Chaos and information in two dimensional turbulence

By performing a large number of fully resolved simulations of incompressible homogeneous and isotropic two dimensional turbulence, we study the scaling behavior of the maximal Lyapunov exponent, the Kolmogorov-Sinai entropy and attractor dimension. The scaling of the maximal Lyapunov exponent is found to be in good agreement with the dimensional predictions. For the cases of the Kolmogorov-Sinai entropy and attractor dimension the simple dimensional predictions are found to be insufficient. A dependence on the system size and the forcing length scale is found, suggesting non-universal behavior. The applicability of these results to atmospheric predictability is also discussed.Comment: 10 pages, 9 figures, In Press Physical Review Fluids, 202

Production and transfer of energy and information in Hamiltonian systems

We present novel results that relate energy and information transfer with sensitivity to initial conditions in chaotic multi-dimensional Hamiltonian systems. We show the relation among Kolmogorov-Sinai entropy, Lyapunov exponents, and upper bounds for the Mutual Information Rate calculated in the Hamiltonian phase space and on bi-dimensional subspaces. Our main result is that the net amount of transfer from kinetic to potential energy per unit of time is a power-law of the upper bound for the Mutual Information Rate between kinetic and potential energies, and also a power-law of the Kolmogorov-Sinai entropy. Therefore, transfer of energy is related with both transfer and production of information. However, the power-law nature of this relation means that a small increment of energy transferred leads to a relatively much larger increase of the information exchanged. Then, we propose an ?experimental? implementation of a 1-dimensional communication channel based on a Hamiltonian system, and calculate the actual rate with which information is exchanged between the first and last particle of the channel. Finally, a relation between our results and important quantities of thermodynamics is presented

Complexity without chaos: Plasticity within random recurrent networks generates robust timing and motor control

It is widely accepted that the complex dynamics characteristic of recurrent neural circuits contributes in a fundamental manner to brain function. Progress has been slow in understanding and exploiting the computational power of recurrent dynamics for two main reasons: nonlinear recurrent networks often exhibit chaotic behavior and most known learning rules do not work in robust fashion in recurrent networks. Here we address both these problems by demonstrating how random recurrent networks (RRN) that initially exhibit chaotic dynamics can be tuned through a supervised learning rule to generate locally stable neural patterns of activity that are both complex and robust to noise. The outcome is a novel neural network regime that exhibits both transiently stable and chaotic trajectories. We further show that the recurrent learning rule dramatically increases the ability of RRNs to generate complex spatiotemporal motor patterns, and accounts for recent experimental data showing a decrease in neural variability in response to stimulus onset

Stability Assessment of Stochastic Differential-Algebraic Systems via Lyapunov Exponents with an Application to Power Systems

