Journal articles: 'Stochastics dynamics'

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IMKELLER, PETER, and ADAM HUGH MONAHAN. "CONCEPTUAL STOCHASTIC CLIMATE MODELS." Stochastics and Dynamics 02, no. 03 (September 2002): 311–26. http://dx.doi.org/10.1142/s0219493702000443.

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From July 9 to 11, 2001, 50 researchers from the fields of climate dynamics and stochastic analysis met in Chorin, Germany, to discuss the idea of stochastic models of climate. The present issue of Stochastics and Dynamics collects several papers from this meeting. In this introduction to the volume, the idea of simple conceptual stochastic climate models is introduced amd recent results in the mathematically rigorous development and analysis of such models are reviewed. As well, a brief overview of the application of ideas from stochastic dynamics to simple models of the climate system is given.

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Yoda, M., and M. Sasai. "2P308 Stochastics dynamics of coupled repressilators." Seibutsu Butsuri 45, supplement (2005): S196. http://dx.doi.org/10.2142/biophys.45.s196_4.

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de Levie, Robert. "Stochastics, the Basis of Chemical Dynamics." Journal of Chemical Education 77, no. 6 (June 2000): 771. http://dx.doi.org/10.1021/ed077p771.

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Lin, Winston T., Hong-Jen Lin, and Yueh H. Chen. "The Dynamics and Stochastics of Currency Betas Based on the Unbiasedness Hypothesis in Foreign Exchange Markets." Multinational Finance Journal 6, no. 3/4 (December 1, 2002): 167–95. http://dx.doi.org/10.17578/6-3/4-2.

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Ovchinnikov, Igor V., Wenyuan Li, Yuquan Sun, Andrew E. Hudson, Karlheinz Meier, Robert N. Schwartz, and Kang L. Wang. "Criticality or Supersymmetry Breaking?" Symmetry 12, no. 5 (May 12, 2020): 805. http://dx.doi.org/10.3390/sym12050805.

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In many stochastic dynamical systems, ordinary chaotic behavior is preceded by a full-dimensional phase that exhibits 1/f-type power spectra and/or scale-free statistics of (anti)instantons such as neuroavalanches, earthquakes, etc. In contrast with the phenomenological concept of self-organized criticality, the recently found approximation-free supersymmetric theory of stochastics (STS) identifies this phase as the noise-induced chaos (N-phase), i.e., the phase where the topological supersymmetry pertaining to all stochastic dynamical systems is broken spontaneously by the condensation of the noise-induced (anti)instantons. Here, we support this picture in the context of neurodynamics. We study a 1D chain of neuron-like elements and find that the dynamics in the N-phase is indeed featured by positive stochastic Lyapunov exponents and dominated by (anti)instantonic processes of (creation) annihilation of kinks and antikinks, which can be viewed as predecessors of boundaries of neuroavalanches. We also construct the phase diagram of emulated stochastic neurodynamics on Spikey neuromorphic hardware and demonstrate that the width of the N-phase vanishes in the deterministic limit in accordance with STS. As a first result of the application of STS to neurodynamics comes the conclusion that a conscious brain can reside only in the N-phase.

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Wissel, Christian. "Metastability, a consequence of stochastics in multiple stable population dynamics." Theoretical Population Biology 36, no. 3 (December 1989): 296–310. http://dx.doi.org/10.1016/0040-5809(89)90036-1.

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Синенко, D. Sinenko, Гараева, G. Garaeva, Еськов, Valeriy Eskov, Ворошилова, and A. Voroshilova. "Stochastic and chaos in evaluation order parameter in regenerative medicine." Complexity. Mind. Postnonclassic 3, no. 4 (July 10, 2014): 87–100. http://dx.doi.org/10.12737/7655.

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The basis of the third global paradigm of theory of chaos and self-organization, which focuses on the assessment of the chaotic dynamics of the state vector of complex biological systems using multi-dimensional phase space of states. The paper presents a comparative description of the effectiveness of the traditional stochastic methods and methods of calculating the parameters of quasi-attractors. It is showed the difference in efficiency (low) of stochastics, which leads to the uncertainty of the 1st kind, and methods of multidimensional phase spaces, providing the solution of system synthesis. Volumes quasi-attractors with kinesotherapy in patients with acute stroke increased 5.3 times in the initial stage of treatment, and then falling off sharply. It is discussed the need for parallel applications and stochastic methods and methods of theory of chaos and self-organization in the study of complex medical and biological systems.

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van Mourik, Anke M., Andreas Daffertshofer, and Peter J. Beek. "Extracting Global and Local Dynamics From the Stochastics of Rhythmic Forearm Movements." Journal of Motor Behavior 40, no. 3 (May 2008): 214–31. http://dx.doi.org/10.3200/jmbr.40.3.214-231.

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Горбунов, D. Gorbunov, Эльман, Kseniya Elman, Гавриленко, T. Gavrilenko, Григоренко, and V. Grigorenko. "Features stochastics and chaos theory processing myogram." Complexity. Mind. Postnonclassic 4, no. 1 (August 23, 2015): 45–53. http://dx.doi.org/10.12737/10864.

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In studies using the method of multi-dimensional phase space. In the study and modeling of complex biological objects (complexity) there is the possibility of introducing traditional physical methods in biological research and new methods based on chaos theory, self-organization. The paper shows the feasibility of applying the method of multi-dimensional phase space as a quantitative measure for the evaluation of chaotic dynamics on the example of the muscles (flexor of the little finger). As a measure of the state of the neuromuscular system of the person (weak muscle tension and strong, almost the maximum force) used quasi-attractors volumes of multidimensional phase space. This enables identification of the actual measurements of the parameters of the functional state with weak muscles (p = 5th Dan) and strong (P = 10 daN) static stress. Was built timebase signal obtained with myograph and were built autocorrelation function A (t) signal. In the end analysis of the biomechanical system based on a comparison of volume quasi-attractor, as well as on the basis of analysis of the Shannon entropy N. kzvaziattraktora volume displacement at low load is slightly less than the same amount of displacement under a heavy load of flexor muscles of the little finger, just as the values of the Shannon entropy at a heavy load is increased as compared with the values obtained by the weak muscle load.

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Еськов, V. Eskov, Джумагалиева, L. Dzhumagalieva, Еськов, Valeriy Eskov, Гудкова, and S. Gudkova. "Medicine and the Chaos Theory in Description of Individual and Particular." Journal of New Medical Technologies 21, no. 3 (September 5, 2014): 27–35. http://dx.doi.org/10.12737/5892.

