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We propose a novel framework for approximating the statistical properties of turbulent flows by combining variational methods for the search of unstable periodic orbits with resolvent analysis for dimensionality reduction. Traditional approaches relying on identifying all short, fundamental Unstable Periodic Orbits to compute ergodic averages via cycle expansion are computationally prohibitive for high-dimensional fluid systems. Our framework stems from the observation in Lasagna, Phys. Rev. E (2020), that a single unstable periodic orbit with a period sufficiently long to span a large fraction of the attractor captures the statistical properties of chaotic trajectories. Given the difficulty of identifying unstable periodic orbits for high-dimensional fluid systems, approximate trajectories residing in a low-dimensional subspace are instead constructed using resolvent modes, which inherently capture the temporal periodicity of unstable periodic orbits. The amplitude coefficients of these modes are adjusted iteratively with gradient-based optimisation to minimise the violation of the projected governing equations, producing trajectories that approximate, rather than exactly solve, the system dynamics. A first attempt at utilising this framework on a chaotic system is made here on the Lorenz 1963 equations, where resolvent analysis enables an exact dimensionality reduction from three to two dimensions. Key observables averaged over these trajectories produced by the approach as well as probability distributions and spectra rapidly converge to values obtained from long chaotic simulations, even with a limited number of iterations. This indicates that exact solutions may not necessary to approximate the system's statistical behaviour, as the trajectories obtained from partial optimisation provide a sufficient ``sketch'' of the attractor in state space.

We investigate the dynamics of phase oscillators in the fully disordered Kuramoto model with couplings of defined asymmetry. The mean-field dynamics is reduced to a self-consistent stochastic single-oscillator problem which we analyze perturbatively and by numerical simulations. We elucidate the influence of the asymmetry on the correlation and response function of the system as well as on the distribution of the order parameter. The so-called volcano transition is shown to be robust with respect to a small degree of coupling asymmetry but to disappear when the antisymmetry in the couplings outweighs the symmetry.

We study the emergent dynamics of quantum self-sustained oscillators induced by the simultaneous presence of attraction and repulsion in the coupling path. We consider quantum Stuart-Landau oscillators under attractive-repulsive coupling and construct the corresponding quantum master equation in the Lindblad form. We discover an interesting symmetry-breaking transition from quantum limit cycle oscillation to quantum inhomogeneous steady state; This transition is contrary to the previously known symmetry-breaking transition from quantum homogeneous to inhomogeneous steady state. The result is supported by the analysis on the noisy classical model of the quantum system in the weak quantum regime. Remarkably, we find the generation of entanglement associated with the symmetry-breaking transition that has no analogue in the classical domain. This study will enrich our understanding of the collective behaviors shown by coupled oscillators in the quantum domain., Comment: 8 pages, 7 figures, To appear in Phys. Rev. E

The energy surplus resulting from radiative forcing causes warming of the Earth system. This initial warming drives a myriad of changes including in sea surface temperatures (SSTs), leading to different radiative feedbacks. The relationship between the radiative feedbacks and the pattern of SST changes is referred to as the "pattern effect". The current approach to study the pattern effect relies on diagnosing the response of atmosphere-only models to perturbations in the SST boundary condition. Here, we argue that the fluctuation-dissipation relation (FDR), together with coarse-graining procedures, is a computationally cheap and theoretically grounded alternative to model experiments. We introduce a protocol to study the pattern effect and present its application in a state-of-the-art coupled climate model. By focusing on the coupled dynamics, we unveil the role of the slow ocean component in setting the pattern effect. We present a new "sensitivity map", representing a first, qualitative prediction of the response of the average top-of-the-atmosphere (TOA) radiative flux to perturbations in the SST field. We find negative sensitivity throughout the tropics, in contrast to the current understanding of a positive-negative dipole of sensitivity in the tropical Pacific. Considering only the shortest time scales, the response is dominated by the fast atmospheric variability and we recover results in qualitative agreement with the literature. Therefore, the difference between our results and previous studies, largely comes from including the atmosphere-ocean coupling. The framework offers a conceptually novel perspective on the pattern effect: feedbacks in the coupled system are encoded in a temporally and spatially dependent response operator, rather than time-independent maps as for previous studies.

Learning in zero-sum games studies a situation where multiple agents competitively learn their strategy. In such multi-agent learning, we often see that the strategies cycle around their optimum, i.e., Nash equilibrium. When a game periodically varies (called a ``periodic'' game), however, the Nash equilibrium moves generically. How learning dynamics behave in such periodic games is of interest but still unclear. Interestingly, we discover that the behavior is highly dependent on the relationship between the two speeds at which the game changes and at which players learn. We observe that when these two speeds synchronize, the learning dynamics diverge, and their time-average does not converge. Otherwise, the learning dynamics draw complicated cycles, but their time-average converges. Under some assumptions introduced for the dynamical systems analysis, we prove that this behavior occurs. Furthermore, our experiments observe this behavior even if removing these assumptions. This study discovers a novel phenomenon, i.e., synchronization, and gains insight widely applicable to learning in periodic games., Comment: 8 pages, 5 figures (main); 7 pages, 1 figure (appendix)

Quantitative forecasting of average rainfall into the next season remains highly challenging, but in some favourable isolated cases may be possible with a series of relatively simple steps. We chose to explore predictions of austral springtime rainfall in SE Australia regions based on the surrounding ocean surface temperatures during the winter. In the first stage, we search for correlations between the target rainfall and both the standard ocean climate indicators as well as the time series of surface temperature data expanded in terms of Empirical Orthogonal Functions (EOFs). In the case of the Indian Ocean, during the winter the dominant EOF shows stronger correlation with the future rainfall than the commonly used Indian Ocean Dipole. Information sources with the strongest correlation to the historical rainfall data are then used as inputs into deep learning artificial neural networks. The resulting hindcasts appear accurate for September and October and less reliable for November. We also attempt to forecast the rainfall in several regions for the coming austral spring.

