Lyapunov Time Research Articles

On the timescales in the chaotic dynamics of a 4D symplectic map.

In this work, we investigate different timescales of chaotic dynamics in a multi-parametric 4D symplectic map. We compute the Lyapunov time and a macroscopic timescale, the instability time, for a wide range of values of the system's parameters and many different ensembles of initial conditions in resonant domains. The instability time is obtained by plain numerical simulations and by its estimates from the diffusion time, which we derive in three different ways: through a normal and an anomalous diffusion law and by the Shannon entropy, whose formulation is briefly revisited. A discussion about which of the four approaches provide reliable values of the timescale for a macroscopic instability is addressed. The relationship between the Lyapunov time and the instability time is revisited and studied for this particular system where in some cases, an exponential or polynomial law has been observed. The main conclusion of the present research is that only when the dynamical system behaves as a nearly ergodic one such relationship arises and the Lyapunov and instability times are global timescales, independent of the position in phase space. When stability regions prevent the free diffusion, no correlations between both timescales are observed, they are local and depend on both the position in phase space and the perturbation strength. In any case, the instability time largely exceeds the Lyapunov time. Thus, when the system is far from nearly ergodic, the timescale for predictable dynamics is given by the instability time, being the Lyapunov time its lower bound.

On the predictability of possible storylines for forced complex systems

Abstract It is well-known that even for fairly simple deterministic nonlinear systems, exact prediction of future state is, on average, impossible beyond some small multiple of the Lyapunov time that quantifies the rate of separation of trajectories within an attractor. Nonetheless, it may be possible to find a physical measure that is the distribution of a trajectory within the attractor. In that sense, there can be a still weaker form of predictability. In this paper, we show that this can also fail but an even weaker form of predictability can appear for non-autonomous (i.e. forced) systems in the presence of tipping points. The {\em predictability of possible storylines} appears when one can interpret the frequencies of runs within an ensemble arriving at one of several possible future attractors (storylines) in a probabilistic manner.
As predictability is a major concern and a challenge in climate science, we illustrate this notion of predictability with two climate-related examples: a chaotic energy balance model and a global ocean model featuring a tipping point of the Atlantic Meridional Overturning Circulation.

NEOROCKS color survey: Final results

Context. Near-Earth objects (NEOs) are the most accessible small Solar System bodies by both spacecrafts and ground-based telescopes. Close encounters of these objects with Earth represent opportunities to characterize their physical and mineralogical properties. They are also a constant threat to humanity due to possible impact events with Earth. In this context, the NEOROCKS project has been financed by the European Union’s Horizon 2020 research and innovation program. Aims. We present the final results on photometry of the NEOROCKS project, with the aim of extending the dataset of surface colors for small NEOs with unknown properties and, when possible, characterizing newly discovered NEOs. Methods. Photometric observations were performed using the 1.2 m telescope at the Haute-Provence observatory (in France) in the BVRI filters of the Johnson-Cousins photometric systems between May 2022 and June 2023. The stability and dynamics of objects from the NEOROCKS database was investigated by numerical integration. Results. We obtained new surface colors for 83 NEOs. Overall, the NEOROCKS color database contains 170 objects. The majority of the objects in the dataset with diameters D<500 m belong to a group of silicate bodies. We estimated the unbalanced percentage between S- and C-type objects as an observational bias due to reflective proprieties of the surface of objects. The average of Lyapunov time of about 100 years is evidence of highly chaotic orbits of objects from the color database of NEOROCKS. Asteroid 2011 OL51 has a reasonable probability of being a parent body contributor to the October Capricornidis meteor shower. Asteroids 2004 HK33, 2022 VV (D-type), 2003 WR21, and 2017 SE1 (A-type) belong to end-member classes and have ΔV<7 km/s; thus, they are possible candidates for in situ investigations.

Physical parameters and orbital evolution of asteroids in retrograde orbits

Context. We studied the dynamical orbital and physical evolution of all 21 numbered and 13 selected unnumbered asteroids in retrograde orbits. Aims. Based on all published observations of studied asteroids in retrograde orbits, we computed their starting orbital elements, absolute magnitudes, and diameters, together with the non-gravitational parameters A2 and da/dt. Methods. Using publicly available orbital computation methods, we studied the dynamical evolution of orbital elements and the physical parameters of asteroids: rotation period, spin direction, and the non-gravitational parameters A2. Results. Lyapunov times (LT) for studied asteroids are short, from 60 to 36 496 yr, with a mean of 5978. Without considering non-gravitational effects, LT is longer: values range from 328 to 63 165 yr, with a mean of 6392. Over the next 10 Myr and beyond, the rotation period of the studied asteroids P decreases by approximately 8%. Moreover, 15% of the clones slow down. Conclusions. The starting spin distribution becomes flatter, with only one large maximum in the range (0–5) deg containing 279 clones (i.e. approximately 13%). However, the non-gravitational parameter da/dt has a maximum value of around (0–0.04)× 10−5 au day−2 .