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The article presents three approaches (deterministic, stochastic and chaotic – self-organizing) for studying biomedical systems. The authors show that complex biosystems cann’t be described by deterministic and stochastics because of constant changing parameters xi of a state vector of such systems x=x(t). The fundamental distinguish of deterministic and stochastic systems from chaotic – self-organizing is continuous movement x(t) in phase space of states. The authors also present complex of objects which the authors have been studying for the last 30 years and which conform the type III systems. The particular features of the personalized medicine are presented, that denies possibility of identification of body state at one measurement (a point in a phase space). It is connected with the fact that there is a uniform distribution x(t) in time-domain xi which is revealed in continuous change of distribution functions f(x) for different discrete recording time-domain x(t) at all xi. The authors assert that behavior dynamics of neural networks is similar to work of neuroemulators that is terminated by certainty in quasi-attractor’s volumes.

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Берестин, D. Berestin, Попов, Yuriy Popov, Вохмина, Yu Vokhmina, Хадарцева, and K. Khadartseva. "Possibilities of stochastic processing of system parameters with chaotic dynamics." Complexity. Mind. Postnonclassic 3, no. 2 (May 21, 2014): 59–67. http://dx.doi.org/10.12737/5519.

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The paper presents the first attempt to combine methods of stochastics (mathematical statistics) and methods of theory of chaos and self-organization for studying such complex (chaotic) processes as postural tremor. It was established that when re-registering tremor in each subject by n=15 or n=30 obtained tremorograms do not exhibit normal distribution, and non-parametric distributions show distinctions at pairwise comparison on Wilcoxon test (only 2 or 3 pairs from 210 may belong to the same general population). Static physical load sharply changes this picture and the number of such ("similar") pairs increases. The estimation method for effect of a load on tremor is proposed. Simultaneously, within calculating quasi-attractors there is a clear picture of division of chaotic dynamics of tremor parameters with load and without load. Prospects of a new method application in physiological measurements are discussed. Limited method of stochas- tics in description of complexity is underlined, and necessity of calculation quasi-attractor´s parameters in phase space of state is proved.

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Lerner, Vladimir S. "Macrodynamic Cooperative Complexity of Information Dynamics." Open Systems & Information Dynamics 15, no. 03 (September 2008): 231–79. http://dx.doi.org/10.1142/s1230161208000195.

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The introduced concept and measure of macrocomplexity (MC) arise in an irreversible macrodynamic cooperative process and determine the process components ability to assemble into an integrated system. MC serves as a common indicator of the origin of the cooperative complexity measured by the specific entropy speed per assembled volume, rather than the entropy, as it has been accepted before. The MC cooperative mechanism is studied using the variation principle (VP) of information macrodynamics (IMD), applied to a complex system with the macrolevel dynamics and random microlevel stochastics and different forms of physical and virtual transitions of information. This principle describes a transition from a local unstable process movement to a local stable process — the former is associated with the current influx of information that precedes cooperation, and the latter is continued after the cooperation with incoming information and its accumulation. This transition enables the production of the cooperative phenomena and, in particular, the contributions from different superimposing processes, which are measured by the MC. MC, arising as an indicator of these phenomena at the unification (or decomposition) of the system processes, is defined by the invariant information measure, allowing for both analytical formulation and computer evaluation. An optimal multi-dimensional consolidation process, satisfying the VP, forms the information hierarchical network (IN) consisting of the model eigenvalues sequential cooperation in triples. The MC of such an optimal cooperative triplet structure is measured by the IN triplet code (as an algorithm of the minimal program, which evaluates the IN hierarchical structure by the triplet information contributions in bits of information). The distributed IN allows for the automatic arrangement and measurement of the MC-local complexities in a multi-dimensional cooperative process, taking into account the MC time-space locations and mutual dependencies, and providing the MC hierarchical invariant information measure by quantity and quality in the triplet code. The MC is connected to Kolmogorov (K) complexity, which measures a deterministic order over a stochastic disorder by a minimal program. The MC specific consists of providing a computable complexity measure of a cooperative dynamic irreversible process, as an opposite to the K incomputability.

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LERNER, VLADIMIR S. "MACRODYNAMIC COOPERATIVE COMPLEXITY OF BIOSYSTEMS." Journal of Biological Systems 14, no. 01 (March 2006): 131–68. http://dx.doi.org/10.1142/s0218339006001714.

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The introduced dynamic macrocomplexity (MC) is an indicator of the phenomena and parameters of the cooperative dynamic process in a complex system with the macrolevel's dynamics and random microlevel's stochastics. The MC, arising from the unification (or decomposition) of the system's elements, is defined by the information measure, allowing for both analytical formulation and computer evaluation. The MC cooperative mechanism is analyzed using the minimax variation principle of Information Macrodynamics (IMD). When applied, this principle creates a transition from a local, unstable process' movement to a local, stable process — the former associated with the current influx of information that precedes cooperation and the latter continuing after the cooperation with incoming information and its accumulation. This transition enables producing the cooperative phenomena, in particular, the contributions from different superimposing processes, measured by the MC. The IMD optimal consolidation process forms the information hierarchical network (IN) consisting of the model eigenvalues' sequential cooperation by three. The MC of such optimal cooperative triplets' structure is measured by the IN's triplet's code as an algorithm of the minimal program. The MC covers the Kolmogorov's complexity, which measures a deterministic order over a stochastic disorder by a minimal program, as well as the statistical complexity. The MC specific consists of providing a precise complexity measure of a dynamic irreversible process, which evaluates the aforementioned forms of complexities in bits of information. The geometrical space curvature conceals information of the cooperative structures in the cells' form, whose code, in particular, measures the cooperative complexity. The process' trajectory, located in this geometrical space, acquires a sequence of the code cells.

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Azimova, Natalia, Svetlana Kholodova, Maria Bedoidze, Jakhangul Zairova, and Alexander Ermakov. "Statistical assessment of biogenic risk for the human population caused by COVID-19." E3S Web of Conferences 371 (2023): 05073. http://dx.doi.org/10.1051/e3sconf/202337105073.

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In the paper an appreciable hierarchy of mathematical models is proposed. It describes the actual dynamics of COVID-19 thickness during 02/12/2020 – 09/22/2021. The incidence sub-model reflects reliably regular (aperiodic and periodic), as well as random components. It is established that the dynamics of the epidemy is essentially seasonal thrice a year. Model elaborated enables to clarify and explain weak weekly fluctuations in the death rate dynamics. It turned out that the maximum risk of death is at 15 and 22 days of disease duration. It means that this virus will presumably be a "satellite" of the human population with corresponded mortality at 1.75%. Calculations performed enable to estimate the level of stochastics in disease and death dynamics. It is near to the amplitude of periodic variation. Computer experiments with developed model predict the global dynamics of the incidence of COVID-19. New epidemic data can show the prospect of improving our model to regard the competitiveness between new sporadically emerging virus strains.