Biological activities are often seen entrained onto the day-night and other celestial mechanical cycles (e.g., seasonal and lunar), but studies on the origin of life have largely not accounted for such periodic external environmental variations. We argue that this may be an important omission, because the signature replication behaviour of life represents temporal memory in the dynamics of ecosystems, that signifies the absence of mixing properties (i.e., the dynamics are not fully chaotic), and entrainment onto regular, periodic external perturbative influences has been proven capable of suppressing chaos, and thus may bring otherwise unstable chemical reaction sets into viability, as precursors to abiogenesis. As well, external perturbations may be necessary to prevent an open dissipative (bio)chemical system from collapsing into the opposite extreme -- the point attractor of thermal equilibrium. In short, life may precariously rest on the edge of chaos, and open-loop periodic perturbation rooted in celestial mechanics (and should be simulated in laboratory experiments in origin-of-life studies) may help with the balancing. Such considerations, if pertinent, would also be consequential to exobiology, e.g., in regard to tidal-locking properties of potential host worlds., Comment: 6 pages

This study examines the dynamical properties of the Ikeda map, with a focus on bifurcations and chaotic behavior. We investigate how variations in dissipation parameters influence the system, uncovering shrimp-shaped structures that represent intricate transitions between regular and chaotic dynamics. Key findings include the analysis of period-doubling bifurcations and the onset of chaos. We utilize Lyapunov exponents to distinguish between stable and chaotic regions. These insights contribute to a deeper understanding of nonlinear and chaotic dynamics in optical systems.

In nontwist systems, primary shearless curves act as barriers to chaotic transport. Surprisingly, the onset of secondary shearless curves has been reported in a few twist systems. Meanwhile, we found that, in twist systems, the onset of these secondary shearless curves is a standard process that may appear as control parameters are varied in situations where there is resonant mode coupling. Namely, we analyze these shearless bifurcations in two-harmonic systems for the standard map, the Ullmann map, and for the Walker-Ford Hamiltonian flow. The onset of shearless curves is related to bifurcations of periodic points. Furthermore, depending on the bifurcation, these shearless curves can emerge alone or in pairs, and in some cases, deform into separatrices.

A method for the automatic classification of the orbits of magnetic field lines into topologically distinct classes using the Vietoris-Rips persistent homology is presented. The input to the method is the Poincare map orbits of field lines and the output is a separation into three classes: islands, chaotic layers, and invariant tori. The classification is tested numerically for the case of a toy model of a perturbed tokamak represented initially in its geometric coordinates. The persistent $H_1$ data is demonstrated to be sufficient to distinguish magnetic islands from the other orbits. When combined with persistent $H_0$ information, describing the average spacing between points on the Poincare section, the larger chaotic orbits can then be separated from very thin chaotic layers and invariant tori. It is then shown that if straight field line coordinates exist for a nearby integrable field configuration, the performance of the classification can be improved by transforming into this natural coordinate system. The focus is the application to toroidal magnetic confinement but the method is sufficiently general to apply to generic $1\frac{1}{2}$d Hamiltonian systems., Comment: 41 pages, 30 Figures

In the context of non-Hermitian photonics, we consider a nonlinear optical trimer with three lossy waveguides with complex couplings. This non-Hermitian trimer exhibits stable stationary and oscillatory regimes in a wide range of values of the coupling-loss parameters. Moreover, chaotic dynamics through period-doubling is confirmed via Lyapunov exponent measurements and the underlying bifurcation structure is found by semi-analytical continuation of solutions. The interplay of chaos, due to Kerr nonlinearity, and non-Hermiticity, due to the combined dissipation and complex coupling, provides a different perspective in the area of nonlinear and non-Hermitian optics.

Observational evidence strongly supports the existence of a Super Massive Black Hole (SMBH) at the Galactic center, surrounded by dense stellar clusters. Modeling galactic centers with intricate structures like shells and rings pose challenges, prompting the use of simplified models such as a spherical monopole potential with a multipolar halo mass distribution. This approach, employing a multipolar expansion model, provides versatility for numerical analyses, revealing the complex dynamics of stars in this region. Pseudo-Potentials like Paczynsky-Wiita and Artemova-Bjornsson-Novikov are utilized to simulate the impacts of strong gravity from non-rotating and rotating compact objects respectively, elucidating their influence on stellar dynamics. Chaos naturally arises due to non-central forces, visualized using the Poincar\'e section technique. Of particular importance is the utilization of the Smaller Alignment Index (SALI), a powerful nonlinear dynamical tool, which categorizes particle orbits as escaping, regular, sticky, or chaotic. We exhaustively examine all combinations of multipolar moments up to the octupolar term along with spin using this tool, which had not been studied earlier. SALI provides a straightforward yet efficient method for assessing the interplay between the system's different multipolar moments, their combinations, and spin. Thus, our findings offer insights into the dynamics of compact objects enshrouded in a halo mass distribution and lay the groundwork for understanding complex astrophysical systems in galactic centers., Comment: 16 pages, 7 figures

