control strategies are proposed. In the first scenario, the amplitude of control forces reads a constant value that should be precisely determined. In the second strategy, this amplitude can be proportional to the deflection of the plate. The performance of each controller is investigated and it is found that the second scenario would be more efficient.     

Chaos control in duffing system

Analytical and numerical results concerning the inhibition of chaos in Duffing’s equation with two weak forcing excitations are presented. We theoretically give parameter-space regions by using Melnikov’s function, where chaotic states can be suppressed. The intervals of initial phase difference between the two excitations for which chaotic dynamics can be eliminated are given. Meanwhile, the influence of the phase difference on Lyapunov exponents for different frequencies is investigated. Numerical simulation results show the consistence with the theoretical analysis and the chaotic motions can be controlled to period-motions by adjusting parameter of suppressing excitation.     

A self-consistent field method applied to the dynamics of viscous suspensions

A method for the approximate solution of the problem of many bodies of spherical form in a viscous fluid is developed in the Stokes approximation. Using a purely hydrodynamic approach, based on the use of the concept of a self-consistent field, the classical boundary value problem is reduced to a formal procedure for solving a linear system of algebraic equations in the tensor coefficients, which occur in the solution obtained for the velocity field and pressure of the liquid. A procedure for the approximate solution of this system of equations is constructed for the case of dilute suspensions, when the ratio of the size of the dispersed particles to the characteristic distance between them is a small parameter. Finally, the initial boundary value problem is reduced to solving a recurrent system of equations, in which each subsequent approximation for all the required quantities depends solely on the previous approximations. The system of recurrent equations obtained can be solved analytically in any specified approximation with respect to a small parameter. It is shown that this system of equations contains in itself all possible physical formulations of the problems, and, within the frameworks of the mathematical procedure constructed, they are distinguished solely by a set of specified and required functions. The practical possibilities of the method are in no way limited by the number of dispersed particles in the fluid.     

Symplectic integrators with adaptive time step applied to runaway electron dynamics

Numerical Algorithms - Studying the dynamics of runaway electrons has theoretical and practical significance. As the system is highly relativistic, multi-scale and nonlinear, accurate and efficient...     

Chaos control of Chen chaotic dynamical system

This paper is devoted to study the problem of controlling chaos in Chen chaotic dynamical system. Two different methods of control, feedback and nonfeedback methods are used to suppress chaos to unstable equilibria or unstable periodic orbits (UPO). The Lyapunov direct method and Routh–Hurwitz criteria are used to study the conditions of the asymptotic stability of the steady states of the controlled system. Numerical simulations are presented to show these results.     

Chaos prediction and control in MEMS resonators

The chaotic dynamics of a micro mechanical resonator with electrostatic forces on both sides is investigated. Using the Melnikov function, an analytical criterion for homoclinic chaos in the form of an inequality is written in terms of the system parameters. Detailed numerical studies including phase portrait, Poincare map and bifurcation diagram confirm the analytical prediction and reveal the effect of excitation amplitude on the system transition to chaos. Moreover a robust adaptive fuzzy control algorithm previously proposed by the authors is applied for controlling the chaotic motion. Additional numerical simulations show the effectiveness of the proposed control approach.     

A discrete dynamics approach to sparse calculation and applied in ontology science

ABSTRACT

In the era of big data, with the increase of data processing information and the increase of data complexity, higher requirements are put on the tools and algorithms of data processing. As a tool for structured information representation, ontology has been used in engineering fields such as chemistry, biology, pharmacy, and materials. As a dynamic structure, the increasing concepts contributes to a gradual increase of a single ontology. In order to solve the problem of computational complexity decreasing in the procedure of similarity calculating, the techniques of dimensionality reduction and sparse computing are applied to ontology learning. This article presents discrete dynamics approach showing several tricks on applying the sparse computing method to ontology learning, and verify its efficiency through experiments.     

Forward dynamics applied to a three-dimensional continuum-mechanical model of the upper limb

This paper introduces, for the first time, a methodology to achieve a forward dynamics simulation of the musculoskeletal system using three-dimensional continuum-mechanical skeletal muscle models. This is achieved by coupling one- and three-dimensional skeletal muscle models. The feasibility of this methodology is demonstrated through a forward dynamics simulation of the upper limb involving the biceps and triceps muscle. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On homotopy analysis method applied to linear optimal control problems

In this paper, the homotopy analysis method (HAM) is employed to solve the linear optimal control problems (OCPs), which have a quadratic performance index. The study examines the application of the homotopy analysis method in obtaining the solution of equations that have previously been obtained using the Pontryagin’s maximum principle (PMP). The HAM approach is also applied in obtaining the solution of the matrix Riccati equation. Numerical results are presented for several test examples involving scalar and 2nd-order systems to demonstrate the applicability and efficiency of the method.     

