Low-dimensional Dynamical Systems Research Articles

Evidence for crisis-induced intermittency during geomagnetic superchron transitions.

The geomagnetic field's dipole undergoes polarity reversals in irregular time intervals. Particularly long periods without reversals (of the order of 10^{7} yr), called superchrons, have occurred at least three times in the Phanerozoic (since 541 million years ago). We provide observational evidence for high non-Gaussianity in the vicinity of a transition to and from a geomagnetic superchron, consisting of a sharp increase in high-order moments (skewness and kurtosis) of the dipole's distribution. Such an increase in the moments is a universal feature of crisis-induced intermittency in low-dimensional dynamical systems undergoing global bifurcations. This implies a temporal variation of the underlying parameters of the physical system. Through a low-dimensional system that models the geomagnetic reversals, we show that the increase in the high-order moments during transitions to geomagnetic superchrons is caused by the progressive destruction of global periodic orbits exhibiting both polarities as the system approaches a merging bifurcation. We argue that the non-Gaussianity in this system is caused by the redistribution of the attractor around local cycles as global ones are destroyed.

A reduced-order model of thermoacoustic instability in solid rocket motors

A finite-dimensional dynamical system is derived from a coupled system of partial differential equations modeling the thermoacoustic oscillations inside a solid rocket motor. The governing equations for the coupling between the chamber acoustics and combustion of the solid propellant are derived from conservation laws and existing combustion models. Model reduction is carried out for the unsteady burning rate model based on a pseudo-spectral truncation, leading to a three-dimensional dynamical system. The resulting low-dimensional dynamical system is shown to be able to capture the main bifurcation behavior of the original infinite-dimensional system. The reduced-order model can be used to predict the nonlinear thermoacoustic behavior of a solid rocket motor at reduced computation costs.

Bayesian framework for simulation of dynamical systems from multidimensional data using recurrent neural network.

We suggest a new method for building data-driven dynamical models from observed multidimensional time series. The method is based on a recurrent neural network with specific structure, which allows for the joint reconstruction of both a low-dimensional embedding for dynamical components in the data and an evolution operator. The key link of the method is a Bayesian optimization of both model structure and the hypothesis about the data generating law, which is needed for constructing the cost function for model learning. First, the performance of the method is successfully tested in the situation when a signal from a low-dimensional dynamical system is hidden in noisy multidimensional observations. Second, the method is used for building the data-driven model of the low frequency variability (LFV) in the quasigeostrophic model of the Earth's midlatitude atmosphere-a high-dimensional chaotic system. It is demonstrated that the key regimes of the atmospheric LFV are reproduced correctly in data simulations by means of the obtained model.

Adaptive path-integral autoencoder: representation learning and planning for dynamical systems**This article is an updated version of: Ha J-S, Park Y-J, Chae H-J, Park S-S and Choi H-L 2018 Adaptive path-integral autoencoders: representation learning and planning for dynamical systems Advances in Neural Information Processing Systems vol 31, ed S Bengio et al (Red Hook, NY: Curran Associates, Inc.) pp 8927–38.

We present a representation learning algorithm that learns a low-dimensional latent dynamical system from high-dimensional sequential raw data, e.g. video. The framework builds upon recent advances in amortized inference methods that use both an inference network and a refinement procedure to output samples from a variational distribution given an observation sequence, and takes advantage of the duality between control and inference to approximately solve the intractable inference problem using the path integral control approach. The learned dynamical model can be used to predict and plan the future states; we also present the efficient planning method that exploits the learned low-dimensional latent dynamics. Numerical experiments show that the proposed path-integral control based variational inference method leads to tighter lower bounds in statistical model learning of sequential data. The supplementary video (https://youtu.be/xCp35crUoLQ) and the implementation code (https://github.com/yjparkLiCS/18-NIPS-APIAE) are available online.

Propagation of orientation selectivity in a spiking network model of layered primary visual cortex.

