Nonlinear Sciences
- [1] arXiv:2405.13840 [pdf, ps, html, other]
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Title: Asymptotic behaviour of the confidence region in orbit determination for hyperbolic maps with a parameterSubjects: Chaotic Dynamics (nlin.CD); Mathematical Physics (math-ph); Dynamical Systems (math.DS)
When dealing with an orbit determination problem, uncertainties naturally arise from intrinsic errors related to observation devices and approximation models. Following the least squares method and applying approximation schemes such as the differential correction, uncertainties can be geometrically summarized in confidence regions and estimated by confidence ellipsoids. We investigate the asymptotic behaviour of the confidence ellipsoids while the number of observations and the time span over which they are performed simultaneously increase. Numerical evidences suggest that, in the chaotic scenario, the uncertainties decay at different rates whether the orbit determination is set up to recover the initial conditions alone or along with a dynamical or kinematical parameter, while in the regular case there is no distinction. We show how to improve some of the results in \cite{maro.bonanno}, providing conditions that imply a non-faster-than-polynomial rate of decay in the chaotic case with the parameter, in accordance with the numerical experiments. We also apply these findings to well known examples of chaotic maps, such as piecewise expanding maps of the unit interval or affine hyperbolic toral transformations. We also discuss the applicability to intermittent maps.
- [2] arXiv:2405.13917 [pdf, ps, html, other]
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Title: Numerical spectral analysis of standing waves in quantum hydrodynamics with viscositySubjects: Chaotic Dynamics (nlin.CD); Analysis of PDEs (math.AP)
We study the spectrum of the linearization around standing wave profiles for two quantum hydrodynamics systems with linear and nonlinear viscosity. The essential spectrum for such profiles is stable; we investigate the point spectrum using an Evans function technique. For both systems we show numerically that there exists a real unstable eigenvalue, thus providing numerical evidence for spectral instability.
- [3] arXiv:2405.14217 [pdf, ps, other]
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Title: Symmetry and symmetry-breaking in soil pores and climate change mitigation: What fractal geometry can tell us?Subjects: Pattern Formation and Solitons (nlin.PS)
Soil is a critical component of terrestrial ecosystems, directly influencing global biogeochemical cycles. Despite its importance, the complex architecture of soil pores and their impact on greenhouse gas emissions remain poorly understood. This perspective aims to address this gap by applying symmetry and symmetry-breaking concepts through fractal geometry to elucidate the structural and functional complexities of soil pores. We highlight how fractal parameters can quantify the self-similar nature of soil pore structures, revealing their size, shape, and connectivity. These geometric attributes influence soil properties such as permeability and diffusivity, which are essential for understanding gas exchange and microbial activity within the soil matrix. Furthermore, we emphasize the effects of various land management practices, including tillage and wetting-drying cycles, on soil pore complexity using three-dimensional multi-fractal analysis. Literature indicates that different agricultural practices significantly alter pore heterogeneity and connectivity, affecting greenhouse gas emissions. Conventional tillage decreases pore connectivity and increases randomness, whereas no-tillage preserves larger, more complex pore structures. We propose that integrating combinatorial, geometric, and functional symmetry concepts offers a comprehensive framework for examining the structure-property-function relationships in soil. This novel approach could enhance our understanding of soil's role in the global cycle of greenhouse gases and provide insights into sustainable land management practices aimed at mitigating climate change.
- [4] arXiv:2405.14607 [pdf, ps, html, other]
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Title: Discontinuous transition to chaos in a canonical random neural networkComments: Submitted to Physical Review ESubjects: Chaotic Dynamics (nlin.CD); Neurons and Cognition (q-bio.NC)
We study a paradigmatic random recurrent neural network introduced by Sompolinsky, Crisanti, and Sommers (SCS). In the infinite size limit, this system exhibits a direct transition from a homogeneous rest state to chaotic behavior, with the Lyapunov exponent gradually increasing from zero. We generalize the SCS model considering odd saturating nonlinear transfer functions, beyond the usual choice $\phi(x)=\tanh x$. A discontinuous transition to chaos occurs whenever the slope of $\phi$ at 0 is a local minimum (i.e.,~for $\phi'''(0)>0$). Chaos appears out of the blue, by an attractor-repeller fold. Accordingly, the Lyapunov exponent stays away from zero at the birth of chaos.