Key network motifs searching in complex networks is one of the crucial aspects of network analysis. There has been a series of insightful findings and valuable applications for various scenarios through the analysis of network structures. However, in dynamic systems, slight changes in the choice of dynamic equations and parameters can alter the significance of motifs. The known methods are insufficient to address this issue effectively. In this paper, we introduce a concept of perturbation energy based on the system's Jacobian matrix, and define motif centrality for dynamic systems by seamlessly integrating network topology with dynamic equations. Through simulations, we observe that the key motifs obtained by the proposed energy method present better effective and accurate than them by integrating network topology methods, without significantly increasing algorithm complexity. The finding of key motifs can be used to apply for system control, such as formulating containment policies for the spread of epidemics and protecting fragile ecosystems. Additionally, it makes substantial contribution to a deeper understanding of concepts in physics, such as signal propagation and system's stability., Comment: 15 pages, 7 figures

Cross-Correlation random matrices have emerged as a promising indicator of phase transitions in spin systems. The core concept is that the evolution of magnetization encapsulates thermodynamic information [R. da Silva, Int. J. Mod. Phys. C, 2350061 (2023)], which is directly reflected in the eigenvalues of these matrices. When these evolutions are analyzed in the mean-field regime, an important question arises: Can the Langevin equation, when translated into maps, perform the same function? Some studies suggest that this method may also capture the chaotic behavior of certain systems. In this work, we propose that the spectral properties of random matrices constructed from maps derived from deterministic or stochastic differential equations can indicate the critical or chaotic behavior of such systems. For chaotic systems, we need only the evolution of iterated Hamiltonian equations, and for spin systems, the Langevin maps obtained from mean-field equations suffice, thus avoiding the need for Monte Carlo (MC) simulations or other techniques., Comment: 19 pages, 5 figures

Although stable solutions of dynamical systems are typically considered more important than unstable ones, unstable solutions have a critical role in the dynamical integrity of stable solutions. In fact, usually, basins of attraction boundaries are composed of unstable solutions and their stable manifolds. This study proposes a method for estimating unstable limit cycles surrounding stable equilibrium points. The method exploits the shape of the decrement of trajectories converging towards the equilibrium. For the method, trajectories obtained from small perturbations from the equilibrium state are sufficient to estimate the unstable limit cycle roughly. No mathematical model of the system dynamics is needed for the computation, which requires only a single trajectory in the phase space. As such, the method is computationally very rapid and potentially implementable in real structures.

The interplay between order and disorder is crucial across various fields, especially in understanding oscillatory phenomena. Periodic oscillations are frequently observed in heterogeneous catalysis, yet their underlying mechanisms need deeper exploration. Here, we investigate how periodic oscillations arise during methane oxidation catalysed by palladium nanoparticles (Pd NPs), utilizing a suite of complementary operando techniques across various spatial scales. We found that reaction intensity and collective dynamic modes can be tuned by the reactant gas-flow rate. At lower gas-flow rates, we observed periodic facet reconstruction of Pd NPs correlated with repeated bubbling behaviour at the Pd/PdO interface, without evident global oscillatory responses. Conversely, at higher gas-flow rates, Pd NPs undergo chaotic transformations between metallic and oxidized states, resulting in overall oscillation. Integrating our observations at different gas-flow rates, we attributed the emergence of global oscillation to thermal coupling regulated by gas flow and connected local and global dynamics through a weak synchronization mechanism. This work demonstrates the correlations between open surfaces and interfaces, chaos and regularity, and dissipative processes and coupling behaviour. Our findings offer critical insights into the complexity behind catalytic oscillations and provide guidance for modulating oscillatory behaviours in catalytic processes, with significant implications for both science and industry.

Recent results suggest that highly active, chaotic, non-equilibrium states of living fluids might share much in common with high Reynolds number, inertial turbulence. We now show, by using a hydrodynamical model, the onset of intermittency and the consequent multiscaling of Eulerian and Lagrangian structure functions as a function of the bacterial activity. Our results bridge the worlds of low and high Reynolds number flows as well as open up intriguing possibilities of what makes flows intermittent.

The dynamics of a single-mode semiconductor laser induced by the optical injection of a frequency comb are analyzed through the lens of complexity. In particular, the focus is dynamics outside the injection-locking region. Permutation entropy (P.E.) and chaos bandwidth (CBW) are used to quantify complexity. Various numerically simulated system outputs across the frequency detuning range and for different injection strengths are considered. Numerically simulated outputs from a single-mode injection system in the absence of comb injection are compared to comment on the origin of the reported complexity. Furthermore, experimental outputs are used to confirm the findings. Despite the presence of periodic dynamics originating from the frequency comb, the presented system generates remarkably high complex outputs with P.E. up to 0.99 maintained over an extended period of observation and long timescales with CBW up to 25 GHz., Comment: 7 pages, 5 figures, 33 references

We report a new mechanism through which extreme events with a dragon king-like distribution emerge in a network of locally coupled Hindmarsh-Rose bursting neurons. We establish and substantiate the fact that depending on the choice of initial conditions, the neurons in the network are divided into clusters and whenever these clusters are phase synchronized intermittently, extreme events originate in the collective observable. This mechanism, which we name as intermittent cluster synchronization is proposed as the new precursor for the generation of extreme events in this system. These results are also true for electrical diffusive coupling. The distribution of the local maxima shows long tailed non-Gaussian while the interevent interval follows the Weibull distribution. The goodness of fit are corroborated using probability-probability plot and quantile-quantile plot. These extreme events become rarer and rarer with the increase in the number of different initial conditions., Comment: 11 pages, 12 figures