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15

Горбунов, D. Gorbunov, Эльман, Kseniya Elman, Гараева, G. Garaeva, Еськов, Valeriy Eskov, Третьяков, and S. Tretyakov. "Physiotherapy for hypertensive disease from the perspectives of chaotic dynamics of cardiovascular system parameters in patients." Journal of New Medical Technologies. eJournal 8, no. 1 (November 5, 2014): 0. http://dx.doi.org/10.12737/7242.

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The problem of one-type uncertainty is solved when cardiovascular system parameters in hypertensive patients undergoing physiotherapy aren’t differentiated by stochastics, but these differences are clearly revealed by the methods of neurocomputing and calculation of parameters of quasi-attractors. Simultaneously, the solution of system synthesis problem is possible, i.e. identification of more important diagnostic characters xi from the whole set of state vector of cardiovascular system x(t) in hypertensive patients. Efficiency of physiotherapy is estimated at two stages of course of treatment: in the initial state (initial physiotherapy) and after the termination of course of treatment. The dynamics of motion of quasi-attractors in phase space of states in hypertensive patients is shown. Thus, the problem of elimination of one-type uncertainty in studying the effi-ciency of curative measures is solved.

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Нерсисян, N. Nersisyan, Шакирова, L. Shakirova, Нифонтова, O. Nifontova, Рассадина, and Yu Rassadina. "Dynamics of spectral power of heart rate variability in students the latitudinal movement." Journal of New Medical Technologies. eJournal 10, no. 1 (May 19, 2016): 0. http://dx.doi.org/10.12737/18602.

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In the conditions of sanatorium treatment the parameters of the cardiovascular system of schoolchildren with the latitudinal displacements were analyzed. Analysis of parameters of cardiovascular system of children in sanatoria from the position of the stochastics showed that the behavior of RR-intervals is still chaotic. The results of the study revealed that short-term treatment reduces the size of quasi-attractor vector of conditions of the human body and partially normalizes the indicators of the cardio-respiratory system of children. However, after the rest of the distance rX increasing, says lack of formation of mechanisms of adaptation of students, as well as significant tension of regulatory processes. The use of the method of calculation of matrices mega-factory of distances in m-dimensional phase space provides some quantitative evaluation of adaptive reserves of the body. This allows us to objectively assess the dynamics of reserve possibilities of organism and their prognostic significance.

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JUMARIE, GUY. "STOCHASTICS OF ORDER n IN BIOLOGICAL SYSTEMS: APPLICATIONS TO POPULATION DYNAMICS, THERMODYNAMICS, NONEQUILIBRIUM PHASE AND COMPLEXITY." Journal of Biological Systems 11, no. 02 (June 2003): 113–37. http://dx.doi.org/10.1142/s021833900300083x.

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In the present paper, a modeling in the complex space is combined with complex-valued fractional Brownian motion to get some new results in biological systems. The rational of this approach is as follows. Biological dynamics which evolve continuously in time but are not time differentiable, necessarily exhibit random properties. These random features appear also as a result of the randomness of the proper time of biological systems. Usually, this is taken into account by using white noises that is to say fractals of order two. Fractals of order n larger than two are more suitable for increments with large amplitudes, and they may be introduced by using either real-valued fractal noises with long range memory or Brownian motions with independent increments, which are necessarily complex-valued. In the later case, we are then led to describe biological systems in the complex plane. After some background on the complex-valued fractional Brownian motion, we shall deal successively with population growth, information thermodynamics of order n, nonequilibrium phase transition via fractal noises and complexity of Markovian processes via the concept of informational divergence.

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Westerkamp, Margret, Igor Ovchinnikov, Philipp Frank, and Torsten Enßlin. "Dynamical Field Inference and Supersymmetry." Entropy 23, no. 12 (December 8, 2021): 1652. http://dx.doi.org/10.3390/e23121652.

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Knowledge on evolving physical fields is of paramount importance in science, technology, and economics. Dynamical field inference (DFI) addresses the problem of reconstructing a stochastically-driven, dynamically-evolving field from finite data. It relies on information field theory (IFT), the information theory for fields. Here, the relations of DFI, IFT, and the recently developed supersymmetric theory of stochastics (STS) are established in a pedagogical discussion. In IFT, field expectation values can be calculated from the partition function of the full space-time inference problem. The partition function of the inference problem invokes a functional Dirac function to guarantee the dynamics, as well as a field-dependent functional determinant, to establish proper normalization, both impeding the necessary evaluation of the path integral over all field configurations. STS replaces these problematic expressions via the introduction of fermionic ghost and bosonic Lagrange fields, respectively. The action of these fields has a supersymmetry, which means there exists an exchange operation between bosons and fermions that leaves the system invariant. In contrast to this, measurements of the dynamical fields do not adhere to this supersymmetry. The supersymmetry can also be broken spontaneously, in which case the system evolves chaotically. This affects the predictability of the system and thereby makes DFI more challenging. We investigate the interplay of measurement constraints with the non-linear chaotic dynamics of a simplified, illustrative system with the help of Feynman diagrams and show that the Fermionic corrections are essential to obtain the correct posterior statistics over system trajectories.

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Григоренко, V. Grigorenko, Горбунов, D. Gorbunov, Еськов, Valeriy Eskov, Шадрин, and G. Shadrin. "Analysis of Myograms Acording to the Stochastics and the Chaos Theory – Self-Organization." Journal of New Medical Technologies 22, no. 2 (February 25, 2015): 32–38. http://dx.doi.org/10.12737/11829.

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The paper shows the feasibility of applying the method of multi-dimensional phase space as a quantitative measure for the evaluation of chaotic dynamics on the example of the muscles (flexor of the little finger). The method of multi-dimensional phase space was used. In the study and modeling of complex biological objects (complexity) there is the possibility of introducing traditional physical methods in biological research and new methods based on the chaos theory and self-organization. As a measure of the state of the neuromuscular system of the person (weak muscle tension and strong, almost the maximum force), the authors used quasi-attractors volumes of multidimensional phase space. This enables identification of the actual measurements of the parameters of the functional state with weak muscles (Fi = 5 daN) and strong (Fi = 10 daN) static stress. The authors built a timebase signal received from the electromyograph and the autocorrelation function A(t) of signal. A biomechanical analysis of the state of the system is carried out on the basis of comparison of the volume VG quasi attractor, as well as on the basis of the analysis of the Shannon entropy E. Volume of quasi attractor VG displacements under low load is slightly less than the same volume displacement VG with strong exertion of the muscles of the flexor of the little finger, exactly the same as the values of the Shannon entropy under a heavy load increases compared to the values obtained under low load the muscles.

