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Kadakia, N, Rey D, Ye J, Abarbanel HDI.  2017.  Symplectic structure of statistical variational data assimilation. Quarterly Journal of the Royal Meteorological Society. 143:756-771.   10.1002/qj.2962   Abstract

Data assimilation variational principles (4D-Var) exhibit a natural symplectic structure among the state variables x(t) and. x(t). We explore the implications of this structure in both Lagrangian coordinates {x(t), x(t)} andHamiltonian canonical coordinates {x(t), p(t)} through a numerical examination of the chaotic Lorenz 1996 model in ten dimensions. We find that there are a number of subtleties associated with discretization, boundary conditions, and symplecticity, suggesting differing approaches when working in the the Lagrangian versus the Hamiltonian description. We investigate these differences in detail, and accordingly develop a protocol for searching for optimal trajectories in a Hamiltonian space. We find that casting the problem into canonical coordinates can, in some situations, considerably improve the quality of predictions.

Tenny, R, Tsimring LS, Larson L, Abarbanel HDI.  2003.  Using distributed nonlinear dynamics for public key encryption. Physical Review Letters. 90   10.1103/PhysRevLett.90.047903   AbstractWebsite

We introduce a new method for asymmetric (public key/private key) encryption exploiting properties of nonlinear dynamical systems. A high-dimensional dissipative nonlinear dynamical system is distributed between transmitter and receiver, so we call the method distributed dynamics encryption (DDE). The transmitter dynamics is public, and the receiver dynamics is hidden. A message is encoded by modulation of parameters of the transmitter, and this results in a shift of the overall system attractor. An unauthorized receiver does not know the hidden dynamics in the receiver and cannot decode the message. We present an example of DDE using a coupled map lattice.

Abarbanel, HDI, Kennel MB, Illing L, Tang S, Chen HF, Liu JM.  2001.  Synchronization and communication using semiconductor lasers with optoelectronic feedback. IEEE Journal of Quantum Electronics. 37:1301-1311.   10.1109/3.952542   AbstractWebsite

Semiconductor lasers provide an excellent opportunity for communication using chaotic waveforms. We discuss the characteristics and the synchronization of two semiconductor lasers with optoelectronic feedback. The systems exhibit broadband chaotic intensity oscillations whose dynamical dimension generally increases with the time delay in the feedback loop. We explore the robustness of this synchronization with parameter mismatch in the lasers, with mismatch in the optoelectronic feedback delay, and with the strength of the coupling between the systems. Synchronization is robust to mismatches between the intrinsic parameters of the lasers, but it is sensitive to mismatches of the time delay in the transmitter and receiver feedback loops. An open-loop receiver configuration Is suggested, eliminating feedback delay mismatch issues. Communication strategies for arbitrary amplitude of modulation onto the chaotic signals are discussed, and the bit-error rate for one such scheme is evaluated as a function of noise in the optical channel.

Varona, P, Torres JJ, Huerta R, Abarbanel HDI, Rabinovich MI.  2001.  Regularization mechanisms of spiking-bursting neurons. Neural Networks. 14:865-875.   10.1016/s0893-6080(01)00046-6   AbstractWebsite

An essential question raised after the observation of highly variable bursting activity in individual neurons of Central Pattern Generators (CPGs) is how an assembly of such cells can cooperatively act to produce regular signals to motor systems. It is well known that some neurons in the lobster stomatogastric ganglion have a highly irregular spiking-bursting behavior when they are synaptically isolated from any connection in the CPG. Experimental recordings show that periodic stimuli on a single neuron can regulate its firing activity. Other evidence demonstrates that specific chemical and/or electrical synapses among neurons also induce the regularization of the rhythms. In this paper we present a modeling study in which a slow subcellular dynamics, the exchange of calcium between an intracellular store and the cytoplasm, is responsible for the origin and control of the irregular spiking-bursting activity. We show this in simulations of single cells under periodic driving and in minimal networks where the cooperative activity can induce regularization. While often neglected in the description of realistic neuron models, subcellular processes with slow dynamics may play an important role in information processing and short-term memory of spiking-bursting neurons. (C) 2001 Elsevier Science Ltd. All rights reserved.