An interesting feature in dissipative nonlinear systems is the emergence of characteristic domains in parameter space that exhibit periodic temporal evolution, known as shrimp-shaped domains. We investigate the parameter space of the dissipative asymmetric kicked rotor map and show that, in the regime of strong dissipation, the shrimp-shaped domains repeat themselves as the nonlinearity parameter increases while maintaining the same period. We analyze the dependence of the length of each periodic domain with the nonlinearity parameter, revealing that it follows a power law with the same exponent regardless of the dissipation parameter. Additionally, we find that the distance between adjacent shrimp-shaped domains is scaling invariant with respect to the dissipation parameter. Furthermore, we show that for weaker dissipation, a multistable scenario emerges within the periodic domains. We find that as the dissipation gets weaker, the ratio of multistable parameters for each periodic domain increases, and the area of the periodic basin decreases as the nonlinearity parameter increases.

The treatment of Hall-effect thrusters as nonlinear, dynamical systems has emerged as a new perspective to understand and analyze data acquired from the thrusters. The acquisition of high-speed data that can resolve the characteristic high-frequency oscillations of these thruster enables additional levels of classification in these thrusters. Notably, these signals may serve as unique indicators for the full state of the system that can aid digital representations of thrusters and predictions of thruster dynamics. In this work, a Reservoir Computing framework is explored to build surrogate models from experimental time-series measurements of a Hall-effect thruster. Such a framework has shown immense promise for predicting the behavior of low-dimensional yet chaotic dynamical systems. In particular, the surrogates created by the Reservoir Computing framework are capable of both predicting the observed behavior of the thruster and estimating the values of certain measurements from others, known as inference.

In society, it is common to face problems that require collaboration with other people, from everyday challenges to complex tasks, such as group projects at work. In this context, the search for more effective problem-solving strategies becomes a topic of great interest. This paper presents a new dynamic that integrates coevolution into the NK model together with an Erd\"os-R\'enyi random network, allowing more than one neighbor update. We explore how this coevolution can be employed as an optimization strategy for solving group problems. In recent coevolution models, only one agent is removed and another is added to the neighborhood of influence. Here, we allow $L$ agents to be changed, which had a large impact on the system performance. In the analysis of the results, we consider rewiring as a way for the target agent to obtain information from individuals or groups that were not part of its neighborhood. Our simulations demonstrate that coevolution can produces gain on the computational cost., Comment: 9 pages, 8 figures

We provide conditions on the coupling function such that a system of 4 globally coupled identical oscillators has chaotic attractors, a pair of Lorenz attractors or a 4-winged analogue of the Lorenz attractor. The attractors emerge near the triple instability threshold of the splay-phase synchronization state of the oscillators. We provide theoretical arguments and verify numerically, based on the pseudohyperbolicity test, that the chaotic dynamics are robust with respect to small, e.g. time-dependent, perturbations of the system. The robust chaoticity should also be inherited by any network of weakly interacting systems with such attractors., Comment: 6 pages, 4 figures

We study Ruelle-Pollicott resonances in translationally invariant quantum many-body systems via spectra of momentum-resolved operator propagator on infinite systems. Momentum dependence gives insight into decay of correlation functions, showing that, depending on their symmetries, different correlation functions in general decay with different rates. Focusing on the kicked Ising model the spectrum seems to be typically composed of an annular random matrix like ring whose size we theoretically predict, and few isolated resonances. We identify several interesting regimes, including a mixing regime with a power-law decay of correlation functions. In that regime we also observe a huge difference in time-scales of different correlation functions. An exact expression for the singular values of the operator propagator is conjectured, showing that it becomes singular at a special point., Comment: 12 pages

We show that bifurcations of periodic orbits with multipliers $(-1,i,-i)$ can lead to the birth of pseudohyperbolic (i.e., robustly chaotic) Lorenz-like attractors of three different types: one is a discrete analogue of the classical Lorenz attractor, and the other two are new. We call them two- and four-winged ``Sim\'o angels''. These three attractors exist in an orientation-reversing, three-dimensional, quadratic H\'enon map. Our analysis is based on a numerical study of a normal form for this bifurcation, a three-dimensional system of differential equations with a Z4-symmetry. We investigate bifurcations in the normal form and describe those responsible for the emergence of the Lorenz attractor and the continuous-time version of the Simo angels. Both for the normal form and the 3D H\'enon map, we have found open regions in the parameter space where the attractors are pseudohyperbolic, implying that for every parameter value from these regions every orbit in the attractor has positive top Lyapunov exponent., Comment: 40 pages, 24 figures

Classical chaos arises from the inherent non-linearity of dynamical systems. However, quantum maps are linear; therefore, the definition of chaos is not straightforward. To address this, we study a quantum system that exhibits chaotic behavior in its classical limit: the kicked top model, whose classical dynamics are governed by Hamilton's equations on phase space, whereas its quantum dynamics are described by the Schr\"odinger equation in Hilbert space. We explore the critical degree of non-linearity signifying the onset of chaos in the kicked top by modifying the original Hamiltonian so that the non-linearity is parametrized by a quantity $p$. We find two distinct behaviors of the modified kicked top depending on the value of $p$. Chaos intensifies as $p$ varies within the range of $1\leq p \leq 2$, whereas it diminishes for $p > 2$, eventually transitioning to a purely regular oscillating system as $p$ tends to infinity. We also comment on the complicated phase space structure for non-chaotic dynamics. Our investigation sheds light on the relationship between non-linearity and chaos in classical systems, offering insights into their dynamic behavior.

