Lyapunov Indices Research Articles

Investigation on the Impact of Excitation Amplitude on AFM-TM Microcantilever Beam System's Dynamic Characteristics and Implementation of an Equivalent Circuit.

Alterations in the dynamical properties of an atomic force microscope microcantilever beam system in tapping mode can appreciably impact its measurement precision. Understanding the influence mechanism of dynamic parameter changes on the system's motion characteristics is vital to improve the accuracy of the atomic force microscope in tapping mode (AFM-TM). In this study, we categorize the mathematical model of the AFM-TM microcantilever beam system into systems 1 and 2 based on actual working conditions. Then, we analyze the alterations in the dynamic properties of both systems due to external excitation variations using bifurcation diagrams, phase trajectories, Lyapunov indices, and attraction domains. The numerical simulation results show that when the dimensionless external excitation g < 0.183, the motion state of system 2 is period 1. When g < 0.9, the motion state of system 1 is period 1 motion. Finally, we develop the equivalent circuit model of the AFM-TM microcantilever beam and perform related software simulations, along with practical circuit experiments. Our experimental results indicate that the constructed equivalent circuit can effectively analyze the dynamic characteristics of the AFM-TM microcantilever beam system in the presence of complex external environmental factors. It is observed that the practical circuit simulation attenuates high-frequency signals, resulting in a 31.4% reduction in excitation amplitude compared to numerical simulation results. This provides an essential theoretical foundation for selecting external excitation parameters for AFM-TM cantilever beams and offers a novel method for analyzing the dynamics of micro- and nanomechanical systems, as well as other nonlinear systems.

Poincaré Maps and Aperiodic Oscillations in Leukemic Cell Proliferation Reveal Chaotic Dynamics.

Biological systems are dynamic systems featuring two very common characteristics; Initial conditions and progression over time. Conceptualizing this on tumour models it can lead to important conclusions about disease progression, as well as the disease’s “starting point”. In the present study we tried to answer two questions: (a) which are the evolving properties of proliferating tumour cells that started from different initial conditions and (b) we have attempted to prove that cell proliferation follows chaotic orbits and it can be described by the use of Poincaré maps. As a model we have used the acute lymphoblastic leukemia cell line CCRF-CEM. Measurements of cell population were taken at certain time points every 24 h or 48 h. In addition to the population measurements flow cytometry studies have been conducted in order to examine the apoptotic and necrotic rate of the system and also the DNA content of the cells as they progress through. The cells exhibited a proliferation rate of nonlinear nature with aperiodic oscillatory behavior. In addition to that, the (positive) Lyapunov indices and the Poincaré representations in phase-space that we performed confirmed the presence of chaotic orbits. Several studies have dealt with the complex dynamic behaviour of animal populations, but few with cellular systems. This type of approach could prove useful towards the understanding of leukemia dynamics, with particular interest in the understanding of leukemia onset and progression.

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ НЕЛИНЕЙНЫХ КОЛЕБАНИЙ ПЛАСТИНКИ ПРИ ВОЗДЕЙСТВИИ ЦВЕТНОГО ШУМА С УЧЕТОМ КОНТАКТНОГО ВЗАИМОДЕЙСТВИЯ С БАЛКОЙ