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Fernández, Rocío, Javier Alcocer, Alfonso Lugo, Luis A. Oseguera, and Sandra Guadarrama-Hernández. "Seasonal and Interannual Dynamics of Pelagic Rotifers in a Tropical, Saline, Deep Lake." Diversity 14, no. 2 (February 5, 2022): 113. http://dx.doi.org/10.3390/d14020113.

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This is the first long-term study (monthly samples at two 4-year intervals: 1998 to 2001 and 2013 to 2016) on rotifers in a saline, deep lake. The pelagic rotifer assemblage of Lake Alchichica is simple and comprised by two species, both new and most likely endemic: Brachionus sp. Mexico (related to B. plicatilis) and Hexarthra sp. (related to H. jenkinae). Similar low species richness and composition are found in other saline lakes associated with salinity. Rotifers in Lake Alchichica were an irregular component of the zooplankton community. Rotifers’ overall abundance (471 ± 1211 ind m−2) and biomass (24 ± 63 mg DW m−2) were low; Brachionus sp. Mexico and Hexarthra sp. contributed similarly to the annual mean abundance (54% and 46%, respectively) and biomass (53% and 47%, respectively). Abundance and biomass were tightly coupled, but there was no regular pattern in their seasonal dynamics. When co-existing, Brachionus sp. Mexico showed a higher abundance than Hexarthra sp. The dominant (≈80%) phytoplankton biomass in Lake Alchichica, the large (35–63 µm) diatom Cyclotella alchichicana, is inedible for rotifers, thus rotifers most probably relied only on nanophytoplankton (≤20 µm). Seasonal and interannual differences in rotifers seem related to food availability (oligotrophy) and probably to biotic interactions (e.g., competition). Rotifer abundance and biomass values in 1998–2001 went down to 12.5% in 2013–2016. Climate change and stochastics events leading to pulses of the rotifers’ food, and biotic interactions seem to be the most plausible explanation.

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Lessio, Federico, and Alberto Alma. "Models Applied to Grapevine Pests: A Review." Insects 12, no. 2 (February 16, 2021): 169. http://dx.doi.org/10.3390/insects12020169.

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This paper reviews the existing predictive models concerning insects and mites harmful to grapevine. A brief conceptual description is given on the definition of a model and about different types of models: deterministic vs. stochastics, continuous vs. discrete, analytical vs. computer-based, and descriptive vs. data-driven. The main biological aspects of grapevine pests covered by different types of models are phenology, population growth and dynamics, species distribution, and invasion risk. A particular emphasis is put on forecasting epidemics of plant disease agents transmitted by insects with sucking-piercing mouthparts. The most investigated species or groups are the glassy-winged sharpshooter Homalodisca vitripennis (Germar) and other vectors of Xylella fastidiosa subsp. fastidiosa, a bacterium agent of Pierce’s disease; the European grape berry moth, Lobesia botrana (Denis and Schiffermuller); and the leafhopper Scaphoideus titanus Ball, the main vector of phytoplasmas agents of Flavescence dorée. Finally, the present and future of decision-support systems (DSS) in viticulture is discussed.

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Meshcheryakov, V. A., and V. V. Weber. "SETPOINT OPTIMIZATION FOR THE CONTROL SYSTEM OF THE MOTOR GRADER IN HEAVYLOAD MODE." Vestnik SibADI 15, no. 4 (September 12, 2018): 502–13. http://dx.doi.org/10.26518/2071-7296-2018-4-502-513.

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Introduction. The authors suggest the optimal tuning method foran automatic control system of the heavy motor grader blade. The demonstrated system regulates the value of digging force. Moreover, the setpoint optimization criterion is the maximum of production rate.Materials and methods. The optimal setpoint is generating to the signal estimations, specifically to the measured digging force and the wheels slip ratio during the previous stroke of the motor grader.There searchincludes:development of the blade control system functional diagram including the setpoint former;meaningful estimation of the slip ratio signal for control purposes;development of the setpoint forming algorithm for a microprocessor control unit;program realization of the motor grader workflow model and simulation;development of the algorithm to compose the lookup table containing optimal setpoint values and based on simulation results;dependence of optimal setpoint on the incoming signal estimations.Results. The lookup table of optimal setpoint values is obtained, the estimations of digging force and wheels slip ratio are presented. In addition, the authors suggest the control system structure with the optimal setpoint former and also develop the former operation algorithm.Discussion and conclusion. The optimal setpoint values are theoretically validated for the motor grader control system. The tuning method for an automatic control system of the heavy motor grader blade has the following characteristics as:the optimal control criterion as the production rate;process dynamics and stochastics;the excessive slippage time ratio.

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Ferrandis, Eduardo. "On the stochastic approach to marine population dynamics." Scientia Marina 71, no. 1 (March 30, 2007): 153–74. http://dx.doi.org/10.3989/scimar.2007.71n1153.

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Ma, Yi-An, and Hong Qian. "A thermodynamic theory of ecology: Helmholtz theorem for Lotka–Volterra equation, extended conservation law, and stochastic predator–prey dynamics." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 471, no. 2183 (November 2015): 20150456. http://dx.doi.org/10.1098/rspa.2015.0456.

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We carry out mathematical analyses, à la Helmholtz’s and Boltzmann’s 1884 studies of monocyclic Newtonian dynamics, for the Lotka–Volterra (LV) equation exhibiting predator–prey oscillations. In doing so, a novel ‘thermodynamic theory’ of ecology is introduced. An important feature, absent in the classical mechanics, of ecological systems is a natural stochastic population dynamic formulation of which the deterministic equation (e.g. the LV equation studied) is the infinite population limit. Invariant density for the stochastic dynamics plays a central role in the deterministic LV dynamics. We show how the conservation law along a single trajectory extends to incorporate both variations in a model parameter α and in initial conditions: Helmholtz’s theorem establishes a broadly valid conservation law in a class of ecological dynamics. We analyse the relationships among mean ecological activeness θ , quantities characterizing dynamic ranges of populations A and α , and the ecological force F α . The analyses identify an entire orbit as a stationary ecology, and establish the notion of an ‘equation of ecological states’. Studies of the stochastic dynamics with finite populations show the LV equation as the robust, fast cyclic underlying behaviour. The mathematical narrative provides a novel way of capturing long-term dynamical behaviours with an emergent conservative ecology .

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Glushkov, A. V., E. R. Gubanova, O. Yu Khetselius, G. P. Prepelitsa, A. A. Svinarenko, Yu Ya Bunyakova, and V. V. Buyadzhi. "Analysis and forecast of the environmental radioactivity dynamics based on methods of chaos theory: general scheme and some application." Ukrainian hydrometeorological journal, no. 16 (October 29, 2017): 40–45. http://dx.doi.org/10.31481/uhmj.16.2015.05.