Chaos control in AFM system using sliding mode control by backstepping design

This paper presents a robust algorithm to control the chaotic atomic force microscope system (AFMs) by backstepping design procedure. The proposed feedback controller is composed by a sliding mode control (SMC) and a backstepping feedback, so its implementation is quite simple and can be made on the basis of the measured signal. The developed control scheme allows chaos suppression despite uncertainties in the model as well as system external disturbances. The concept of extended system is used such that a continuous sliding mode control effort is generated using backstepping scheme. It is guaranteed that under the proposed control law, uncertain AFMs can asymptotically track target orbits. The converging speed of error states can be arbitrary turned by assigning the corresponding dynamics of the sliding surfaces. Numerical simulations demonstrate its advantages by stabilizing the unstable periodic orbits of the AFMs and this method can also be easily extended to elimination chaotic motion in any types of chaotic AFMs.     

Assessment of errors in the averaging method as applied to linear viscoelastic dynamics

Errors of the averaging method are assessed as applied to the oscillations of a undimensional, linear viscoelastic oscillator.M. V. Lomonosov State University, Moscow. Translated from Mekhanika Polimerov, No. 5, pp. 943–944, September–October, 1972.     

Rapid-equilibrium Approximation applied to mathematical models of Tracer dynamics in biochemical reaction systems

A method is presented which allows to reduce the size of the differential equation system describing the dynamics of radioactive tracers in biochemical reaction systems endowed with time hierarchy, i.e., encompassing fast and slow reaction rates which may differ by several orders of magnitude. The basic idea of this approach is to apply the rapid-equilibrium approximation commonly used in the mathematical modelling of metabolic systems (cf. [1] for lumping into single “pool variables” all those reactive groups which can be labeled and which are connected by fast reversible reactions. The set of remaining slow reactions governing the dynamics of the reduced system of pool variables depends on the special way of labeling chosen. Thus, the reduction procedure permits to judge the maximal possible number of flux rates which can be estimated from stationary or time-dependent tracer experiments by regression analysis (fitting). Moreover, the method can be employed for identifying near-equilibrium and nonequilibrium reactions. An illustration of the reduction algorithm is provided by considering the distribution of ¹⁴C-tracers in the pentose phosphate pathway.     

Chaotic dynamics applied to signal complexity in phase space and in time domain

In the past few years fractal analysis techniques have gained increasing attention in signal and image processing in Medicine. We concentrate on using fractal techniques for analysis of encephalographic data (EEG). Better understanding of general principles that govern discrete dynamics of these signals can help to reveal ‘the signatures' of different physiological and pathological states. Fractal complexity of the signal in time domain, calculated using Higuchi's algorithm, seems to be the simplest method and may also be used in other biomedical applications.     

On non-linear dynamics and control designs applied to the ideal and non-ideal variants of the Fitzhugh–Nagumo (FN) mathematical model

The Fitzhugh–Nagumo (FH) mathematical model is considered a simplification of the Hodgkin–Huxley (HH) model. This paper analyzes the non-linear dynamics of the Fitzhugh–Nagumo (FN) mathematical model, and still presents some modifications in the governing equations of the system in order to transform it into a non-ideal one (taking into account that an energy source has limited power supply). We also developed an optimal linear control design and used Sinhas’s theory for the membrane’s action potential in order to stabilize the variation of this potential.     

Chaos synchronization in RCL-shunted Josephson junction via active control

This paper investigates the synchronization of coupled RCL-shunted Josephson junction that is of interest in high-frequency applications. A nonlinear controller is developed in order to achieve the desired behavior. The synchronization is obtained using the slave–master technique and the controller ensures that the states of the controlled chaotic slave system exponentially synchronize with the state of the master system. Numerical simulations are illustrate and verify the proposed method.     

Chaos control by harmonic excitation with proper random phase

Chaos control may have a dual function: to suppress chaos or to generate it. We are interested in a kind of chaos control by exerting a weak harmonic excitation with random phase. The dual function of chaos control in a nonlinear dynamic system, whether a suppressing one or a generating one, can be realized by properly adjusting the level of random phase and determined by the sign of the top Lyapunov exponent of the system response. Two illustrative examples, a Duffing oscillator subject to a harmonic parametric control and a driven Murali-Lakshmanan-Chua (MLC) circuit imposed with a weak harmonic control, are presented here to show that the random phase plays a decisive role for control function. The method for computing the top Lyapunov exponent is based on Khasminskii's formulation for linearized systems. Then, the obtained results are further verified by the Poincare map analysis on dynamical behavior of the system, such as stability, bifurcation and chaos. Both two methods lead to fully consistent results.     

Chaos control and synchronization of a modified chaotic system

By replacing a quadratic nonlinear term in Lü system with a piecewise linear signum (PWL) function, a new simplified three-dimensional piecewise continuous autonomous system (a modified Lü system) is introduced. The qualitative properties of the modified Lü system are studied. Based on these properties, the feedback control law is applied to suppress chaos to one of the three equilibria. Several different synchronized methods, such as the active control, one way coupling by active control, and the adaptive active control are applied to achieve the state synchronization of two identical modified Lü systems. These results show that after the simplification, the modified Lü system can still keep the basic and typical nonlinear phenomena. Compared with the original Lü system, the modified Lü system has a lot of advantages, by which the modified Lü system can be more easily implemented by theoretical analysis, and more practicable made by secret communications.