Szucs, A, Varona P, Volkovskii AR, Abarbanel HDI, Rabinovich MI, Selverston AI.  2000.  Interacting biological and electronic neurons generate realistic oscillatory rhythms. Neuroreport. 11:563-569. AbstractWebsite

Small assemblies of neurons such as central pattern generators (CPG) are known to express regular oscillatory firing patterns comprising bursts of action potentials. In contrast, individual CPG neurons isolated from the remainder of the network can generate irregular firing patterns. In our study of cooperative behavior in CPGs we developed an analog electronic neuron (EN) that reproduces firing patterns observed in lobster pyloric CPG neurons. Using a tuneable artificial synapse we connected the EN bidirectionally to neurons of this CPG. We found that the periodic bursting oscillation of this mixed assembly depends on the strength and sign of the electrical coupling. Working with identified, isolated pyloric CPG neurons whose network rhythms were impaired, the EN/biological network restored the characteristic CPG rhythm both when the EN oscillations are regular and when they are irregular. NeuroReport 11:563-569 (C) 2000 Lippincon Williams & Wilkins.

Pinto, RD, Varona P, Volkovskii AR, Szucs A, Abarbanel HDI, Rabinovich MI.  2000.  Synchronous behavior of two coupled electronic neurons. Physical Review E. 62:2644-2656.   10.1103/PhysRevE.62.2644   AbstractWebsite

We report on experimental studies of synchronization phenomena in a pair of analog electronic neurons (ENs). The ENs were designed to reproduce the observed membrane voltage oscillations of isolated biological neurons from the stomatogastric ganglion of the California spiny lobster Panulirus interruptus. The ENs are simple analog circuits which integrate four-dimensional differential equations representing fast and slow subcellular mechanisms that produce the characteristic regular/chaotic spiking-bursting behavior of these cells. In this paper we study their dynamical behavior as we couple them in the same configurations as we have done for their counterpart biological neurons. The interconnections we use for these neural oscillators are both direct electrical connections and excitatory and inhibitory chemical connections: each realized by analog circuitry and suggested by biological examples. We provide here quantitative evidence that the ENs and the biological neurons behave similarly when coupled in the same manner. They each display well defined bifurcations in their mutual synchronization and regularization. We report briefly on an experiment on coupled biological neurons and four-dimensional ENs, which provides further ground for testing the validity of our numerical and electronic models of individual neural behavior. Our experiments as a whole present interesting new examples of regularization and synchronization in coupled nonlinear oscillators.

Sushchik, M, Rulkov N, Larson L, Tsimring L, Abarbanel H, Yao K, Volkovskii A.  2000.  Chaotic pulse position modulation: A robust method of communicating with chaos. IEEE Communications Letters. 4:128-130.   10.1109/4234.841319   AbstractWebsite

In this letter we investigate a communication strategy for digital ultra-wide bandwidth impulse radio, where the separation between the adjacent pulses is chaotic arising from a dynamical system with irregular behavior. A pulse position method is used to modulate binary information onto the carrier. The receiver is synchronized to the chaotic pulse train, thus providing the time reference for information extraction. We characterize the performance of this scheme in terms of error probability versus E-b/N-o by numerically simulating its operation in the presence of noise and filtering.

Rabinovich, MI, Abarbanel HDI.  1998.  The role of chaos in neural systems. Neuroscience. 87:5-14.   10.1016/s0306-4522(98)00091-8   AbstractWebsite

The ideas of dynamical chaos have altered our understanding of the origin of random appearing behavior in man fields of physics and engineering. In the 1980s and 1990s these new viewpoints about apparent random oscillations arising in deterministic systems were investigated in neurophysiology and have led to quite successful reports of chaos in experimental and theoretical investigations. This paper is a "view" paper addressing the role of chaos in living systems, not just reviewing the evidence for its existence, and in particular we ask about the utility of chaotic behavior in nervous systems. From our point of view chaotic oscillations of individual neurons may nor be essential for the observed activity of neuronal assemblies bur may, instead, be responsible for the multitude of regular regimes of operation that can be accomplished by elements which are chaotic. The organization of chaotic elements in assemblies where their synchronization can result in organized adaptive and reliable activities may lead to general principles used by nature in accomplishing critical functional goals. (C) 1998 IBRO. Published by Elsevier Science Ltd.