Recently, machine learning techniques, particularly deep learning, have demonstrated superior performance over traditional time series forecasting methods across various applications, including both single-variable and multi-variable predictions. This study aims to investigate the capability of i) Next Generation Reservoir Computing (NG-RC) ii) Reservoir Computing (RC) iii) Long short-term Memory (LSTM) for predicting chaotic system behavior, and to compare their performance in terms of accuracy, efficiency, and robustness. These methods are applied to predict time series obtained from four representative chaotic systems including Lorenz, R\"ossler, Chen, Qi systems. In conclusion, we found that NG-RC is more computationally efficient and offers greater potential for predicting chaotic system behavior., Comment: 8 pages, 15 figures

Complexity of two-level systems, e.g. spins, qubits, magnetic moments etc, are analysed based on the so-called correlational entropy in the case of pure quantum systems and the thermal entropy in case of thermal equilibrium that are suitable quantities essentially free from basis dependence. The complexity is defined as the difference between the Shannon-entropy and the second order R\'enyi-entropy, where the latter is connected to the traditional participation measure or purity. It is shown that the system attains maximal complexity for special choice of control parameters, i.e. strength of disorder either in the presence of noise of the energy states or the presence of disorder in the off diagonal coupling. It is shown that such a noise or disorder dependence provides a basis free analysis and gives meaningful insights. We also look at similar entropic complexity of spins in thermal equilibrium for a paramagnet at finite temperature, $T$ and magnetic field $B$, as well as the case of an Ising model in the mean-field approximation. As a result all examples provide important evidence that the investigation of the entropic complexity parameters help to get deeper understanding in the behavior of these systems., Comment: 8 pages, 9 figures

This paper presents a novel approach for propagating uncertainties in dynamical systems building on high-order Taylor expansions of the flow and moment-generating functions (MGFs). Unlike prior methods that focus on Gaussian distributions, our approach leverages the relationship between MGFs and distribution moments to extend high-order uncertainty propagation techniques to non-Gaussian scenarios. This significantly broadens the applicability of these methods to a wider range of problems and uncertainty types. High-order moment computations are performed one-off and symbolically, reducing the computational burden of the technique to the calculation of Taylor series coefficients around a nominal trajectory, achieved by efficiently integrating the system's variational equations. Furthermore, the use of the proposed approach in combination with event transition tensors, allows for accurate propagation of uncertainties at specific events, such as the landing surface of a celestial body, the crossing of a predefined Poincar\'e section, or the trigger of an arbitrary event during the propagation. Via numerical simulations we demonstrate the effectiveness of our method in various astrodynamics applications, including the unperturbed and perturbed two-body problem, and the circular restricted three-body problem, showing that it accurately propagates non-Gaussian uncertainties both at future times and at event manifolds.

In this study, given the inherent nature of dissipation in realistic dynamical systems, we explore the effects of dissipation within the context of fractional dynamics. Specifically, we consider the dissipative versions of two well known fractional maps: the Riemann-Liouville (RL) and the Caputo (C) fractional standard maps (fSMs). Both fSMs are two-dimensional nonlinear maps with memory given in action-angle variables $(I_n,\theta_n)$; $n$ being the discrete iteration time of the maps. In the dissipative versions these fSMs are parameterized by the strength of nonlinearity $K$, the fractional order of the derivative $\alpha\in(1,2]$, and the dissipation strength $\gamma\in(0,1]$. In this work we focus on the average action $\left< I_n \right>$ and the average squared action $\left< I_n^2 \right>$ when~$K\gg1$, i.e. along strongly chaotic orbits. We first demonstrate, for $|I_0|>K$, that dissipation produces the exponential decay of the average action $\left< I_n \right> \approx I_0\exp(-\gamma n)$ in both dissipative fSMs. Then, we show that while $\left< I_n^2 \right>_{RL-fSM}$ barely depends on $\alpha$ (effects are visible only when $\alpha\to 1$), any $\alpha< 2$ strongly influences the behavior of $\left< I_n^2 \right>_{C-fSM}$. We also derive an analytical expression able to describe $\left< I_n^2 \right>_{RL-fSM}(K,\alpha,\gamma)$., Comment: 8 pages, 7 figures

The border-collision normal form is a piecewise-linear family of continuous maps that describe the dynamics near border-collision bifurcations. Most prior studies assume each piece of the normal form is invertible, as is generic from an abstract viewpoint, but in applied problems one piece of the map often has degenerate range, corresponding to a zero determinant. This provides simplification, yet even in two dimensions the dynamics can be incredibly rich. The purpose of this paper is to determine broadly how the dynamics of the two-dimensional border-collision normal form with a zero determinant differs for different values of its parameters. We identify parameter regions of period-adding, period-incrementing, mode-locking, and component doubling of chaotic attractors, and characterise the dominant bifurcation boundaries. The intention is for the results to enable border-collision bifurcations in mathematical models to be analysed more easily and effectively, and we illustrate this with a flu epidemic model and two stick-slip friction oscillator models. We also describe three novel bifurcation structures that remain to be explored.