[EN] In this paper, we discuss stochastic differential-algebraic equations (SDAEs) and the asymptotic stability assessment for such systems via Lyapunov exponents (LEs). We focus on index-1 SDAEs and their reformulation as ordinary stochastic differential equations (SDEs). Via ergodic theory, it is then feasible to analyze the LEs via the random dynamical system generated by the underlying SDEs. Once the existence of well-defined LEs is guaranteed, we proceed to the use of numerical simulation techniques to determine the LEs numerically. Discrete and continuous QR decomposition-based numerical methods are implemented to compute the fundamental solution matrix and to use it in the computation of the LEs. Important computational features of both methods are illustrated via numerical tests. Finally, the methods are applied to two applications from power systems engineering, including the single-machine infinite-bus (SMIB) power system model.A.G.-Z. was supported by Secretaria Nacional de Ciencia y Tecnologia SENESCYT (Ecuador), through the scholarship "Becas de Fomento al Talento Humano", and Deutsche Forschungsgemeinschaft through Collaborative Research Centre Transregio. SFB TRR 154. P.F.-d.-C. was partially supported by grant no. RTI2018-102256-B-I00 (Spain). J.-C.C. acknowledges the support by the Spanish Ministerio de Economia, Industria y Competitividad (MINECO), the Agencia Estatal de Investigacion (AEI), and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-P. V.M. was partially supported by Deutsche Forschungsgemeinschaft through the Excellence Cluster Math+ in Berlin, and Priority Program 1984 "Hybride und multimodale Energiesysteme: Systemtheoretische Methoden fur die Transformation und den Betrieb komplexer Netze".González-Zumba, A.; Fernández De Córdoba, P.; Cortés, J.; Mehrmann, V. (2020). Stability Assessment of Stochastic Differential-Algebraic Systems via Lyapunov Exponents with an Application to Power Systems. Mathematics. 8(9):1-26. https://doi.org/10.3390/math8091393S12689Schein, O., & Denk, G. (1998). Numerical solution of stochastic differential-algebraic equations with applications to transient noise simulation of microelectronic circuits. Journal of Computational and Applied Mathematics, 100(1), 77-92. doi:10.1016/s0377-0427(98)00138-1Winkler, R. (2004). Stochastic differential algebraic equations of index 1 and applications in circuit simulation. Journal of Computational and Applied Mathematics, 163(2), 435-463. doi:10.1016/j.cam.2003.12.017CONG, N. D., & THE, N. T. (2012). LYAPUNOV SPECTRUM OF NONAUTONOMOUS LINEAR STOCHASTIC DIFFERENTIAL ALGEBRAIC EQUATIONS OF INDEX-1. Stochastics and Dynamics, 12(04), 1250002. doi:10.1142/s0219493712500025Küpper, D., Kværnø, A., & Rößler, A. (2011). A Runge-Kutta method for index 1 stochastic differential-algebraic equations with scalar noise. BIT Numerical Mathematics, 52(2), 437-455. doi:10.1007/s10543-011-0354-0Benettin, G., Galgani, L., Giorgilli, A., & Strelcyn, J.-M. (1980). Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory. Meccanica, 15(1), 9-20. doi:10.1007/bf02128236Benettin, G., Galgani, L., Giorgilli, A., & Strelcyn, J.-M. (1980). Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application. Meccanica, 15(1), 21-30. doi:10.1007/bf02128237Dieci, L., & Van Vleck, E. S. (2002). Lyapunov Spectral Intervals: Theory and Computation. SIAM Journal on Numerical Analysis, 40(2), 516-542. doi:10.1137/s0036142901392304Dieci, L., & Van Vleck, E. S. (2006). Lyapunov and Sacker–Sell Spectral Intervals. Journal of Dynamics and Differential Equations, 19(2), 265-293. doi:10.1007/s10884-006-9030-5Linh, V. H., & Mehrmann, V. (2009). Lyapunov, Bohl and Sacker-Sell Spectral Intervals for Differential-Algebraic Equations. Journal of Dynamics and Differential Equations, 21(1), 153-194. doi:10.1007/s10884-009-9128-7Linh, V. H., Mehrmann, V., & Van Vleck, E. S. (2010). QR methods and error analysis for computing Lyapunov and Sacker–Sell spectral intervals for linear differential-algebraic equations. Advances in Computational Mathematics, 35(2-4), 281-322. doi:10.1007/s10444-010-9156-1Dieci, L., Russell, R. D., & Van Vleck, E. S. (1997). On the Compuation of Lyapunov Exponents for Continuous Dynamical Systems. SIAM Journal on Numerical Analysis, 34(1), 402-423. doi:10.1137/s0036142993247311Talay, D. (1990). Second-order discretization schemes of stochastic differential systems for the computation of the invariant law. Stochastics and Stochastic Reports, 29(1), 13-36. doi:10.1080/17442509008833606Dieci, L., Russell, R. D., & Van Vleck, E. S. (1994). Unitary Integrators and Applications to Continuous Orthonormalization Techniques. SIAM Journal on Numerical Analysis, 31(1), 261-281. doi:10.1137/0731014YU. RYAGIN, M., & RYASHKO, L. B. (2004). THE ANALYSIS OF THE STOCHASTICALLY FORCED PERIODIC ATTRACTORS FOR CHUA’S CIRCUIT. International Journal of Bifurcation and Chaos, 14(11), 3981-3987. doi:10.1142/s0218127404011600Definition and Classification of Power System Stability IEEE/CIGRE Joint Task Force on Stability Terms and Definitions. (2004). IEEE Transactions on Power Systems, 19(3), 1387-1401. doi:10.1109/tpwrs.2004.825981Verdejo, H., Vargas, L., & Kliemann, W. (2012). Stability of linear stochastic systems via Lyapunov exponents and applications to power systems. Applied Mathematics and Computation, 218(22), 11021-11032. doi:10.1016/j.amc.2012.04.063Verdejo, H., Escudero, W., Kliemann, W., Awerkin, A., Becker, C., & Vargas, L. (2016). Impact of wind power generation on a large scale power system using stochastic linear stability. Applied Mathematical Modelling, 40(17-18), 7977-7987. doi:10.1016/j.apm.2016.04.020Wadduwage, D. P., Wu, C. Q., & Annakkage, U. D. (2013). Power system transient stability analysis via the concept of Lyapunov Exponents. Electric Power Systems Research, 104, 183-192. doi:10.1016/j.epsr.2013.06.011Milano, F., & Zarate-Minano, R. (2013). A Systematic Method to Model Power Systems as Stochastic Differential Algebraic Equations. IEEE Transactions on Power Systems, 28(4), 4537-4544. doi:10.1109/tpwrs.2013.2266441Geurts, B. J., Holm, D. D., & Luesink, E. (2019). Lyapunov Exponents of Two Stochastic Lorenz 63 Systems. Journal of Statistical Physics, 179(5-6), 1343-1365. doi:10.1007/s10955-019-02457-