A theory of contact interaction of a plate locally supported by a beam, under the influence of external lateral load and external additive color noise (pink, red, white) was constructed. Also described design is in a stationary temperature field. For the plate, the Kirchhoff kinematic model was adopted; for the beam, Euler - Bernoulli, the physical nonlinearity is taken into account according to the theory of small elastic-plastic deformations. The temperature field is taken into account according to the Duhamel - Neumann theory, and there are no restrictions on the temperature distribution over the plate thickness and the height of the beam. The temperature field is determined from the solution of the three-dimensional (plate) and two-dimensional (beam) heat conduction equations. The theory of B.Ya. Cantor. The heat conduction equations are solved by the finite difference method of the second and fourth order of accuracy. The system of differential equations is reduced to the Cauchy problem by the Bubnov - Galerkin methods in higher approximations and finite differences in spatial variables. Next, the Cauchy problem is solved by the fourth-order Runge - Kutta method and the Newmark method. At each time step, the iterative procedure of I. Birger was applied. The results of a numerical experiment are given. To analyze the results, the methods of nonlinear dynamics were used (construction of signals, phase portraits, Poincare sections, Fourier power spectra and Morlet wavelet spectra, analysis of the sign of Lyapunov indices by three methods: Wolf, Kantz, Rosenstein). The effect of color noise on the contact interaction between the plate and the beam has been studied. It has been established that red additive noise has the most significant effect on the oscillation pattern of the lamellar-beam structure compared to pink and white noise.

Phase Dynamics of Metal Fragmentation during Megaplastic (Severe) Deformation

The fragmentation of metals during megaplastic (severe) deformation is described in terms of a two-defect model. The process kinetics has been studied qualitatively. A general expression for the Lyapunov indices, which determine the stability of limiting (stationary) structures, is derived. Critical conditions for controlling parameters are found, and the diagrams that determine the stability of stationary states are constructed.

ПАРАМЕТРИЧЕСКИЕ КОЛЕБАНИЯ И КОНТАКТНОЕ ВЗАИМОДЕЙСТВИЕ В КОМПОНЕНТАХ МИКРОМЕХАНИЧЕСКИХ ДАТЧИКОВ ИНЕРЦИАЛЬНОЙ ИНФОРМАЦИИ

The paper considers constructing a mathematical model of components of inertial information transducers, accounting for parametric oscillations and their contact interaction. A mathematical model of components of inertial information transducers is presented in the form of a multilayered system, consisting of a plate stiffened with a set of spaced beams. Contact interaction of the beamplate structure is analyzed using the methods of nonlinear dynamics: signals, phase portraits and Fourier-specters were constructed, wavelet-transforms and the analysis of the signs of the greatest of Lyapunov indices were used. The effects of the number of stiffening ribs (one, two and three beams) on the character of oscillations of the elements is studied. Zones of synchronization and intermittence of frequencies are found. The adequacy of the solution is corroborated by comparing the results obtained using various numerical methods: finite difference method and Bubnov−Galerkin method with higher approximations along the spatial coordinate, Runge − Kutta-type time methods. Keywords: micromechanical sensors of inertial information, contact interaction of plates and beams, parametric oscillations, Bubnov − Galerkin method, Runge − Kutta method, wavelet analysis.

Self-Organization and Fractality in the Metabolic Process of Glycolysis

Within a mathematical model, the metabolic process of glycolysis is studied. The general scheme of glycolysis is considered as a natural result of the biochemical evolution. By using the theory of dissipative structures, the conditions of self-organization of the given process are sought. The autocatalytic processes resulting in the conservation of cyclicity in the dynamics of the process are determined. The conditions of breaking of the synchronization of the process, increase in the multiplicity of a cyclicity, and appearance of chaotic modes are studied. The phase-parametric diagrams of a cascade of bifurcations, which characterize the transition to chaotic modes according to the Feigenbaum scenario and the intermittence, are constructed. The strange attractors formed as a result of the funnel effect are found. The complete spectra of Lyapunov indices and divergences for the obtained modes are calculated. The values of KS-entropy, horizons of predictability, and Lyapunov dimensions of strange attractors are determined. Some conclusions concerning the structural-functional connections in glycolysis and their influence on the stability of the metabolic process in a cell are presented.