Model adaptive phase space reconstruction.

Phase space reconstruction (PSR) methods allow for the analysis of low-dimensional data with methods from dynamical systems theory, but their application to prediction models, such as those from machine learning (ML), is limited. Therefore, we here present a model adaptive phase space reconstruction (MAPSR) method that unifies the process of PSR with the modeling of the dynamical system. MAPSR is a differentiable PSR based on time-delay embedding and enables ML methods for modeling. The quality of the reconstruction is evaluated by the prediction loss. The discrete-time signal is converted into a continuous-time signal to achieve a loss function, which is differentiable with respect to the embedding delays. The delay vector, which stores all potential embedding delays, is updated along with the trainable parameters of the model to minimize prediction loss. Thus, MAPSR does not rely on any threshold or statistical criterion for determining the dimension and the set of delay values for the embedding process. We apply the MAPSR method to uni- and multivariate time series stemming from chaotic dynamical systems and a turbulent combustor. We find that for the Lorenz system, the model trained with the MAPSR method is able to predict chaotic time series for nearly seven to eight Lyapunov time scales, which is found to be much better compared to other PSR methods [AMI-FNN (average mutual information-false nearest neighbor) and PECUZAL (Pecora-Uzal) methods]. For the univariate time series from the turbulent combustor, the long-term cumulative prediction error of the MAPSR method for the regime of chaos stays between other methods, and for the regime of intermittency, MAPSR outperforms other PSR methods.

Chaotic route to classical thermalization: A real-space analysis.

Most of the previous studies on classical thermalization focus on the wave-vector space, encountering limitations when extended beyond quasi-integrable regions. In this study, we propose a scheme to study the thermalization of the classical Hamiltonian chain of interacting oscillators in real space by developing a thermalization indicator proposed by Parisi [Europhys. Lett. 40, 357 (1997)0295-507510.1209/epl/i1997-00471-9], which approaches zero in the thermal state. Upon reaching the steady state characterized by the generalized Gibbs ensemble for a harmonic chain, a quench protocol is implemented to change the Hamiltonian to a nonintegrable form instantaneously, thereby preparing nonequilibrium initial states. This approach enables investigations of thermalization in real space, particularly valuable for exploring regions beyond quasi-integrability. For the FPUT-β lattice, we observe that the thermalization time as a function of the nonintegrable strength follows a -2 scaling law in the quasi-integrable region and -1/4 in the strongly integrable region. Moreover, numerical results reveal the thermalization time is proportional to the Lyapunov time, which bridges microscopic chaotic dynamics and the macroscopic thermalization process.

River Ecomorphodynamic Models Exhibit Features of Nonlinear Dynamics and Chaos

AbstractModeling the nonlinear interactions between flow, sediment, and vegetation is essential for improving our understanding and prediction of river system dynamics. Using simple numerical models, we simulate the key flow‐sediment‐vegetation interaction where the disturbance is intrinsically generated by the presence of vegetation. In this case, biomass growth modifies the flow field, induces bed scour, and thus potentially causes vegetation uprooting when erosion exceeds root depth. Our results show that this nonlinear feedback produces deterministic chaos under a wide range of conditions, with complex aperiodic dynamics generated by a period‐doubling route to chaos. Moreover, our results suggest relatively small values of Lyapunov time, spanning 2–4 growth‐flood cycles, which significantly restrict the predictability of riverbed evolution. Although further spatial and temporal processes may add complexity to the system, these results call for the use of ensemble methods and associated uncertainty estimates in ecomorphodynamic models.

Analysis of a new three-dimensional jerk chaotic system with transient chaos and its adaptive backstepping synchronous control

A new three-dimensional Jerk chaotic system with line equilibrium points is proposed. The system is researched in detail by the Lyapunov exponent graph, bifurcation diagram, phase diagram, and time domain waveform diagram, which show that the system has rich dynamical behaviors, such as eight types of coexisting attractors, extreme multistability of four different attractor states, and offset boosting in two directions. In addition, the system also has six types of transient chaos, which greatly increase the complexity of the system. We study the variation of the spectral entropy (SE) and C0 complexity when the system takes different initial values. Also, in this paper, the initial conditions under which the system is in a synchronized state are determined by initial values with higher complexity. The correctness of the theoretical analysis and numerical simulation is verified by circuit simulation and hardware experiments. Finally, the new system achieves synchronization control utilizing a designed adaptive backstepping controller, laying the foundation for its subsequent use in secure communications.