Neurons in different layers of sensory cortex generally have different functional properties. But what determines firing rates and tuning properties of neurons in different layers? Orientation selectivity in primary visual cortex (V1) is an interesting case to study these questions. Thalamic projections essentially determine the preferred orientation of neurons that receive direct input. But how is this tuning propagated though layers, and how can selective responses emerge in layers that do not have direct access to the thalamus? Here we combine numerical simulations with mathematical analyses to address this problem. We find that a large-scale network, which just accounts for experimentally measured layer and cell-type specific connection probabilities, yields firing rates and orientation selectivities matching electrophysiological recordings in rodent V1 surprisingly well. Further analysis, however, is complicated by the fact that neuronal responses emerge in a dynamic fashion and cannot be directly inferred from static neuroanatomy, as some connections tend to have unintuitive effects due to recurrent interactions and strong feedback loops. These emergent phenomena can be understood by linearizing and coarse-graining. In fact, we were able to derive a low-dimensional linear dynamical system effectively describing stimulus-driven activity layer by layer. This low-dimensional system explains layer-specific firing rates and orientation tuning by accounting for the different gain factors of the aggregate system. Our theory can also be used to design novel optogenetic stimulation experiments, thus facilitating further exploration of the interplay between connectivity and function.

Beyond Boltzmann-Gibbs-Shannon in Physics and Elsewhere.

The pillars of contemporary theoretical physics are classical mechanics, Maxwell electromagnetism, relativity, quantum mechanics, and Boltzmann–Gibbs (BG) statistical mechanics –including its connection with thermodynamics. The BG theory describes amazingly well the thermal equilibrium of a plethora of so-called simple systems. However, BG statistical mechanics and its basic additive entropy started, in recent decades, to exhibit failures or inadequacies in an increasing number of complex systems. The emergence of such intriguing features became apparent in quantum systems as well, such as black holes and other area-law-like scenarios for the von Neumann entropy. In a different arena, the efficiency of the Shannon entropy—as the BG functional is currently called in engineering and communication theory—started to be perceived as not necessarily optimal in the processing of images (e.g., medical ones) and time series (e.g., economic ones). Such is the case in the presence of generic long-range space correlations, long memory, sub-exponential sensitivity to the initial conditions (hence vanishing largest Lyapunov exponents), and similar features. Finally, we witnessed, during the last two decades, an explosion of asymptotically scale-free complex networks. This wide range of important systems eventually gave support, since 1988, to the generalization of the BG theory. Nonadditive entropies generalizing the BG one and their consequences have been introduced and intensively studied worldwide. The present review focuses on these concepts and their predictions, verifications, and applications in physics and elsewhere. Some selected examples (in quantum information, high- and low-energy physics, low-dimensional nonlinear dynamical systems, earthquakes, turbulence, long-range interacting systems, and scale-free networks) illustrate successful applications. The grounding thermodynamical framework is briefly described as well.

The Lorenz Renormalization Conjecture

Universality in low-dimensional dynamical systems is the remarkable phenomenon that the geometry of a system at a phase transition, as well as the bifurcation patterns of nearby systems, is determined by its topology. This can be explained by an associated renormalization operator being a contraction on topological conjugacy classes. For more physically realistic systems this paradigm is too simple. In the context of Lorenz dynamics, which is associated with homoclinic bifurcations of flows, the renormalization behavior is much richer. Depending on topology, one finds either the traditional universality or two more intricate classes of dynamics. A conjecture describing the dynamics of the Lorenz renormalization operator in terms of these three classes of behavior is stated and the consequences each case has on the dynamics of Lorenz maps is discussed. Numerical evidence supporting the conjecture is provided.

A Study of the Cross-Scale Causation and Information Flow in a Stormy Model Mid-Latitude Atmosphere.