Lyapunov Indices and the Poincare Mapping in a Study of the Stability of the Krebs Cycle

On the basis of a mathematical model, we continue the study of the metabolic Krebs cycle (or the tricarboxylic acid cycle). For the rst time, we consider its consistency and stability, which depend on the dissipation of a transmembrane potential formed by the respiratory chain in the plasmatic membrane of a cell. The phase-parametric characteristic of the dynamics of the ATP level depending on a given parameter is constructed. The scenario of formation of multiple autoperiodic and chaotic modes is presented. Poincare sections and mappings are constructed. The stability of modes and the fractality of the obtained bifurcations are studied. The full spectra of Lyapunov indices, divergences, KS-entropies, horizons of predictability, and Lyapunov dimensionalities of strange attractors are calculated. Some conclusions about the structural-functional connections determining the dependence of the cell respiration cyclicity on the synchronization of the functioning of the tricarboxylic acid cycle and the electron transport chain are presented.

Self-organization and chaos in the metabolism of a cell

Aim. To study the dynamics of auto-oscillations arising at the level of enzyme-substrate interaction in a cell and to find the conditions for the self-organization and the formation of chaos in the metabolic process. Methods. A mathematical model of the metabolic process of steroids transformation in Arthrobacter globiformis. The mathematical apparatus of nonlinear dynamics. Results. The bifurcations resulting in the appearance of strange attractors in the metabolic process are determined. The projections of the phase portraits of attractors are constructed for some chosen modes. The total spectra of Lyapunov's indices are calculated. The structural stability of the attractors obtained is studied. By the general scenario of formation of regular and strange attractors, the structural-functional connections in the metabolic process in the cell are found. Their physical nature is investigated. Conclusions. The presented model explains the mechanism of formation of auto-oscillations observed in the A. globiformis cells and demonstrates a possibility of the mathematical modeling of metabolic processes for the physical explanation of the self-organization of a cell and its vital activity.

The Structure of a Chaos of Strange Attractors within a Mathematical Model of the Metabolism of a Cell

This work continues the study of the earlier constructed mathematical model of the metabolic process running in a cell. We will consider autooscillations arising on the level of enzyme-substrate interactions in the nutrient and respiratory chains, which leads to the self-organization in autocatalysis of the integral metabolic process in cells. The autooscillations organize themselves in the total metabolic process of cells at autocatalysis. The behavior of the phase-parametric characteristic under a high dissipation of the kinetic membrane potential is analyzed. All possible oscillatory modes of the system and the scenario of formation and destruction of regular and strange attractors are studied. The bifurcations of the transitions “order-chaos”, “chaos-order”, “chaos-chaos” and “order-order” are calculated. The total spectra of Lyapunov indices and the divergences for all types of attractors on a part of the phase-parametric characteristic under consideration are determined. For various types of strange attractors, their Lyapunov dimensions, Kolmogorov–Sinai-entropies (KS-entropies), and “predictability horizons” are calculated. Some conclusions about the structure of the chaos of strange attractors and its influence on the stability of the metabolic process in a cell are drawn.

A Dynamics Optimization of Scheduling for Excavating Development Project in Complicated Ore Mining below Railway Special Line

Disturbance of ground structures by the excavation of underground engineering is a non-linear mechanical behavior process. To find time sequence with the smallest disturbance is important for the safety of mining below railway, river, and building (three-underground mining). A numerical simulation was carried out by 3DEC in mining area. The maximum Lyapunov indices are obtained from the time curves of the maximum unbalanced force. The reasonable time sequence schemes of development project construction under railway in the east ore body of XinQiao Pyrite Mine are compared. The results show that the effect of the data analysis method about the information of unbalanced force-time record based on the chaotic dynamics theory is very positive. It can minimize the risk of scheme with the priority of single mechanical (displacement) index, direct the actual construction effectively and guarantee the safety of ‘three-underground’ complex ore mining by preventive actions in supporting, shortening construction time and other strong measures.