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We present firstly a new whole technique of analysis, processing and forecasting environmental radioactivity dynamics, which has been earlier developed for the atmospheric pollution dynamics analysis and investigation of chaotic feature sin dynamics of the typical hydroecological systems. The general formalism include: a) A general qualitative analysis of dynamical problem of the environmental radioactivity dynamics (including a qualitative analysis from the viewpoint of ordinary differential equations, the “Arnold-analysis”); b) checking for the presence of a chaotic (stochastic) features and regimes (the Gottwald-Melbourne’s test; the method of correlation dimension); c) Reducing the phase space (choice of the time delay, the definition of the embedding space by methods of correlation dimension algorithm and false nearest neighbor points); d) Determination of the dynamic invariants of a chaotic system (Computation of the global Lyapunov dimension λa; determination of the Kaplan-York dimension dL and average limits of predictability Prmax on the basis of the advanced algorithms; e) A non-linear prediction (forecasting) of an dynamical evolution of the system. The last block indeed includes new (in a theory of environmental radioactivity dynamics) methods and algorithms of nonlinear prediction such as methods of predicted trajectories, stochastic propagators and neural networks modelling, renorm-analysis with blocks of the polynomial approximations, wavelet-expansions etc.

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Santonja, Francisco-José, and Leonid Shaikhet. "Analysing Social Epidemics by Delayed Stochastic Models." Discrete Dynamics in Nature and Society 2012 (2012): 1–13. http://dx.doi.org/10.1155/2012/530472.

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We investigate the dynamics of a delayed stochastic mathematical model to understand the evolution of the alcohol consumption in Spain. Sufficient condition for stability in probability of the equilibrium point of the dynamic model with aftereffect and stochastic perturbations is obtained via Kolmanovskii and Shaikhet general method of Lyapunov functionals construction. We conclude that alcohol consumption in Spain will be constant (with stability) in time with around 36.47% of nonconsumers, 62.94% of nonrisk consumers, and 0.59% of risk consumers. This approach allows us to emphasize the possibilities of the dynamical models in order to study human behaviour.

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Yamamoto, Takehiro, Shinpei Hirota, and Takuya Fujiwara. "A Stochastic Rotation Dynamics Model for Dilute Spheroidal Colloid Suspensions." Nihon Reoroji Gakkaishi 44, no. 3 (2016): 185–88. http://dx.doi.org/10.1678/rheology.44.185.

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H. Hirpara, Ravish, and Shambhu N. Sharma. "On the Stochastic Filtering Theory of a Power System Dynamics." Transactions of the Institute of Systems, Control and Information Engineers 29, no. 1 (2016): 9–17. http://dx.doi.org/10.5687/iscie.29.9.

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Zhang, Yimin, Qiaoling Liu, and Bangchun Wen. "Dynamic Research of a Nonlinear Stochastic Vibratory Machine." Shock and Vibration 9, no. 6 (2002): 277–81. http://dx.doi.org/10.1155/2002/734102.

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This paper presents the dynamics problems of stochastic vibratory machine systems. The random responses of the vibratory machine systems with stochastic parameters subjected to random excitation are researched using a stochastic perturbation method. The numerical results are obtained. The dynamic characteristics of nonlinear stochastic vibratory machine are analyzed.

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Kurth, Jutta G., Thorsten Rings, and Klaus Lehnertz. "Testing Jump-Diffusion in Epileptic Brain Dynamics: Impact of Daily Rhythms." Entropy 23, no. 3 (March 5, 2021): 309. http://dx.doi.org/10.3390/e23030309.

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Stochastic approaches to complex dynamical systems have recently provided broader insights into spatial-temporal aspects of epileptic brain dynamics. Stochastic qualifiers based on higher-order Kramers-Moyal coefficients derived directly from time series data indicate improved differentiability between physiological and pathophysiological brain dynamics. It remains unclear, however, to what extent stochastic qualifiers of brain dynamics are affected by other endogenous and/or exogenous influencing factors. Addressing this issue, we investigate multi-day, multi-channel electroencephalographic recordings from a subject with epilepsy. We apply a recently proposed criterion to differentiate between Langevin-type and jump-diffusion processes and observe the type of process most qualified to describe brain dynamics to change with time. Stochastic qualifiers of brain dynamics are strongly affected by endogenous and exogenous rhythms acting on various time scales—ranging from hours to days. Such influences would need to be taken into account when constructing evolution equations for the epileptic brain or other complex dynamical systems subject to external forcings.

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Zhang, Yingqi, Wei Cheng, Xiaowu Mu, and Caixia Liu. "Stochasticℋ∞Finite-Time Control of Discrete-Time Systems with Packet Loss." Mathematical Problems in Engineering 2012 (2012): 1–15. http://dx.doi.org/10.1155/2012/897481.

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This paper investigates the stochastic finite-time stabilization andℋ∞control problem for one family of linear discrete-time systems over networks with packet loss, parametric uncertainties, and time-varying norm-bounded disturbance. Firstly, the dynamic model description studied is given, which, if the packet dropout is assumed to be a discrete-time homogenous Markov process, the class of discrete-time linear systems with packet loss can be regarded as Markovian jump systems. Based on Lyapunov function approach, sufficient conditions are established for the resulting closed-loop discrete-time system with Markovian jumps to be stochasticℋ∞finite-time boundedness and then state feedback controllers are designed to guarantee stochasticℋ∞finite-time stabilization of the class of stochastic systems. The stochasticℋ∞finite-time boundedness criteria can be tackled in the form of linear matrix inequalities with a fixed parameter. As an auxiliary result, we also give sufficient conditions on the robust stochastic stabilization of the class of linear systems with packet loss. Finally, simulation examples are presented to illustrate the validity of the developed scheme.

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Dahani, Khawla, and Rajae Aboulaich. "Dynamic Stochastic General Equilibrium model for the Islamic economy." Investment Management and Financial Innovations 15, no. 3 (October 2, 2018): 370–82. http://dx.doi.org/10.21511/imfi.15(3).2018.30.

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This article is concerned with the debate around the economic knowledge evolution and the role of ethics in economy. It reports on the 2008 crisis, the research literature reveals two main problems: the efficiency of the economic modeling and the failure of the ethical system.The authors explore the use of the new Dynamic Stochastic General Equilibrium “DSGE” model in the case of Islamic economy, it can enable to develop a new approach, taking into account the criticism of the models used before the crisis, and giving more importance to the ethical principles.The question is to know if the principles of Islamic economy feed into a sustainable economic system.The characteristic of this model lies in the consideration of Islamic principles, namely the abolition of interest rates and their replacement by the rate of return of the capital. In this perspective, it is supposed that the intervention of the monetary authorities is done by an unconventional approach. The model also distinguishes itself by the integration of Zakat. The model is applied in the case of Morocco.The results of simulations show that the introduction of these Islamic principles has no negative effects on the macroeconomic and financial conditions of Morocco and that the stability of the economic system is maintained.