A fundamental problem regarding the storm–jet stream interaction in the extratropical atmosphere is how energy and information are exchanged between scales. While energy transfer has been extensively investigated, the latter has been mostly overlooked, mainly due to a lack of appropriate theory and methodology. Using a recently established rigorous formalism of information flow, this study attempts to examine the problem in the setting of a three-dimensional quasi-geostrophic zonal jet, with storms excited by a set of optimal perturbation modes. We choose for this study a period when the self-sustained oscillation is in quasi-equilibrium, and when the energetics mimick the mid-latitude atmospheric circulation where available potential energy is cascaded downward to smaller scales, and kinetic energy is inversely transferred upward toward larger scales. By inverting a three-dimensional elliptic differential operator, the model is first converted into a low-dimensional dynamical system, where the components correspond to different time scales. The information exchange between the scales is then computed through ensemble prediction. For this particular problem, the resulting cross-scale information flow is mostly from smaller scales to larger scales. That is to say, during this period, this model extratropical atmosphere is dominated by a bottom-up causation, as collective patterns emerge out of independent entities and macroscopic thermodynamic properties evolve from random molecular motions. This study makes a first step toward an important field in understanding the eddy–mean flow interaction in weather and climate phenomena such as atmospheric blocking, storm track, North Atlantic Oscillation, to name a few.

Strong regularity

This is an introduction of a book called regularity, to appear at Ast\'erisque, containing: 1) Yoccoz' proof of Jakobson theorem www.college-de-france.fr/media/jean-christophe-yoccoz/UPL7416254474776698194_Jakobson_jcy.pdf 2) Berger's proof of the abundance of non-uniformly hyperbolic H\'enon like endomorphisms arxiv.org/abs/0903.1473 It gives an overview of the main examples and conjectures of non-uniformly hyperbolic set for low dimensional dynamical systems. It compares the proofs of parameter selections based on the concept of binding with those based on the one of strong regularity.

Cutting and shuffling with diffusion: Evidence for cut-offs in interval exchange maps.

Low-dimensional dynamical systems are fruitful models for mixing in fluid and granular flows. We study a one-dimensional discontinuous dynamical system (termed "cutting and shuffling" of a line segment), and we present a comprehensive computational study of its finite-time mixing properties including the effect of diffusion. To explore a large parameter space, we introduce fit functions for two mixing metrics of choice: the number of cutting interfaces (a standard quantity in dynamical systems theory of interval exchange transformations) and a mixing norm (a more physical measure of mixing). We compute averages of the mixing metrics across different permutations (shuffling protocols), showing that the latter averages are a robust descriptor of mixing for any permutation choice. If the decay of the normalized mixing norm is plotted against the number of map iterations rescaled by the characteristic e-folding time, then universality emerges: mixing norm decay curves across all cutting and shuffling protocols collapse onto a single stretched-exponential profile. Next, we predict this critical number of iterations using the average length of unmixed subsegments of continuous color during cutting and shuffling and a Batchelor-scale-type diffusion argument. This prediction, called a "stopping time" for finite Markov chains, compares well with the e-folding time of the stretched-exponential fit. Finally, we quantify the effect of diffusion on cutting and shuffling through a Péclet number (a dimensionless inverse diffusivity), showing that the system transitions more sharply from an unmixed initial state to a mixed final state as the Péclet number becomes large. Our numerical investigation of cutting and shuffling of a line segment in the presence of diffusion thus presents evidence for the latter phenomenon, known as a "cut-off" for finite Markov chains, in interval exchange maps.

Finite-time scaling in local bifurcations

Finite-size scaling is a key tool in statistical physics, used to infer critical behavior in finite systems. Here we have made use of the analogous concept of finite-time scaling to describe the bifurcation diagram at finite times in discrete (deterministic) dynamical systems. We analytically derive finite-time scaling laws for two ubiquitous transitions given by the transcritical and the saddle-node bifurcation, obtaining exact expressions for the critical exponents and scaling functions. One of the scaling laws, corresponding to the distance of the dynamical variable to the attractor, turns out to be universal, in the sense that it holds for both bifurcations, yielding the same exponents and scaling function. Remarkably, the resulting scaling behavior in the transcritical bifurcation is precisely the same as the one in the (stochastic) Galton-Watson process. Our work establishes a new connection between thermodynamic phase transitions and bifurcations in low-dimensional dynamical systems, and opens new avenues to identify the nature of dynamical shifts in systems for which only short time series are available.