Study on the chaotic behavior of mining rock seepage system

One dimensional non-steady, non-Darcy flow of water in a rock stratum was reduced into a system described by six ordinary differential equations involving five controlling parameters. Through response computations and time series analysis, chaotic behavior in the reduced system was discussed in details. Firstly, the dynamical response of the reduced system under a set of parameters was calculated, and the power spectrum of the attractor was obtained through fast Lagrangian transformation; then the phase space was reconstructed by fixing embedding dimension to be 6 and delay time to range from 1 to 20, and the correlation dimension of the attractor was calculated based on the curves under the coordinates of logarithm of correlation integral vs. logarithm of covering radius; and lastly, the Lyapunov indices of the attractor were calculated by using Gram-Schmit’s orthogonalization method. The results show that the power spectrum of the attractor is continuous; the correlation dimension of the attractor is equal to 2.36; among the Lyapunov indices, LE1, LE2, LE3 are positive, LE5, LE6 are negative, and LE4 fluctuates near zero. All the analysis indicates that there may exist chaos in the system of non-steady, non-Darcy flow.

Stabilizability of the linear algebro-differential one-input control systems

Consideration was given to the controllable system of ordinary linear differential equations with the matrix at the derivative of the desired vector function that is identically degenerate in the domain of definition. For the one-input systems, the questions of stabilizability and solvability of the control problem by the Lyapunov indices were studied for the stationary and nonstationary cases. Analysis was based on the assumptions providing existence of the so-called "equivalent form" where the "algebraic" and "differential" parts are separated. An arbitrarily high index of insolvability and variable rank of the matrix at the derivative were admitted.

Lyapunov indices with two nearby trajectories in a curved spacetime

We compare three methods for computing invariant Lyapunov exponents (LEs) in general relativity. They involve the geodesic deviation vector technique (M1), the two-nearby-orbits method with projection operations and with coordinate time as an independent variable (M2), and the two-nearby-orbits method without projection operations and with proper time as an independent variable (M3). An analysis indicates that M1 and M3 do not need any projection operation. In general, the values of LEs from the three methods are almost the same. As an advantage, M3 is simpler to use than M2. In addition, we propose to construct the invariant fast Lyapunov indictor (FLI) with two-nearby-trajectories and give its algorithm in order to quickly distinguish chaos from order. Taking a static axisymmetric spacetime as a physical model, we apply the invariant FLIs to explore the global dynamics of phase space of the system where regions of chaos and order are clearlyidentified.

Application of a Graphical Test for Analyzing Dynamic Systems with a Joker

A method for analyzing dynamic systems with a joker is proposed which makes it possible to establish the presence of a joker and determine its type from a series of observations of a single system variable. The possibility of obtaining numerical characteristics such as an estimate of the fractal dimensionality, Lyapunov indices, etc., in systems with a joker is examined. The correlation dimensionality is estimated by calculation for Henon mappings with an interval joker.

Parametric instability and Hamiltonian chaos in cavity semiclassical electrodynamics

We study the nonlinear dynamics of the interaction of two-level atoms and a selected mode of a high-Q cavity with frequency modulation analytically and numerically. In the absence of modulation, the corresponding semiclassical Heisenberg equations for the expectation values of the collective atomic observables and the field-mode amplitudes allow, in the rotating wave approximation and in the strong-coupling limit, an exact solution with arbitrary detuning. Using this solution, we detect the coherent effect of trapping of the population of atomic levels and of trapping of the number of photons in the cavity. The explanation for this effect lies in the destructive interference of the atomic dipoles and the field mode. The integrable version of the system of equations exhibits a separatrix near which a stochastic layer is formed when modulation is introduced. The width of the layer is found to gradually increase with degree of modulation, and finally it fills the entire energy-permissible volume of the phase space. We show that the rotating wave approximation does not hinder the formation of Hamiltonian chaos in cavity semiclassical electrodynamics. The calculation of the maximum Lyapunov indices of nonlinear (in this approximation) equations of motion as functions of the modulation frequency δ and the frequency of natural Rabi oscillations of the atom-field system, Ω, suggests that Hamiltonian chaos appears first in the area of the fundamental parametric resonance, δ/2Ω≃1. Parametric instability increases with increasing modulation and decreasing detuning from the atom-field resonance, generating at exact resonance new areas of chaos corresponding to multiple parametric resonances. The results of numerical experiments and estimates of the characteristic parameters show that Rydberg atoms placed in a high-Q microwave cavity are possible objects for observing parametric instability and dynamical chaos.