Predictability of monthly streamflow by considering complexity measures

Natural fluid flow systems exhibit turbulent and chaotic behavior that determines their high-level complexity. Chaos has an accurate mathematical definition, while turbulence is a property of fluid flow without an accurate mathematical definition. Using the Kolmogorov complexity (KC) and its derivative (KC spectrum), permutation entropy (PE), and Lyapunov exponent (LE), we considered how chaos affected the predictability of natural fluid flow systems. This paper applied these measures to investigate the turbulent, complex and chaotic behaviors of monthly streamflow of rivers from Bosnia and Herzegovina, the United States, and the Mendoza Basin (Argentina) and evaluated their time horizons using the Lyapunov time (LT). Based on the measures applied for river streamflow, we derived four modes of the interrelationship between turbulence, complexity, and chaos. Finally, using the measures, we clustered rivers with similar time horizons representing their predictability.

Deep learning-based state prediction of the Lorenz system with control parameters.

Nonlinear dynamical systems with control parameters may not be well modeled by shallow neural networks. In this paper, the stable fixed-point solutions, periodic and chaotic solutions of the parameter-dependent Lorenz system are learned simultaneously via a very deep neural network. The proposed deep learning model consists of a large number of identical linear layers, which provide excellent nonlinear mapping capability. Residual connections are applied to ease the flow of information and a large training dataset is further utilized. Extensive numerical results show that the chaotic solutions can be accurately forecasted for several Lyapunov times and long-term predictions are achieved for periodic solutions. Additionally, the dynamical characteristics such as bifurcation diagrams and largest Lyapunov exponents can be well recovered from the learned solutions. Finally, the principal factors contributing to the high prediction accuracy are discussed.

A hybrid proper orthogonal decomposition and next generation reservoir computing approach for high-dimensional chaotic prediction: Application to flow-induced vibration of tube bundles.

Chaotic time series prediction is a central science problem in diverse areas, ranging from engineering, economy to nature. Classical chaotic prediction techniques are limited to short-term prediction of low- or moderate-dimensional systems. Chaotic prediction of high-dimensional engineering problems is notoriously challenging. Here, we report a hybrid approach by combining proper orthogonal decomposition (POD) with the recently developed next generation reservoir computing (NGRC) for the chaotic forecasting of high-dimensional systems. The hybrid approach integrates the synergistic features of the POD for model reduction and the high efficiency of NGRC for temporal data analysis, resulting in a new paradigm on data-driven chaotic prediction. We perform the first chaotic prediction of the nonlinear flow-induced vibration (FIV) of loosely supported tube bundles in crossflow. Reducing the FIV of a continuous beam into a 3-degree-of-freedom system using POD modes and training the three time coefficients via a NGRC network with three layers, the hybrid approach can predict time series of a weakly chaotic system with root mean square prediction error less than 1% to 19.3 Lyapunov time, while a three Lyapunov time prediction is still achieved for a highly chaotic system. A comparative study demonstrates that the POD-NGRC outperforms the other existing methods in terms of either predictability or efficiency. The efforts open a new avenue for the chaotic prediction of high-dimensional nonlinear dynamic systems.

Data-driven modeling and forecasting of chaotic dynamics on inertial manifolds constructed as spectral submanifolds.

We present a data-driven and interpretable approach for reducing the dimensionality of chaotic systems using spectral submanifolds (SSMs). Emanating from fixed points or periodic orbits, these SSMs are low-dimensional inertial manifolds containing the chaotic attractor of the underlying high-dimensional system. The reduced dynamics on the SSMs turn out to predict chaotic dynamics accurately over a few Lyapunov times and also reproduce long-term statistical features, such as the largest Lyapunov exponents and probability distributions, of the chaotic attractor. We illustrate this methodology on numerical data sets including delay-embedded Lorenz and Rössler attractors, a nine-dimensional Lorenz model, a periodically forced Duffing oscillator chain, and the Kuramoto-Sivashinsky equation. We also demonstrate the predictive power of our approach by constructing an SSM-reduced model from unforced trajectories of a buckling beam and then predicting its periodically forced chaotic response without using data from the forced beam.

Dynamical chaos in the integrable Toda chain induced by time discretization.