We investigate the ability to discover data assimilation (DA) schemes meant for chaotic dynamics with deep learning (DL). The focus is on learning the analysis step of sequential DA, from state trajectories and their observations, using a simple residual convolutional neural network, while assuming the dynamics to be known. Experiments are performed with the Lorenz 96 dynamics, which display spatiotemporal chaos and for which solid benchmarks for DA performance exist. The accuracy of the states obtained from the learned analysis approaches that of the best possibly tuned ensemble Kalman filter (EnKF), and is far better than that of variational DA alternatives. Critically, this can be achieved while propagating even just a single state in the forecast step. We investigate the reason for achieving ensemble filtering accuracy without an ensemble. We diagnose that the analysis scheme actually identifies key dynamical perturbations, mildly aligned with the unstable subspace, from the forecast state alone, without any ensemble-based covariances representation. This reveals that the analysis scheme has learned some multiplicative ergodic theorem associated to the DA process seen as a non-autonomous random dynamical system.

Discerning chaos in quantum systems is an important problem as the usual route of Lyapunov exponents in classical systems is not straightforward in quantum systems. A standard route is the comparison of statistics derived from model physical systems to those from random matrix theory (RMT) ensembles, of which the most popular is the nearest-neighbour-spacings distribution (NNSD), which almost always shows good agreement with chaotic quantum systems. However, even in these cases, the long-range statistics (like number variance, spectral rigidity etc.), which are also more difficult to calculate, often show disagreements with RMT. As such, a more stringent test for chaos in quantum systems, via an analysis of intermediate-range statistics is needed, which will additionally assess the extent of agreement with RMT universality. In this paper, we deduce the effective level-repulsion parameters and the corresponding Wigner-surmise-like results of the next-nearest-neighbor spacing distribution (nNNSD) for integrable systems (semi-Poissonian statistics) as well as the three classical quantum-chaotic Wigner-Dyson regimes, by stringent comparisons to numerical RMT models and benchmarking against our exact analytical results for $3\times 3$ Gaussian matrix models, along with a semi-analytical form for the nNNSD in the Orthogonal-to-Unitary symmetry crossover. To illustrate the robustness of these RMT based results, we test these predictions against the nNNSD obtained from quantum chaotic models as well as disordered lattice spin models. This reinforces the Bohigas-Giannoni-Schmit and the Berry-Tabor conjectures, extending the associated universality to longer range statistics. In passing, we also highlight the equivalence of nNNSD in the apparently distinct Orthogonal-to-Unitary and diluted-Symplectic-to-Unitary crossovers., Comment: 19 Pages, 15 Figures

This work explores the conditions under which global contraction manifests in the leaky continuous time reservoirs, thus guaranteeing Generalized Synchronization. Results on continuous time reservoirs make use of the logarithmic norm of the connectivity matrix. Further analysis yields some simple guidelines on how to better construct the connectivity matrix in these systems. Additionally, we outline how the Universal Approximation Property of discrete time reservoirs is readily satisfied by virtue of the activation function being contracting, and how continuous time reservoirs may inherit a limited form of universal approximation by virtue of them overlapping with Neural Ordinary Differential Equations. The ability of the Reservoir Computing framework to universally approximate topological conjugates, along with their fast training, make them a compelling data-driven, black-box surrogate of dynamical systems, and a potential mechanism for developing digital twins.

We examine three billiard systems -- the cardioid, diamond (Superman), and Sinai billiards -- all of which are known to be classically chaotic. We compute their classical Lyapunov exponents, and using out-of-time-order correlators (OTOCs) in the semi-classical regime, we also derive their quantum Lyapunov exponents. We observe that the classical and quantum Lyapunov exponents are in agreement, strengthening the role of OTOCs as a diagnostic for quantum chaos in billiard systems. At very low temperatures, the OTOC of the Sinai billiard shows sharp growth, a phenomenon absent in the other two billiards. We identify the source of this anomalous behaviour in the geometry of the ground state wave function of the Sinai billiard, which is more sensitive to the curvature of the billiard compared to the other billiards. We also remark on the late-time behaviour of the OTOCs and how the scrambling/Ehrenfest time is related to the temperature of quantum billiards., Comment: 35 pages, 21 figures

In this work, we investigate the implications of the differential Hebbian learning rule known as Input-Correlations (ICO) learning in the classification of synchronization in coupled nonlinear oscillator systems. We are investigating the parity-time symmetric coupled Duffing oscillator system with nonlinear dissipation/amplification. In our investigation of the temporal dynamics of this system, it is observed that the system exhibits chaotic as well as quasiperiodic dynamics. On further investigation, it is found that the chaotic dynamics is distorted anti-phase synchronized, whereas the quasiperiodic dynamics is desynchronized. So, on the application of the ICO learning in these two parametric regimes, we observe that the weight associated with the stimulus remains constant when the oscillators are anti-phase synchronized, in spite of there being distortion in the synchronization. But when the oscillators exhibit quasiperiodic dynamics, there is erratic evolution of the weight with time. So, from this, it could be ascertained that the ICO learning could be made use of in the classification of synchronization dynamics in nonlinear systems., Comment: 6 pages, 5 figures

The Cislunar region is crucial for expanding human presence in space in the forthcoming decades. This paper presents a comprehensive review of recent and anticipated Earth-Moon missions, and ongoing space domain awareness initiatives. An introduction to the dynamics as well as periodic trajectories in the Cislunar realm is presented. Then, a review of modern Cislunar programs as well as smaller missions are compiled to provide insights into the key players pushing towards the Moon. Trends of Cislunar missions and practices are identified, including the identification of regions of interest, such as the South Pole and the Near-rectilinear halo orbit. Finally, a review of the current state and short-comings of space domain awareness (SDA) in the region is included, utilizing the regions of interest as focal points for required improvement. The SDA review is completed through the analysis of the Artemis 1 trajectory., Comment: Progress in Aerospace Sciences, July 2024