Effect of Random Parameter Switching on Commensurate Fractional Order Chaotic Systems

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.The paper explores the effect of random parameter switching in a fractional order (FO) unified chaotic system which captures the dynamics of three popular sub-classes of chaotic systems i.e. Lorenz, Lu and Chen's family of attractors. The disappearance of chaos in such systems which rapidly switch from one family to the other has been investigated here for the commensurate FO scenario. Our simulation study show that a noise-like random variation in the key parameter of the unified chaotic system along with a gradual decrease in the commensurate FO is capable of suppressing the chaotic fluctuations much earlier than that with the fixed parameter one. The chaotic time series produced by such random parameter switching in nonlinear dynamical systems have been characterized using the largest Lyapunov exponent (LLE) and Shannon entropy. The effect of choosing different simulation techniques for random parameter FO switched chaotic systems have also been explored through two frequency domain and three time domain methods. Such a noise-like random switching mechanism could be useful for stabilization and control of chaotic oscillation in many real-world applications

Local characterization of transient chaos on finite times in open systems

To characterize local finite-time properties associated with transient chaos in open dynamical systems, we introduce an escape rate and fractal dimensions suitable for this purpose in a coarse-grained description. We numerically illustrate that these quantifiers have a considerable spread across the domain of the dynamics, but their spatial variation, especially on long but non-asymptotic integration times, is approximately consistent with the relationship that was recognized by Kantz and Grassberger for temporally asymptotic quantifiers. In particular, deviations from this relationship are smaller than differences between various locations, which confirms the existence of such a dynamical law and the suitability of our quantifiers to represent underlying dynamical properties in the non-asymptotic regime.Comment: 23 pages, 8 figures. Revision based on referee reports: minor corrections and a new analysis of the deviation from the Kantz-Grassberger relationshi

Dynamic Modeling and Stability Analysis of Stochastic Multi-Physical Systems Applied to Electric Power Systems