We use the Toda chain model to demonstrate that numerical simulation of integrable Hamiltonian dynamics using time discretization destroys integrability and induces dynamical chaos. Specifically, we integrate this model with various symplectic integrators parametrized by the time step τ and measure the Lyapunov time TΛ (inverse of the largest Lyapunov exponent Λ). A key observation is that TΛ is finite whenever τ is finite but diverges when τ→0. We compare the Toda chain results with the nonintegrable Fermi-Pasta-Ulam-Tsingou chain dynamics. In addition, we observe a breakdown of the simulations at times TB≫TΛ due to certain positions and momenta becoming extremely large ("Not a Number"). This phenomenon originates from the periodic driving introduced by symplectic integrators and we also identify the concrete mechanism of the breakdown in the case of the Toda chain.

Confined chaos and the chaotic angular motion of Atlas, a Saturn’s inner satellite

ABSTRACT The dynamics of the Solar system exhibit inherent chaos and instability. Mathematical tools, such as the maximum Lyapunov characteristic exponent (LCE) and Lyapunov time (TL), play a crucial role in providing a qualitative understanding of chaos within celestial objects, such as asteroids and moonlets. Celestial bodies with relatively small Lyapunov times have garnered significant research interest due to their stable orbits, a phenomenon referred to as stable or confined chaos. Notable examples include Saturn’s satellites: Atlas, with a Lyapunov time on the order of 10 yr, Prometheus, and Pandora. This work aims to study the chaotic behaviour of the Atlas satellite and its relatively small TL. We present a three-dimensional model approach designed to isolate the radial contribution from the LCE and assess its influence within the LCE. Our investigation focuses on the Saturn system, comprising Saturn itself, along with its satellites Atlas, Prometheus, Pandora, and Mimas. To estimate the radial contribution of the LCE, we find the projection of the radial vector of a ghost Atlas (a slightly displaced Atlas) onto the Atlas radial vector, which allows us to calculate the difference between the radial vectors. This methodology enables us to estimate the radial contribution of the LCE and calculate the Lyapunov time. Remarkably, our results demonstrate that orbits remain confined even for integration times exceeding TL. Furthermore, we investigate the temporal behaviour of Atlas’ angular position in its orbit, potentially shedding light on chaotic angular dynamics.

Chaotic Diffusion in a Triaxial Galactic Model: an Example of Global Stable Chaos

In this work we focus on the chaotic diffusion in the phase space of a triaxial potential resembling an elliptical galaxy. The transport process is studied in two different action-like starting spaces in order to cope with circulating and noncirculating orbits. Estimates of the diffusion rate obtained by means of the variance approach are discussed in detail and their limitations are exposed. After revisiting the Shannon-entropy-based method from a conceptual point of view in the framework of simple arguments taken from the information theory, we apply it to measure changes in the unperturbed actions or integrals of motion of the system for different sets of small ensembles of random initial conditions. For such sets of ensembles, estimates of the Lyapunov times are also provided. The results show that, within the chaotic component of the phase space, the Lyapunov times are shorter than any physical time scale as the Hubble time, but the diffusion times are much larger than the latter. Thus, we conclude that stable chaos dominates the dynamics of realistic galactic models.

Computation of soliton structure and analysis of chaotic behaviour in quantum deformed Sinh-Gordon model.

Soliton dynamics and nonlinear phenomena in quantum deformation has been investigated through conformal time differential generalized form of q deformed Sinh-Gordon equation. The underlying equation has recently undergone substantial amount of research. In Phase 1, we employed modified auxiliary and new direct extended algebraic methods. Trigonometric, hyperbolic, exponential and rational solutions are successfully extracted using these techniques, coupled with the best possible constraint requirements implemented on parameters to ensure the existence of solutions. The findings, then, are represented by 2D, 3D and contour plots to highlight the various solitons' propagation patterns such as kink-bright, bright, dark, bright-dark, kink, and kink-peakon solitons and solitary wave solutions. It is worth emphasizing that kink dark, dark peakon, dark and dark bright solitons have not been found earlier in literature. In phase 2, the underlying model is examined under various chaos detecting tools for example lyapunov exponents, multistability and time series analysis and bifurcation diagram. Chaotic behavior is investigated using various initial condition and novel results are obtained.

Thermalization slowing down in multidimensional Josephson junction networks.