Dynamical transitions of a low-dimensional model for Rayleigh–Bénard convection under a vertical magnetic field

In this article, we study the dynamic transitions of a low-dimensional dynamical system for the Rayleigh–Bénard convection subject to a vertically applied magnetic field. Our analysis follows the dynamical phase transition theory for dissipative dynamical systems based on the principle of exchange of stability and the center manifold reduction. We find that, as the Rayleigh number increases, the system undergoes two successive transitions: the first one is a well-known pitchfork bifurcation, whereas the second one is structurally more complex and can be of different type depending on the system parameters. More precisely, for large magnetic field, the second transition is of continuous type and gives to a stable limit cycle; on the other hand, for low magnetic field or small height-to-width aspect ratio, a jump transition occurs where an unstable periodic orbit eventually collides with the stable steady state, leading to the loss of stability at the critical Rayleigh number. Finally, numerical results are presented to corroborate the analytic predictions.

On the Validity of Linear Response Theory in High-Dimensional Deterministic Dynamical Systems

This theoretical work considers the following conundrum: linear response theory is successfully used by scientists in numerous fields, but mathematicians have shown that typical low-dimensional dynamical systems violate the theory's assumptions. Here we provide a proof of concept for the validity of linear response theory in high-dimensional deterministic systems for large-scale observables. We introduce an exemplary model in which observables of resolved degrees of freedom are weakly coupled to a large, inhomogeneous collection of unresolved chaotic degrees of freedom. By employing statistical limit laws we give conditions under which such systems obey linear response theory even if all the degrees of freedom individually violate linear response. We corroborate our result with numerical simulations.

Robust Guidance Strategy for Target Circulation by Controlled UAV

A controlled fixed-wing unmanned aerial vehicle is modeled as a simplified low-dimensional nonlinear dynamical system with additive disturbances. The guidance strategy presented, considering simultaneously inputs and states constraints, is shown to be robust to additive perturbations representing norm-bounded uncertainties associated with modeling errors resulting from the lower order model approximation, together with unknown but limited atmospheric disturbances. Mathematical proofs are provided ensuring robustness in achieving target circulation with a minimum prescribed accuracy.

Empirical evaluated SDE modelling for dimensionality-reduced systems and its predictability estimates

This paper develops and validates a method of empirical modelling for a dimensionality-reduced system of a nonlinear dynamical system based on the framework of the stochastic differential equation (SDE). Following the mathematical theorem corresponding to some inverse problem of the probability theory, we derive the empirically evaluating formulae for the drift vector and diffusion matrix. Focusing on a low-dimensional dynamical system of the Lorenz system, we empirically reconstruct an SDE that approximates the original time-series on the projected 2-dimensional plane. The distribution of the ensemble variance of solutions generated by the numerical SDE well agrees with that of the trajectories of the projected time-series, which indicates the ability of the SDE modelling to represent local predictability. Moreover, we also compare our SDE constructing method with the conventional Mori–Zwanzig projected operator method, which is used to derive a generalised Langevin equation for dimensionality-reduced systems, to assess the applicability of the obtained SDE model derived by the presented method.

Pair creation, motion, and annihilation of topological defects in two-dimensional nematic liquid crystals.

We present a framework for the study of disclinations in two-dimensional active nematic liquid crystals and topological defects in general. The order tensor formalism is used to calculate exact multiparticle solutions of the linearized static equations inside a planar uniformly aligned state so that the total charge has to vanish. Topological charge conservation then requires that there is always an equal number of q=1/2 and q=-1/2 charges. Starting from a set of hydrodynamic equations, we derive a low-dimensional dynamical system for the parameters of the static solutions, which describes the motion of a half-disclination pair or of several pairs. Within this formalism, we model defect production and annihilation, as observed in experiments. Our dynamics also provide an estimate for the critical density at which production and annihilation rates are balanced.