The chaotic dynamics of quantum many-body systems are expected to quickly randomize any structured initial state, delocalizing it in Fock space. In this work, we study the spreading of an initial product state in Hilbert space under dual-unitary dynamics, captured by the inverse participation ratios and the distribution of overlaps (bit-string probabilities). We consider the self-dual kicked Ising model, a minimal model of many-body quantum chaos which can be seen as either a periodically driven Floquet model or a dual-unitary quantum circuit. Both analytically and numerically, we show that the inverse participation ratios rapidly approach their ergodic values, corresponding to those of Haar random states, and establish the emergence of the Porter-Thomas distribution for the overlap distribution. Importantly, this convergence happens exponentially fast in time, with a time scale that is independent of system size. We inspect the effect of local perturbations that break dual-unitarity and show a slowdown of the spreading in Fock space, indicating that dual-unitary circuits are maximally efficient at preparing random states., Comment: 6+2 pages

We use parsimonious diffusion maps (PDMs) to discover the latent dynamics of high-fidelity Navier-Stokes simulations with a focus on the 2D fluidic pinball problem. By varying the Reynolds number, different flow regimes emerge, ranging from steady symmetric flows to quasi-periodic asymmetric and turbulence. We show, that the proposed non-linear manifold learning scheme, identifies in a crisp manner the expected intrinsic dimension of the underlying emerging dynamics over the parameter space. In particular, PDMs, estimate that the emergent dynamics in the oscillatory regime can be captured by just two variables, while in the chaotic regime, the dominant modes are three as anticipated by the normal form theory. On the other hand, proper orthogonal decomposition (POD)/PCA, most commonly used for dimensionality reduction in fluid mechanics, does not provide such a crisp separation between the dominant modes. To validate the performance of PDMs, we also computed the reconstruction error, by constructing a decoder using Geometric Harmonics. We show that the proposed scheme outperforms the POD/PCA over the whole Reynolds number range. Thus, we believe that the proposed scheme will allow for the development of more accurate reduced order models for high-fidelity fluid dynamics simulators, thus relaxing the curse of dimensionality in numerical analysis tasks such as bifurcation analysis, optimization and control.

An operation that maps one natural number to another can be considered as a dynamical system in $\mathbb{N}^+$. Some of such systems, e.g. the mapping in the so-called 3x+1 problem proposed by Collatz, is conjectured to have a single global attractor, whereas other systems, e.g. linear congruence, could be ergodic. Here we demonstrate that an operation that is based on digital reversal, has a spectrum of dynamical behaviors, including 2-cycle, 12-cycle, periodic attractors with other cycle lengths, and diverging limiting dynamics that escape to infinity. This dynamical system has infinite number of cyclic attractors, and may have unlimited number of cycle lengths. It also has potentially infinite number of diverging trajectories with a recurrent pattern repeating every 8 steps. Although the transient time before settling on a limiting dynamics is relatively short, we speculate that transient times may not have an upper bound., Comment: 5 figures

We study a system of coupled pendula with diffusive interactions, which could depend both on positions and on momenta. The coupling structure is defined by an undirected network, while the dynamic equations are derived from a Hamiltonian; as such, the energy is conserved. We provide sufficient conditions on the energy for bounded motion, identifying the oscillatory regime. We also describe the bifurcations of synchronised states, exploiting the quantities related to the network structure. Moreover, we consider patterns of anti-synchrony arising from the specific properties of the model. Such patterns can be linked to global properties of the system by looking at motion of the centre of mass. Nonetheless, the anti-synchrony patterns play also a relevant role in the bifurcation analysis. For the case of graphs with a large number of nodes, we characterise the parameter range in which the bifurcations occur. We complement the analysis with some numerical simulations showing the interplay between bifurcations of the origin and transitions to chaos of nearby orbits. A key feature is that the observed chaotic regime emerges at low energies.

We introduce new machine-learning techniques for analyzing chaotic dynamical systems. The primary objectives of the study include the development of a new and simple method for calculating the Lyapunov exponent using only two trajectory data points unlike traditional methods that require an averaging procedure, the exploration of phase transition graphs from regular periodic to chaotic dynamics to identify "almost integrable" trajectories where conserved quantities deviate from whole numbers, and the identification of "integrable regions" within chaotic trajectories. These methods are applied and tested on two dynamical systems: "Two objects moving on a rod" and the "Henon-Heiles" systems., Comment: 15 pages, 6 figures

This essay, derived from a lecture at "The Physics Modeling of Thought" workshop in Berlin in winter 2023, explores the mutually beneficial relationship between theoretical neuroscience and statistical physics through the lens of efficient coding and computation in cortical circuits. It highlights how the study of neural networks has enhanced our understanding of complex, nonequilibrium, and disordered systems, while also demonstrating how neuroscientific challenges have spurred novel developments in physics. The paper traces the evolution of ideas from seminal work on chaos in random neural networks to recent developments in efficient coding and the partial suppression of chaotic fluctuations. It emphasizes how concepts from statistical physics, such as phase transitions and critical phenomena, have been instrumental in elucidating the computational capabilities of neural networks. By examining the interplay between order and disorder in neural computation, the essay illustrates the deep connection between theoretical neuroscience and the statistical physics of nonequilibrium systems. This synthesis underscores the ongoing importance of interdisciplinary approaches in advancing both fields, offering fresh perspectives on the fundamental principles governing information processing in biological and artificial systems. This multidisciplinary approach not only advances our understanding of neural computation and complex systems but also points toward future challenges at the intersection of neuroscience and physics.