[ES] La naturaleza aleatoria que caracteriza algunos fenómenos en sistemas físicos reales (e.g., ingeniería, biología, economía, finanzas, epidemiología y otros) nos ha planteado el desafío de un cambio de paradigma del modelado matemático y el análisis de sistemas dinámicos, y a tratar los fenómenos aleatorios como variables aleatorias o procesos estocásticos. Este enfoque novedoso ha traído como consecuencia nuevas especificidades que la teoría clásica del modelado y análisis de sistemas dinámicos deterministas no ha podido cubrir. Afortunadamente, maravillosas contribuciones, realizadas sobre todo en el último siglo, desde el campo de las matemáticas por científicos como Kolmogorov, Langevin, Lévy, Itô, Stratonovich, sólo por nombrar algunos; han abierto las puertas para un estudio bien fundamentado de la dinámica de sistemas físicos perturbados por ruido. En la presente tesis se discute el uso de ecuaciones diferenciales algebraicas estocásticas (EDAEs) para el modelado de sistemas multifísicos en red afectados por perturbaciones estocásticas, así como la evaluación de su estabilidad asintótica a través de exponentes de Lyapunov (ELs). El estudio está enfocado en EDAEs d-index-1 y su reformulación como ecuaciones diferenciales estocásticas ordinarias (EDEs). Fundamentados en la teoría ergódica, es factible analizar los ELs a través de sistemas dinámicos aleatorios (SDAs) generados por EDEs subyacentes. Una vez garantizada la existencia de ELs bien definidas, hemos procedido al uso de técnicas de simulación numérica para determinar los ELs numéricamente. Hemos implementado métodos numéricos basados en descomposición QR discreta y continua para el cómputo de la matriz de solución fundamental y su uso en el cálculo de los ELs. Las características numéricas y computacionales más relevantes de ambos métodos se ilustran mediante pruebas numéricas. Toda esta investigación sobre el modelado de sistemas con EDAEs y evaluación de su estabilidad a través de ELs calculados numéricamente, tiene una interesante aplicación en ingeniería. Esta es la evaluación de la estabilidad dinámica de sistemas eléctricos de potencia. En el presente trabajo de investigación, implementamos nuestros métodos numéricos basados en descomposición QR para el test de estabilidad dinámica en dos modelos de sistemas eléctricos de potencia de una-máquina bus-infinito (OMBI) afectados por diferentes perturbaciones ruidosas. El análisis en pequeña-señal evidencia el potencial de las técnicas propuestas en aplicaciones de ingeniería.[CA] La naturalesa aleatòria que caracteritza alguns fenòmens en sistemes físics reals (e.g., enginyeria, biologia, economia, finances, epidemiologia i uns altres) ens ha plantejat el desafiament d'un canvi de paradigma del modelatge matemàtic i l'anàlisi de sistemes dinàmics, i a tractar els fenòmens aleatoris com a variables aleatòries o processos estocàstics. Aquest enfocament nou ha portat com a conseqüència noves especificitats que la teoria clàssica del modelatge i anàlisi de sistemes dinàmics deterministes no ha pogut cobrir. Afortunadament, meravelloses contribucions, realitzades sobretot en l'últim segle, des del camp de les matemàtiques per científics com Kolmogorov, Langevin, Lévy, Itô, Stratonovich, només per nomenar alguns; han obert les portes per a un estudi ben fonamentat de la dinàmica de sistemes físics pertorbats per soroll. En la present tesi es discuteix l'ús d'equacions diferencials algebraiques estocàstiques (EDAEs) per al modelatge de sistemes multifísicos en xarxa afectats per pertorbacions estocàstiques, així com l'avaluació de la seua estabilitat asimptòtica a través d'exponents de Lyapunov (ELs). L'estudi està enfocat en EDAEs d-index-1 i la seua reformulació com a equacions diferencials estocàstiques ordinàries (EDEs). Fonamentats en la teoria ergòdica, és factible analitzar els ELs a través de sistemes dinàmics aleatoris (SDAs) generats per EDEs subjacents. Una vegada garantida l'existència d'ELs ben definides, hem procedit a l'ús de tècniques de simulació numèrica per a determinar els ELs numèricament. Hem implementat mètodes numèrics basats en descomposició QR discreta i contínua per al còmput de la matriu de solució fonamental i el seu ús en el càlcul dels ELs. Les característiques numèriques i computacionals més rellevants de tots dos mètodes s'illustren mitjançant proves numèriques. Tota aquesta investigació sobre el modelatge de sistemes amb EDAEs i avaluació de la seua estabilitat a través d'ELs calculats numèricament, té una interessant aplicació en enginyeria. Aquesta és l'avaluació de l'estabilitat dinàmica de sistemes elèctrics de potència. En el present treball de recerca, implementem els nostres mètodes numèrics basats en descomposició QR per al test d'estabilitat dinàmica en dos models de sistemes elèctrics de potència d'una-màquina bus-infinit (OMBI) afectats per diferents pertorbacions sorolloses. L'anàlisi en xicotet-senyal evidencia el potencial de les tècniques proposades en aplicacions d'enginyeria.[EN] The random nature that characterizes some phenomena in the real-world physical systems (e.g., engineering, biology, economics, finance, epidemiology, and others) has posed the challenge of changing the modeling and analysis paradigm and treat these phenomena as random variables or stochastic processes. Consequently, this novel approach has brought new specificities that the classical theory of modeling and analysis for deterministic dynamical systems cannot cover. Fortunately, stunning contributions made overall in the last century from the mathematics field by scientists such as Kolmogorov, Langevin, Lévy, Itô, Stratonovich, to name a few; have opened avenues for a well-founded study of the dynamics in physical systems perturbed by noise. In the present thesis, we discuss stochastic differential-algebraic equations (SDAEs) for modeling multi-physical network systems under stochastic disturbances, and their asymptotic stability assessment via Lyapunov exponents (LEs). We focus on d-index-1 SDAEs and their reformulation as ordinary stochastic differential equations (SDEs). Supported by the ergodic theory, it is feasible to analyze the LEs via the random dynamical system (RDSs) generated by the underlying SDEs. Once the existence of well-defined LEs is guaranteed, we proceed to the use of numerical simulation techniques to determine the LEs numerically. Discrete and continuous QR decomposition-based numerical methods are implemented to compute the fundamental solution matrix and use it in the computation of the LEs. Important numerical and computational features of both methods are illustrated through numerical tests. All this investigation concerning systems modeling through SDAEs and their stability assessment via computed LEs finds an appealing engineering application in the dynamic stability assessment of power systems. In this research work, we implement our QR-based numerical methods for testing the dynamic stability in two types of single-machine infinite-bus (SMIB) power system models perturbed by different noisy disturbances. The analysis in small-signal evidences the potential of the proposed techniques in engineering applications.Mi agradecimiento al estado ecuatoriano que, a través del Programa de Becas para el Fortalecimiento y Desarrollo del Talento Humano en Ciencia y Tecnología 2012 de la Secretaría Nacional de Educación Superior, Ciencia y Tecnología (SENESCYT), han financiado mis estudios de doctorado.González Zumba, JA. (2020). Dynamic Modeling and Stability Analysis of Stochastic Multi-Physical Systems Applied to Electric Power Systems [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/158558TESI