Sushchik, MM, Rulkov NF, Abarbanel HDI.  1997.  Robustness and stability of synchronized chaos: An illustrative model. IEEE Transactions on Circuits and Systems I-Fundamental Theory and Applications. 44:867-873.   10.1109/81.633875   AbstractWebsite

Synchronization of two chaotic systems is not guaranteed by having only negative conditional or transverse Lyapunov exponents, If there are transversally unstable periodic orbits or fixed points embedded in the chaotic set of synchronized motions, the presence of even very small disturbances from noise or inaccuracies from parameter mismatch can cause synchronization to break down and lead to substantial amplitude excursions from the synchronized state. Using an example of coupled one dimensional chaotic maps we discuss the conditions required for robust synchronization and study a mechanism that is responsible for the failure of negative conditional Lyapunov exponents to determine the conditions for robust synchronization.

Rabinovich, MI, Abarbanel HDI, Huerta R, Elson R, Selverston AI.  1997.  Self-regularization of chaos in neural systems: Experimental and theoretical results. IEEE Transactions on Circuits and Systems I-Fundamental Theory and Applications. 44:997-1005.   10.1109/81.633889   AbstractWebsite

The results of neurobiological studies in both vertebrates and invertebrates lead to the general question: How is a population of neurons, whose individual activity is chaotic and uncorrelated able to form functional circuits with regular and stable behavior? What are the circumstances which support these regular oscillations? What are the mechanisms that promote this transition? We address these questions using our experimental and modeling studies describing the behavior of groups of spiking-bursting neurons. We show that the role of inhibitory synaptic coupling between neurons is crucial in the self-control of chaos.

Abarbanel, HDI, Korzinov L, Mees AI, Rulkov NF.  1997.  Small force control of nonlinear systems to given orbits. IEEE Transactions on Circuits and Systems I-Fundamental Theory and Applications. 44:1018-1023.   10.1109/81.633894   AbstractWebsite

Using a low frequency nonlinear electrical circuit, se experimentally demonstrate an efficient nonlinear control method based on our theoretical developments. The method works in a state space for. the circuit which is reconstructed from observations of a single voltage. Assuming small control variations from the uncontrolled state, the method is fully nonlinear and ''one step'' optimal. It requires no knowledge of local state space linearizations of the dynamics near the target state. Starting from various initial states within the basin of attraction of the circuit attractor, we control to a period one and to a period two target orbit Each target orbit is an unstable periodic orbit of the uncontrolled system.

Abarbanel, HDI, Katz RA, Galib T, Cembrola J, Frison TW.  1994.  Nonlinear analysis of high-Reynolds-number flows over a buoyant axisymmetrical body. Physical Review E. 49:4003-4018.   10.1103/PhysRevE.49.4003   AbstractWebsite

Data from experiments on the turbulent boundary layer around an axisymmetric vehicle rising under its own buoyancy are described in detail and analyzed using tools developed in nonlinear dynamics. Arguments are given that in this experiment the size of the wall mounted pressure sensors would make the data sensitive to the dynamics of about ten or so coherent structures in the turbulent boundary layer. Analysis of a substantial number of large, well sampled data sets indicates that the (integer) dimension of the embedding space required to capture the dynamics of the observed flows in the laminar regime is very large. This is consistent with there being no pressure fluctuations expected here and the signal being dominated by instrumental ''noise.'' In a consistency check we find that data from the ambient state of the vehicle before buoyant rise occurs and data from an accelerometer mounted in the prow are also consistent with this large dimension. The time scales in those data are also unrelated to fluid dynamic phenomena. In the transition and turbulent regions of the flow we find the pressure fluctuation time scales to be consistent with those of the fluid flow (about 250 musec) and determine the dimension required for embedding the data to be about 7-8 for the transitional region and about 8-9 for the turbulent regime. These results are examined in detail using both global and local false nearest-neighbor methods as well as mutual information aspects of the data. The results indicate that the pressure fluctuations are determined in these regimes by the coherent structures in the turbulent boundary layer. Applications and further investigations suggested by these results are discussed.