Phase dynamics of a relativistic electron driven by the potential field of a wave packet: Poincaré transformation dynamics

We investigate the Poincare transformation dynamics describing the stochastic motion of a relativistic electron in the regular electrostatic field of a wave packet. The conditions of local instability of the phase trajectories are determined: it is shown that the system is a structurally stable Y system with an attracting set of the hyperbolic type. The relationship between the dynamics at the strange attractor characterized by Lyapunov indices and its statistical structure described by the Kolmogorov entropy is discussed. The fractal dimension of the phase space of the system is determined. The results of a numerical simulation of the effect are presented and discussed.

Effect of memory and dynamical chaos in long Josephson junctions

A long Josephson junction in a constant external magnetic field and in the presence of a dc bias current is investigated. It is shown that the system, simulated by the sine-Gordon equation, ``remembers'' a rapidly damping initial perturbation and final asymptotic states are determined exactly with this perturbation. Numerical solving of the boundary sine-Gordon problem and calculations of Lyapunov indices show that this system has a memory even when it is in a state of dynamical chaos, i.e, dynamical chaos does not destroy initial information having a character of rapidly damping perturbation.

Scaling and dissipation in the GOY shell model

This is a paper about multifractal scaling and dissipation in a shell model of turbulence, called the Gledzer–Ohkitani–Yamada (GOY) model. This set of equations describes a one-dimensional cascade of energy towards higher wave vectors. When the model is chaotic, the high-wave-vector velocity is a product of roughly independent multipliers, one for each logarithmic momentum shell. The appropriate tool for studying the multifractal properties of this model is shown to be the energy flux on each shell rather than the velocity on each shell. Using this quantity, one can obtain better measurements of the deviations from Kolmogorov scaling (in the GOY dynamics) than were available up to now. These deviations are seen to depend upon the details of inertial-range structure of the model and hence are not universal. However, once the conserved quantities of the model are fixed to have the same scaling structure as energy and helicity, these deviations seem to depend only weakly upon the scale parameter of the model. The connection between multifractality in the velocity distribution and multifractality in the dissipation is analyzed. Arguments suggest that the connection is universal for models of this character, but the model has a different behavior from that of real turbulence. Also, the scaling behavior of time correlations of shell velocities, of the dissipation, and of Lyapunov indices are predicted. These scaling arguments can be carried over, with little change, to multifractal models of real turbulence.

Dynamic chaos in a laser with a bleachable filter and dimensionality

A theoretical study of the dynamics of a laser with a bleachable filter revealed chaotic lasing regimes and ranges of bistable states of parameters close to those found in reality. It is shown how a transition to chaos occurs as a result of period-doubling bifurcation. A study is reported of the degree of chaos and of the structure of the resultant strange attractor by calculation of its fractal dimensionality and of the Lyapunov indices.

Baire class of the Lyapunov indices

Then their properties were studied as functions on B. In particular, Theorem i, which shows that the functions ~(-),k = i ..... n belong to the second Baire class (see [4, 5]), was proved. Since, by definition, each Baire class is contained in the next one, there naturally arises the problem of unimprovability of this theorem, i.e., about the strict belongingness of the Lyapunov indices to the second Baire class (see [5]). For abstractly defined morphism (I), this problem cannot have a unique solution. Thus, e.g., if the base consists of a single point, then each function on it is of zero class. We can give examples of the morphisms (i) with discontinuous Lyapunov indices. However, the present problem will be solved in this case if we indicate objects for which the functions ~(.), k= I,..., n belong to the second, but do not belong to any preceeding class.