Origami structures have been receiving a lot of attention from engineering and scientific researchers owing to their unique properties such as deployability, multi-stability, negative stiffness, etc. However, dynamic properties of origami structures have not been explored much due to a lack of validated analytical dynamic modeling approaches. Given the range of interesting properties and applications of origami structures, it is important to study the dynamic behavior of origami structures. In this study, a dynamic modeling approach for origami structures is presented considering distributed mass modeling, which has the potential to be a generalizable approach. In the proposed approach, stiffness is modeled using the bar and hinge modeling approach while the mass is modeled using the mass distribution approach. Various candidate mass distribution approaches were investigated by comparing their responses to the finite element method responses for various geometric conditions, loading and boundary conditions, and deformation modes. It was observed that a dynamic modeling approach with triangle circumcenter mass distribution was able to capture most of the dynamics satisfactorily consistently. Subsequently, a Miura-ori specimen was manufactured and its free vibration response was determined experimentally and then compared to the prediction of the analytical model. The comparison demonstrated that the analytical model was able to capture most of the dynamics in the longitudinal direction.

We construct a $d$-dimensional Eddy Damped Quasi-Normal Markovian (EDQNM) Closure Model to study dynamo action in arbitrary dimensions. In particular, we find lower $d_L$ and upper $d_U$ critical dimensions for sustained dynamo action in this incompressible problem. Our model is adaptable for future studies incorporating helicity, compressible effects and a wide range of magnetic Reynolds and Prandtl numbers., Comment: 15 pages, 5 figures

We discover strange nonchaotic attractor (SNA) through experiments in an unforced system comprising turbulent reactive flow. While models suggest SNAs are common in dynamical systems, experimental observations are primarily limited to systems with external forcing. We observe SNA prior to the emergence of periodic oscillations from chaotic fluctuations. In complex systems, self-organization can lead to order, and inherent nonlinearity can induce chaos. The occurrence of SNA, which is nonchaotic yet nonperiodic in one such complex system, is intriguing., Comment: 14 pages, 5 figures

The driven Kerr parametric oscillator, of interest to fundamental physics and quantum technologies, exhibits an excited state quantum phase transition (ESQPT) originating in an unstable classical periodic orbit. The main signature of this type of ESQPT is a singularity in the level density in the vicinity of the energy of the classical separatrix that divides the phase space into two distinct regions. The quantum states with energies below the separatrix are useful for quantum technologies, because they show a cat-like structure that protects them against local decoherence processes. In this work, we show how chaos arising from the interplay between the external drive and the nonlinearities of the system destroys the ESQPT and eventually eliminates the cat states. Our results demonstrate the importance of the analysis of theoretical models for the design of new parametric oscillators with ever larger nonlinearities., Comment: 4 + pages, 4 figures + Supplemental Material

By means of numerical analysis conducted with the aid of the computer, the collective synchronization of coupled phase oscillators in the Kuramoto model in the connected regime of random networks of various sizes is studied. The oscillators synchronize and achieve phase coherence, and this process is not significantly affected by the level of connectivity of the network. If the probability that two oscillators are coupled is around the network connectivity threshold synchronization still occurs, although in a more attenuated way. If the size of the network is sufficiently large the oscillators have a phase transition., Comment: 6 pages, 4 figures

On applying a small bias force, non-equilibrium systems may respond in paradoxical ways such as with giant negative mobility (GNM) -- a large net drift opposite to the applied bias, or giant positive mobility (GPM) -- an anomalously large drift in the same direction as the applied bias. We consider an idealized model of a memory-driven active particle inspired from experiments with walking/superwalking droplets that takes the form of the celebrated Lorenz system. By adding a small bias force to this Lorenz model, we uncover a mechanism for simultaneous emergence of GNM and GPM in the parameter space. Within the chaotic sea of the parameter space, a symmetric pair of coexisting asymmetric limit cycles separate and migrate under applied bias force, resulting in anomalous transport behaviors that are sensitive to the particle's memory. This provides a general dynamical mechanism for the emergence of anomalous transport behaviors for self-driven particles described by low-dimensional autonomous chaotic systems., Comment: 5 pages, 4 figures

In regular dynamics, discrete maps are model presentations of discrete dynamical systems, and they may approximate continuous dynamical systems. Maps are used to investigate general properties of dynamical systems and to model various natural and socioeconomic systems. They are also used in engineering. Many natural and almost all socioeconomic systems possess memory which, in many cases, is power-law-like memory. Generalized fractional maps, in which memory is not exactly the power-law memory but the asymptotically power-law-like memory, are used to model and investigate general properties of these systems. In this paper we extend the definition of the notion of generalized fractional maps of arbitrary positive orders that previously was defined only for maps which, in the case of integer orders, converge to area/volume-preserving maps. Fractional generalizations of H'enon and Lozi maps belong to the newly defined class of generalized fractional maps. We derive the equations which define periodic points in generalized fractional maps. We consider applications of our results to the fractional and fractional difference H'enon and Lozi maps., Comment: 13 pages, 1 figure