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Kuske, R., and D. Yurchenko. "Editorial." European Journal of Applied Mathematics 30, no. 5 (September 9, 2019): 829. http://dx.doi.org/10.1017/s0956792518000694.

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The origin of this special issue took place at the 9th European Nonlinear Dynamics Conference (ENOC 2017) in Budapest, Hungary. Specifically, the mini-symposium on Random Dynamical Systems – Recent Advances and New Directions brought together novel perspectives on analyzing stochastic dynamics with applications including biology, structural dynamics, control, energy and mechanics. The expanded use of stochasticity in more realistic models exposes questions related to bifurcations, meta-stability, tipping and early warning signals, multiscale dynamics, and connections between chaos and stochastic dynamics. The observed phenomena in applications drive new methodologies and analyses, needed to understand the interplay between different sources of stochastic effects and nonlinearities, network structure, multi-mode and multi-scale behavior, non-smooth dynamics, energy transfer, and spatio-temporal phenomena. Of course, a single issue cannot hope to cover all of the new topics in stochastic analysis for applications. Nevertheless, we hope that the collection of applications and stochastic models presented in this issue illustrates some of the exciting advances and perspectives relevant for broad classes of stochastic models and demonstrates the need in advancing the theory of stochastic processes.

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Heitmann, Stewart, and Michael Breakspear. "Putting the “dynamic” back into dynamic functional connectivity." Network Neuroscience 2, no. 2 (June 2018): 150–74. http://dx.doi.org/10.1162/netn_a_00041.

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The study of fluctuations in time-resolved functional connectivity is a topic of substantial current interest. As the term “dynamic functional connectivity” implies, such fluctuations are believed to arise from dynamics in the neuronal systems generating these signals. While considerable activity currently attends to methodological and statistical issues regarding dynamic functional connectivity, less attention has been paid toward its candidate causes. Here, we review candidate scenarios for dynamic (functional) connectivity that arise in dynamical systems with two or more subsystems; generalized synchronization, itinerancy (a form of metastability), and multistability. Each of these scenarios arises under different configurations of local dynamics and intersystem coupling: We show how they generate time series data with nonlinear and/or nonstationary multivariate statistics. The key issue is that time series generated by coupled nonlinear systems contain a richer temporal structure than matched multivariate (linear) stochastic processes. In turn, this temporal structure yields many of the phenomena proposed as important to large-scale communication and computation in the brain, such as phase-amplitude coupling, complexity, and flexibility. The code for simulating these dynamics is available in a freeware software platform, the Brain Dynamics Toolbox.

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Ovchinnikov, Igor V., and Kang L. Wang. "Stochastic dynamics and combinatorial optimization." Modern Physics Letters B 31, no. 31 (November 6, 2017): 1750285. http://dx.doi.org/10.1142/s0217984917502852.

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Natural dynamics is often dominated by sudden nonlinear processes such as neuroavalanches, gamma-ray bursts, solar flares, etc., that exhibit scale-free statistics much in the spirit of the logarithmic Ritcher scale for earthquake magnitudes. On phase diagrams, stochastic dynamical systems (DSs) exhibiting this type of dynamics belong to the finite-width phase (N-phase for brevity) that precedes ordinary chaotic behavior and that is known under such names as noise-induced chaos, self-organized criticality, dynamical complexity, etc. Within the recently proposed supersymmetric theory of stochastic dynamics, the N-phase can be roughly interpreted as the noise-induced “overlap” between integrable and chaotic deterministic dynamics. As a result, the N-phase dynamics inherits the properties of the both. Here, we analyze this unique set of properties and conclude that the N-phase DSs must naturally be the most efficient optimizers: on one hand, N-phase DSs have integrable flows with well-defined attractors that can be associated with candidate solutions and, on the other hand, the noise-induced attractor-to-attractor dynamics in the N-phase is effectively chaotic or aperiodic so that a DS must avoid revisiting solutions/attractors thus accelerating the search for the best solution. Based on this understanding, we propose a method for stochastic dynamical optimization using the N-phase DSs. This method can be viewed as a hybrid of the simulated and chaotic annealing methods. Our proposition can result in a new generation of hardware devices for efficient solution of various search and/or combinatorial optimization problems.

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Wu, Yan-rui, Peng-fei Yang, and You-li Wu. "Stochastic Control-Oriented Modeling of Flexible Air-Breathing Hypersonic Vehicle." Mathematical Problems in Engineering 2018 (2018): 1–12. http://dx.doi.org/10.1155/2018/1648560.

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The flexible dynamics of commonly used air-breathing hypersonic vehicle model are not tractable for control design and the inevitable stochastic perturbations are usually neglected. Aiming at these deficiencies, reduced flexible dynamics are deducted in this paper and a stochastic control-oriented vehicle model is established accordingly. The responses of the original system to the deterministic and the stochastic part of the generalized force, which is treated as the input of the flexible dynamic system, are analyzed. After that, the simplified flexible dynamics is deduced to approximate the responses. The reduced flexible dynamics, which are tractable for control design since they greatly reduce the complexity of the original dynamics, are comprised of a simple function of the determined generalized force and an Ornstein-Uhlenbeck colored noise. Finally, the longitudinal dynamics in parametric strict feedback form are obtained by substituting the reduced flexible dynamics into the original model. The applicability of the simplified flexible dynamics is validated through the numerical simulations.

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ENCISO, GERMAN, RADEK ERBAN, and JINSU KIM. "Identifiability of stochastically modelled reaction networks." European Journal of Applied Mathematics 32, no. 5 (February 15, 2021): 865–87. http://dx.doi.org/10.1017/s0956792520000492.

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Chemical reaction networks describe interactions between biochemical species. Once an underlying reaction network is given for a biochemical system, the system dynamics can be modelled with various mathematical frameworks such as continuous-time Markov processes. In this manuscript, the identifiability of the underlying network structure with a given stochastic system dynamics is studied. It is shown that some data types related to the associated stochastic dynamics can uniquely identify the underlying network structure as well as the system parameters. The accuracy of the presented network inference is investigated when given dynamical data are obtained via stochastic simulations.

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Ye, Hongbo. "On Stochastic-User-Equilibrium-Based Day-to-Day Dynamics." Transportation Science 56, no. 1 (January 2022): 103–17. http://dx.doi.org/10.1287/trsc.2021.1080.