Detection, Prediction and Control of Epileptic Seizures

abstract: From time immemorial, epilepsy has persisted to be one of the greatest impediments to human life for those stricken by it. As the fourth most common neurological disorder, epilepsy causes paroxysmal electrical discharges in the brain that manifest as seizures. Seizures have the effect of debilitating patients on a physical and psychological level. Although not lethal by themselves, they can bring about total disruption in consciousness which can, in hazardous conditions, lead to fatality. Roughly 1\% of the world population suffer from epilepsy and another 30 to 50 new cases per 100,000 increase the number of affected annually. Controlling seizures in epileptic patients has therefore become a great medical and, in recent years, engineering challenge. In this study, the conditions of human seizures are recreated in an animal model of temporal lobe epilepsy. The rodents used in this study are chemically induced to become chronically epileptic. Their Electroencephalogram (EEG) data is then recorded and analyzed to detect and predict seizures; with the ultimate goal being the control and complete suppression of seizures. Two methods, the maximum Lyapunov exponent and the Generalized Partial Directed Coherence (GPDC), are applied on EEG data to extract meaningful information. Their effectiveness have been reported in the literature for the purpose of prediction of seizures and seizure focus localization. This study integrates these measures, through some modifications, to robustly detect seizures and separately find precursors to them and in consequence provide stimulation to the epileptic brain of rats in order to suppress seizures. Additionally open-loop stimulation with biphasic currents of various pairs of sites in differing lengths of time have helped us create control efficacy maps. While GPDC tells us about the possible location of the focus, control efficacy maps tells us how effective stimulating a certain pair of sites will be. The results from computations performed on the data are presented and the feasibility of the control problem is discussed. The results show a new reliable means of seizure detection even in the presence of artifacts in the data. The seizure precursors provide a means of prediction, in the order of tens of minutes, prior to seizures. Closed loop stimulation experiments based on these precursors and control efficacy maps on the epileptic animals show a maximum reduction of seizure frequency by 24.26\% in one animal and reduction of length of seizures by 51.77\% in another. Thus, through this study it was shown that the implementation of the methods can ameliorate seizures in an epileptic patient. It is expected that the new knowledge and experimental techniques will provide a guide for future research in an effort to ultimately eliminate seizures in epileptic patients.Dissertation/ThesisDoctoral Dissertation Electrical Engineering 201