Optimal curing policy for epidemic spreading over a community network with heterogeneous population

The design of an efficient curing policy, able to stem an epidemic process at an affordable cost, has to account for the structure of the population contact network supporting the contagious process. Thus, we tackle the problem of allocating recovery resources among the population, at the lowest cost possible to prevent the epidemic from persisting indefinitely in the network. Specifically, we analyse a susceptible–infected–susceptible epidemic process spreading over a weighted graph, by means of a first-order mean-field approximation. First, we describe the influence of the contact network on the dynamics of the epidemics among a heterogeneous population, that is possibly divided into communities. For the case of a community network, our investigation relies on the graph-theoretical notion of equitable partition; we show that the epidemic threshold, a key measure of the network robustness against epidemic spreading, can be determined using a lower-dimensional dynamical system. Exploiting the computation of the epidemic threshold, we determine a cost-optimal curing policy by solving a convex minimization problem, which possesses a reduced dimension in the case of a community network. Lastly, we consider a two-level optimal curing problem, for which an algorithm is designed with a polynomial time complexity in the network size.

Tight space-noise tradeoffs in computing the ergodic measure

In this paper we obtain tight bounds on the space-complexity of computing the ergodic measure of a low-dimensional discrete-time dynamical system affected by Gaussian noise. If the scale of the noise is , and the function describing the evolution of the system is not itself a source of computational complexity, then the density function of the ergodic measure can be approximated within precision in space polynomial in . We also show that this bound is tight up to polynomial factors. In the course of showing the above, we prove a result of independent interest in space-bounded computation: namely, that it is possible to exponentiate an -matrix to an exponentially large power in space polylogarithmic in . Bibliography: 25 titles.

Chaotic analysis of embodied and situated agents

Embodied and situated view of cognition is a transdisciplinary framework which stresses the importance of real time and dynamical interaction of an agent with the surrounding environment. This article presents a series of evolutionary robotics experiments that operationalize such concept, training miniature two-wheeled mobile robots to autonomously solve a temporal task. In order to provide a numerical description of the robots’ behavior, chaotic measures are estimated on the attractor reconstructed from the recorded positions of the agent. Chaos theory provides a rigorous mathematical framework consistent with an antireductionist approach, useful for understanding embodied and situated systems while avoiding a decomposition of the integrated system brain–body–environment. Time series are analyzed in detail using nonlinear mathematical tools in order to verify the presence of low-dimensional deterministic dynamical systems, a fundamental prerequisite for chaos theory. In particular, the recorded time series are evaluated with nonlinear prediction error to unveil deterministic dynamics, cross-prediction error to determine the stationarity of the signal, and surrogate data testing to verify the existence of nonlinear components in the underlying system. Estimators for quantifying level of chaos and fractal dimension are applied to suitable datasets. Results show that robots governed by a chaotic dynamic are more efficient at adapting to environments never experience during evolution, demonstrating robustness towards novel and unpredictable situations. Furthermore, chaotic measures, in particular fractal dimension, are correlated with the performance if robots exhibit a similar behavioral strategy.

Energy Based Limit Cycle Control of Elastically Actuated Robots

A new control law for elastic joint robots that allows to regulate an energy function of the system to a desired value is presented in this technical note. Being able to either remove energy from the system or inject into it, oscillations can be both damped out and induced. The proposed nonlinear dynamic state feedback controller forces the system to evolve on a submanifold of the configuration space. The reduced dynamics of the system and of the controller itself are similar to a single elastic joint, for which an asymptotically stable limit cycle is obtained regulating an energy function to a positive desired value. When the desired value of the energy function is chosen to be zero, then the asymptotically stable limit cycle reduces to an asymptotically stable equilibrium point. In this case the oscillations are damped out and the desired task-space configuration is reached. The design of the controller extensively uses the concept of conditional stability, so that the limit cycle can be designed for a lower dimensional dynamical system, although it will result to be a limit cycle for the whole system.