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Researchers have proposed many different concepts and models to study day-to-day dynamics. Some models explicitly model travelers’ perceiving and learning on travel costs, and some other models do not explicitly consider the travel cost perception but instead formulate the dynamics of flows as the functions of flows and measured travel costs (which are determined by flows). This paper investigates the interconnection between these two types of day-to-day models, in particular, those models whose fixed points are a stochastic user equilibrium. Specifically, a widely used day-to-day model that combines exponential-smoothing learning and logit stochastic network loading (called the logit-ESL model in this paper) is proved to be equivalent to a model based purely on flows, which is the logit-based extension of the first-in-first-out dynamic of Jin [Jin W (2007) A dynamical system model of the traffic assignment problem. Transportation Res. Part B Methodological 41(1):32–48]. Via this equivalent form, the logit-ESL model is proved to be globally stable under nonseparable and monotone travel cost functions. Moreover, the model of Cantarella and Cascetta is shown to be equivalent to a second-order dynamic incorporating purely flows and is proved to be globally stable under separable link cost functions [Cantarella GE, Cascetta E (1995) Dynamic processes and equilibrium in transportation networks: Towards a unifying theory. Transportation Sci. 29(4):305–329]. Further, other discrete choice models, such as C-logit, path-size logit, and weibit, are introduced into the logit-ESL model, leading to several new day-to-day models, which are also proved to be globally stable under different conditions.

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Nishimura, Haruki, and Mac Schwager. "SACBP: Belief space planning for continuous-time dynamical systems via stochastic sequential action control." International Journal of Robotics Research 40, no. 10-11 (August 13, 2021): 1167–95. http://dx.doi.org/10.1177/02783649211037697.

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We propose a novel belief space planning technique for continuous dynamics by viewing the belief system as a hybrid dynamical system with time-driven switching. Our approach is based on the perturbation theory of differential equations and extends sequential action control to stochastic dynamics. The resulting algorithm, which we name SACBP, does not require discretization of spaces or time and synthesizes control signals in near real-time. SACBP is an anytime algorithm that can handle general parametric Bayesian filters under certain assumptions. We demonstrate the effectiveness of our approach in an active sensing scenario and a model-based Bayesian reinforcement learning problem. In these challenging problems, we show that the algorithm significantly outperforms other existing solution techniques including approximate dynamic programming and local trajectory optimization.

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Carkovs, Jevgeņijs, Jolanta Goldšteine, and Kārlis Šadurskis. "The Holling Type II Population Model Subjected to Rapid Random Attacks of Predator." Journal of Applied Mathematics 2018 (June 20, 2018): 1–10. http://dx.doi.org/10.1155/2018/6146027.

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We present the analysis of a mathematical model of the dynamics of interacting predator and prey populations with the Holling type random trophic function under the assumption of random time interval passage between predator attacks on prey. We propose a stochastic approximation algorithm for quantitative analysis of the above model based on the probabilistic limit theorem. If the predators’ gains and the time intervals between predator attacks are sufficiently small, our proposed method allows us to derive an approximative average dynamical system for mathematical expectations of population dynamics and the stochastic Ito differential equation for the random deviations from the average motion. Assuming that the averaged dynamical system is the classic Holling type II population model with asymptotically stable limit cycle, we prove that the dynamics of stochastic model may be approximated with a two-dimensional Gaussian Markov process with unboundedly increasing variances.

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Sankar, T. S., S. A. Ramu, and R. Ganesan. "Stochastic Finite Element Analysis for High Speed Rotors." Journal of Vibration and Acoustics 115, no. 1 (January 1, 1993): 59–64. http://dx.doi.org/10.1115/1.2930315.

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The general problem of the dynamic response of highspeed rotors is considered in which certain system parameters may have a spatial stochastic variation. In particular the elastic modulus and mass density of a rotating shaft are described through one dimensional stochastic field functions so that the imperfections in manufacture and measurement can be accounted for. The stochastic finite element method is developed so that the variability of the response of the rotor can be interpreted in terms of the variation of the material property. As an illustration the whirl speed analysis is performed to determine the stochastics of whirl speeds and modes through the solution of a random eigenvalue problem associated with a non self-adjoint system. Numerical results are also presented.

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Zhao, Wenqiang, and Yangrong Li. "Existence of Random Attractors for ap-Laplacian-Type Equation with Additive Noise." Abstract and Applied Analysis 2011 (2011): 1–21. http://dx.doi.org/10.1155/2011/616451.

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We first establish the existence and uniqueness of a solution for a stochasticp-Laplacian-type equation with additive white noise and show that the unique solution generates a stochastic dynamical system. By using the Dirichlet forms of Laplacian and an approximation procedure, the nonlinear obstacle, arising from the additive noise is overcome when we make energy estimate. Then, we obtain a random attractor for this stochastic dynamical system. Finally, under a restrictive assumption on the monotonicity coefficient, we find that the random attractor consists of a single point, and therefore the system possesses a unique stationary solution.

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Reed, William J. "Analyzing Catch–Effort Data Allowing for Randomness in the Catching Process." Canadian Journal of Fisheries and Aquatic Sciences 43, no. 1 (January 1, 1986): 174–86. http://dx.doi.org/10.1139/f86-020.

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For many fisheries the only reliable data is a (bivariate) time series of catches and efforts. Most existing methods of analyzing such data implicitly assume that the main source of randomness is in the dynamics of the population, while ignoring randomness in the catching process. The assumption of a deterministic catch production function (usually of the Schaefer form C = qEX) must be contrary to the experience of almost everyone who has ever gone fishing. In this paper a stochastic catch model coupled with a deterministic dynamic model is used in the analysis of catch–effort data and shown to give very plausible results. Estimates (with confidence intervais) of catchability, maximum sustainable yield, and other dynamic model parameters are obtained numerically by the method of maximum likelihood. The incorporation of stochastic dynamics with the stochastic catch model is difficult.

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Wieczorek, Radosław. "Markov chain model of phytoplankton dynamics." International Journal of Applied Mathematics and Computer Science 20, no. 4 (December 1, 2010): 763–71. http://dx.doi.org/10.2478/v10006-010-0058-7.

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Markov chain model of phytoplankton dynamicsA discrete-time stochastic spatial model of plankton dynamics is given. We focus on aggregative behaviour of plankton cells. Our aim is to show the convergence of a microscopic, stochastic model to a macroscopic one, given by an evolution equation. Some numerical simulations are also presented.