Abarbanel, HDI, Carroll TA, Pecora LM, Sidorowich JJ, Tsimring LS.  1994.  Predicting physical variables in time delay embedding. Physical Review E. 49:1840-1853.   10.1103/PhysRevE.49.1840   AbstractWebsite

In time-delay reconstruction of chaotic attractors we can accurately predict the short-term future behavior of the observed variable x(t) = x(n) = x(t0 + tau(s)n) without prior knowledge of the equations of motion by building local or global models in the state space. In many cases we also want to predict variables other than the one which is observed and require methods for determining models to predict these variables in the same space. We present a method which takes measurements of two variables x(n) and z(n) and builds models for the determination of z(n) in the phase-space made out of the x(n) and its time lags. Similarly we show that one may produce models for x(n) in the z(n) space, except where special symmetries prevent this, such as in the familiar Lorenz model. Our algorithm involves building local polynomial models in the reconstructed phase space of the observed variable of low order (linear or quadratic) which approximate the function z(n) = F(x(n)) where x(n) is a vector constructed from a sequence of values of observed variables in a time delay fashion. We train the models on a partial data set of measured values of both x(n) and z(n) and then predict the z(n) in a recovery set of observations of x(n) alone. In all of our analyses we assume that the observed data alone are available to us and that we possess no knowledge of the dynamical equations. We test this method on the numerically generated data set from the Lorenz model and also on a number of experimental data sets from electronic circuits.

Abarbanel, HDI, Brown R, Sidorowich JJ, Tsimring LS.  1993.  The analysis of observed chaotic data in physical systems. Reviews of Modern Physics. 65:1331-1392.   10.1103/RevModPhys.65.1331   AbstractWebsite

Chaotic time series data are observed routinely in experiments on physical systems and in observations in the field. The authors review developments in the extraction of information of physical importance from such measurements. They discuss methods for (1) separating the signal of physical interest from contamination (''noise reduction''), (2) constructing an appropriate state space or phase space for the data in which the full structure of the strange attractor associated with the chaotic observations is unfolded, (3) evaluating invariant properties of the dynamics such as dimensions, Lyapunov exponents, and topological characteristics, and (4) model making, local and global, for prediction and other goals. They briefly touch on the effects of linearly filtering data before analyzing it as a chaotic time series. Controlling chaotic physical systems and using them to synchronize and possibly communicate between source and receiver is considered. Finally, chaos in space-time systems, that is, the dynamics of fields, is briefly considered. While much is now known about the analysis of observed temporal chaos, spatio-temporal chaotic systems pose new challenges. The emphasis throughout the review is on the tools one now has for the realistic study of measured data in laboratory and field settings. lt is the goal of this review to bring these tools into general use among physicists who study classical and semiclassical systems. Much of the progress in studying chaotic systems has rested on computational tools with some underlying rigorous mathematics. Heuristic and intuitive analysis tools guided by this mathematics and realizable on existing computers constitute the core of this review.

Brown, R, Bryant P, Abarbanel HDI.  1991.  Computing the Lyapunov spectrum of a dynamic system from an observed time series. Physical Review A. 43:2787-2806.   10.1103/PhysRevA.43.2787   AbstractWebsite

We examine the question of accurately determining, from an observed time series, the Lyapunov exponents for the dynamical system generating the data. This includes positive, zero, and some or all of the negative exponents. We show that even with very large data sets, it is clearly advantageous to use local neighborhood-to-neighborhood mappings with higher-order Taylor series, rather than just local linear maps as has been done previously. We give examples using up to fifth-order polynomials. We demonstrate this procedure on two familiar maps and two familiar flows: the Henon and Ikeda maps of the plane to itself, the Lorenz system of three ordinary differential equations, and the Mackey-Glass delay differential equation. We stress the importance of maintaining two dimensions for converting the scalar data into time delay vectors: one is a global dimension to ensure proper unfolding of the attractor as a whole, and the other is a local dimension for capturing the local dynamics on the attractor. We show the effects of changing the local and global dimensions, changing the order of the mapping polynomial, and additive (measurement) noise. There will always be some limit to the number of exponents that can be accurately determined from a given finite data set. We discuss a method of determining this limit by numerically obtaining the singularity spectra of the data set and also show how it is often appropriate to make this choice based on the fractal dimension of the attractor. If excessively large dimensions are used, spurious exponents will be generated, and in some cases the accuracy of the true exponents will be affected. We present methods of identifying these spurious exponents by determining the Lyapunov direction vectors at particular points in the data set. We can then use these to identify numerical problems and to associate data-set singularities with particular exponents. The behavior of spurious exponents in the presence of noise is also investigated, and found to be different from that of the true exponents. These provide methods for identifying spurious exponents in the analysis of experimental data where the system dynamics may not be known a priori.