We characterize thermalization slowing down of Josephson junction networks in one, two, and three spatial dimensions for systems with hundreds of sites by computing their entire Lyapunov spectra. The ratio of Josephson coupling E_{J} to energy density h controls two different universality classes of thermalization slowing down, namely, the weak-coupling regime, E_{J}/h→0, and the strong-coupling regime, E_{J}/h→∞. We analyze the Lyapunov spectrum by measuring the largest Lyapunov exponent and by fitting the rescaled spectrum with a general ansatz. We then extract two scales: the Lyapunov time (inverse of the largest exponent) and the exponent for the decay of the rescaled spectrum. The two universality classes, which exist irrespective of network dimension, are characterized by different ways the extracted scales diverge. The universality class corresponding to the weak-coupling regime allows for the coexistence of chaos with a large number of near-conserved quantities and is shown to be characterized by universal critical exponents, in contrast with the strong-coupling regime. We expect our findings, which we explain using perturbation theory arguments, to be a general feature of diverse Hamiltonian systems.

Overview and Perspectives of Chaos Theory and Its Applications in Economics

Starting from the contribution of such thinkers as the famous Giordano Bruno (1583) and the great mathematician and physicist Henri Poincaré (1889) and the surprising discovery of the meteorologist Edward Lorenz (1963), we consider the expansion of the mathematics of chaos in this article, paying attention to topology, qualitative geometry, and Catastrophe Theory, on the one hand, and addressing the possibilities derived from the new Computer Science as Quantum Algorithms and the advances in Artificial Intelligence, on the other. We especially highlight the section on computing chaos, which we consider to be new calculation and analysis instruments, such as machine learning and its algorithm called reservoir computing, through which we can know the dynamics of a chaotic system. With past data, with equations like Karamoto–Sivashinsky, one can improve predictions of the system eight times further ahead than in previous methods. Integrating the machine learning approach and traditional model-based prediction, one could obtain accurate predictions twelve Lyapunov times. As we know, in the framework of chaos theory, it is habitually accepted that the idea of long-term prediction seems impossible because we live under a veil of uncertainty. But with technological advances, the landscape begins to change, both in chaos theory and in its applications, especially in the field of economics, to which we devote particular attention, carrying out as an example the analysis of the evolution of the Madrid Stock Exchange in the 2006–2013 crisis. Above all this, a reflection of a general nature is necessary to enlighten us on the possibility of opening a new horizon.

Variability of echo state network prediction horizon for partially observed dynamical systems.

Study of dynamical systems using partial state observation is an important problem due to its applicability to many real-world systems. We address the problem by studying an echo state network (ESN) framework with partial state input with partial or full state output. Application to the Lorenz system and Chua's oscillator (both numerically simulated and experimental systems) demonstrate the effectiveness of our method. We show that the ESN, as an autonomous dynamical system, is capable of making short-term predictions up to a few Lyapunov times. However, the prediction horizon has high variability depending on the initial condition-an aspect that we explore in detail using the distribution of the prediction horizon. Further, using a variety of statistical metrics to compare the long-term dynamics of the ESN predictions with numerically simulated or experimental dynamics and observed similar results, we show that the ESN can effectively learn the system's dynamics even when trained with noisy numerical or experimental data sets. Thus, we demonstrate the potential of ESNs to serve as cheap surrogate models for simulating the dynamics of systems where complete observations are unavailable.

Dynamics of a data-driven low-dimensional model of turbulent minimal Couette flow

Because the Navier–Stokes equations are dissipative, the long-time dynamics of a flow in state space are expected to collapse onto a manifold whose dimension may be much lower than the dimension required for a resolved simulation. On this manifold, the state of the system can be exactly described in a coordinate system parameterising the manifold. Describing the system in this low-dimensional coordinate system allows for much faster simulations and analysis. We show, for turbulent Couette flow, that this description of the dynamics is possible using a data-driven manifold dynamics modelling method. This approach consists of an autoencoder to find a low-dimensional manifold coordinate system and a set of ordinary differential equations defined by a neural network. Specifically, we apply this method to minimal flow unit turbulent plane Couette flow at $Re=400$ , where a fully resolved solutions requires ${O}(10^5)$ degrees of freedom. Using only data from this simulation we build models with fewer than $20$ degrees of freedom that quantitatively capture key characteristics of the flow, including the streak breakdown and regeneration cycle. At short times, the models track the true trajectory for multiple Lyapunov times and, at long times, the models capture the Reynolds stress and the energy balance. For comparison, we show that the models outperform POD-Galerkin models with $\sim$ 2000 degrees of freedom. Finally, we compute unstable periodic orbits from the models. Many of these closely resemble previously computed orbits for the full system; in addition, we find nine orbits that correspond to previously unknown solutions in the full system.