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SIETTOS, CONSTANTINOS I., IOANNIS G. KEVREKIDIS, and DIMITRIOS MAROUDAS. "COARSE BIFURCATION DIAGRAMS VIA MICROSCOPIC SIMULATORS: A STATE-FEEDBACK CONTROL-BASED APPROACH." International Journal of Bifurcation and Chaos 14, no. 01 (January 2004): 207–20. http://dx.doi.org/10.1142/s0218127404009193.

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We present and illustrate a feedback control-based framework that enables microscopic/stochastic simulators to trace their "coarse" bifurcation diagrams, characterizing the dependence of their expected dynamical behavior on parameters. The framework combines the so-called "coarse time stepper" and arc-length continuation ideas from numerical bifurcation theory with linear dynamic feedback control. An augmented dynamical system is formulated, in which the bifurcation parameter evolution is linked with the microscopic simulation dynamics through feedback laws. The augmentation stably steers the system along both stable and unstable portions of the open-loop bifurcation diagram. The framework is illustrated using kinetic Monte Carlo simulations of simple surface reaction schemes that exhibit both coarse regular turning points and coarse Hopf bifurcations.

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Bod’ová, Katarína, Enikő Szép, and Nicholas H. Barton. "Dynamic maximum entropy provides accurate approximation of structured population dynamics." PLOS Computational Biology 17, no. 12 (December 1, 2021): e1009661. http://dx.doi.org/10.1371/journal.pcbi.1009661.

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Realistic models of biological processes typically involve interacting components on multiple scales, driven by changing environment and inherent stochasticity. Such models are often analytically and numerically intractable. We revisit a dynamic maximum entropy method that combines a static maximum entropy with a quasi-stationary approximation. This allows us to reduce stochastic non-equilibrium dynamics expressed by the Fokker-Planck equation to a simpler low-dimensional deterministic dynamics, without the need to track microscopic details. Although the method has been previously applied to a few (rather complicated) applications in population genetics, our main goal here is to explain and to better understand how the method works. We demonstrate the usefulness of the method for two widely studied stochastic problems, highlighting its accuracy in capturing important macroscopic quantities even in rapidly changing non-stationary conditions. For the Ornstein-Uhlenbeck process, the method recovers the exact dynamics whilst for a stochastic island model with migration from other habitats, the approximation retains high macroscopic accuracy under a wide range of scenarios in a dynamic environment.

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Alshammari, Fehaid Salem, Fahir Talay Akyildiz, Muhammad Altaf Khan, Anwarud Din, and Pongsakorn Sunthrayuth. "A Stochastic Mathematical Model for Understanding the COVID-19 Infection Using Real Data." Symmetry 14, no. 12 (November 29, 2022): 2521. http://dx.doi.org/10.3390/sym14122521.

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Natural symmetry exists in several phenomena in physics, chemistry, and biology. Incorporating these symmetries in the differential equations used to characterize these processes is thus a valid modeling assumption. The present study investigates COVID-19 infection through the stochastic model. We consider the real infection data of COVID-19 in Saudi Arabia and present its detailed mathematical results. We first present the existence and uniqueness of the deterministic model and later study the dynamical properties of the deterministic model and determine the global asymptotic stability of the system for R0≤1. We then study the dynamic properties of the stochastic model and present its global unique solution for the model. We further study the extinction of the stochastic model. Further, we use the nonlinear least-square fitting technique to fit the data to the model for the deterministic and stochastic case and the estimated basic reproduction number is R0≈1.1367. We show that the stochastic model provides a good fitting to the real data. We use the numerical approach to solve the stochastic system by presenting the results graphically. The sensitive parameters that significantly impact the model dynamics and reduce the number of infected cases in the future are shown graphically.

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Pavelev, A. V., and V. V. Semin. "Investigation of non-markovian dynamics of two dipole-dipole interacting Qubits based on numerical solution of the non-linear stochastic schrödinger equation." Computer Optics 43, no. 2 (April 2019): 168–73. http://dx.doi.org/10.18287/2412-6179-2019-43-2-168-173.

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In this paper, we investigate non-markovian dynamics of a system of two interacting qubits. With the help of stochastic calculus we derive the non-Markovian non-linear stochastic Schrödinger equation. This equation is solved by the direct computer simulation. The simulation is used to obtain some dynamic properties of the system.

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Hu, Weipeng, Tao Liu, and Zhengqi Han. "Dynamical Symmetry Breaking of Infinite-Dimensional Stochastic System." Symmetry 14, no. 8 (August 7, 2022): 1627. http://dx.doi.org/10.3390/sym14081627.

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The mapping relationship between the symmetry and the conserved quantity inspired researchers to seek the conserved quantity from the viewpoint of the symmetry for the dynamic systems. However, the symmetry breaking in the dynamic systems is more common than the symmetry in the engineering. Thus, as the supplement of our previous work on the symmetry breaking of infinite-dimensional deterministic dynamic systems, the dynamical symmetry breaking of infinite-dimensional stochastic systems is discussed in this paper. Following a brief review of the stochastic (multi-)symplectic for the dynamic system excited by stochastic white noise, two types of stochastic symmetry breaking factors, including the general stochastic excitation and the general stochastic parameters of the infinite-dimensional dynamic systems, are investigated in detail. We find that both the general stochastic excitation and the general stochastic parameters will not break the local multi-symplectic structure of the dynamic systems. However, the local energy conservation law will be broken by the general stochastic excitation, as well as by the stochastic parameters, which are given by the local energy dissipation in this paper. To illustrate the validity of the analytical results, the stochastic vibration of a clamped single-walled carbon nanotube is investigated and the critical condition of the appearance of chaos is obtained. The theoretical results obtained can be used to guide us to construct the structure-preserving method for the stochastic dynamic systems.

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Pavlos, G. P., D. Kugiumtzis, M. A. Athanasiu, N. Hatzigeorgiu, D. Diamantidis, and E. T. Sarris. "Nonlinear analysis of magnetospheric data Part II. Dynamical characteristics of the AE index time series and comparison with nonlinear surrogate data." Nonlinear Processes in Geophysics 6, no. 2 (June 30, 1999): 79–98. http://dx.doi.org/10.5194/npg-6-79-1999.

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Abstract. In this study we have used dynamical characteristies such as Lyapunov exponents, nonlinear dynamic models and mutual information for the nonlinear analysis of the magnetospheric AE index time series. Similarly with the geometrical characteristic studied in Pavlos et al. (1999b), we have found significant differences between the original time series and its surrogate data. These results also suggest the rejection of the null hypothesis that the AE index belongs to the family of stochastic linear signals undergoing a static nonlinear distortion. Finally, we believe that these results support the hypothesis of nonlinearity and chaos for the magnetospheric dynamics.

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Journal articles on the topic 'Stochastics dynamics'

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