- [5] arXiv:2405.14682 [pdf, ps, html, other]
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Title: Turing instabilities for three interacting speciesComments: 6 pages, 1 figureSubjects: Pattern Formation and Solitons (nlin.PS); Dynamical Systems (math.DS); Spectral Theory (math.SP); Quantitative Methods (q-bio.QM)
In this paper, I prove necessary and sufficient conditions for the existence of Turing instabilities in a general system with three interacting species. Turing instabilities describe situations when a stable steady state of a reaction system (ordinary differential equation) becomes an unstable homogeneous steady state of the corresponding reaction-diffusion system (partial differential equation). Similarly to a well-known inequality condition for Turing instabilities in a system with two species, I find a set of inequality conditions for a system with three species. Furthermore, I distinguish conditions for the Turing instability when spatial perturbations grow steadily and the Turing-Hopf instability when spatial perturbations grow and oscillate in time simultaneously.
New submissions for Friday, 24 May 2024 (showing 5 of 5 entries )
- [6] arXiv:2405.13180 (cross-list from eess.SP) [pdf, ps, html, other]
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Title: Data Assimilation with Machine Learning Surrogate Models: A Case Study with FourCastNetSubjects: Signal Processing (eess.SP); Machine Learning (cs.LG); Chaotic Dynamics (nlin.CD); Atmospheric and Oceanic Physics (physics.ao-ph); Applications (stat.AP)
Modern data-driven surrogate models for weather forecasting provide accurate short-term predictions but inaccurate and nonphysical long-term forecasts. This paper investigates online weather prediction using machine learning surrogates supplemented with partial and noisy observations. We empirically demonstrate and theoretically justify that, despite the long-time instability of the surrogates and the sparsity of the observations, filtering estimates can remain accurate in the long-time horizon. As a case study, we integrate FourCastNet, a state-of-the-art weather surrogate model, within a variational data assimilation framework using partial, noisy ERA5 data. Our results show that filtering estimates remain accurate over a year-long assimilation window and provide effective initial conditions for forecasting tasks, including extreme event prediction.
- [7] arXiv:2405.13182 (cross-list from q-bio.NC) [pdf, ps, html, other]
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Title: Robustly encoding certainty in a metastable neural circuit modelComments: 14 pages, 10 figuresSubjects: Neurons and Cognition (q-bio.NC); Pattern Formation and Solitons (nlin.PS)
Localized persistent neural activity has been shown to serve delayed estimation of continuous variables. Common experiments require that subjects store and report the feature value (e.g., orientation) of a particular cue (e.g., oriented bar on a screen) after a delay. Visualizing recorded activity of neurons according to their feature tuning reveals activity bumps whose centers wander stochastically, degrading the estimate over time. Bump position therefore represents the remembered estimate. Recent work suggests that bump amplitude may represent estimate certainty reflecting a probabilistic population code for a Bayesian posterior. Idealized models of this type are fragile due to the fine tuning common to constructed continuum attractors in dynamical systems. Here we propose an alternative metastable model for robustly supporting multiple bump amplitudes by extending neural circuit models to include quantized nonlinearities. Asymptotic projections of circuit activity produce low-dimensional evolution equations for the amplitude and position of bump solutions in response to external stimuli and noise perturbations. Analysis of reduced equations accurately characterizes phase variance and the dynamics of amplitude transitions between stable discrete values. More salient cues generate bumps of higher amplitude which wander less, consistent with the experimental finding that greater certainty correlates with more accurate memories.
- [8] arXiv:2405.13569 (cross-list from cond-mat.stat-mech) [pdf, ps, html, other]
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Title: Synchronization through frequency shufflingJournal-ref: Phys. Rev. E 109, L052302 (2024)Subjects: Statistical Mechanics (cond-mat.stat-mech); Adaptation and Self-Organizing Systems (nlin.AO)
A wide variety of engineered and natural systems are modelled as networks of coupled nonlinear oscillators. In nature, the intrinsic frequencies of these oscillators are not constant in time. Here, we probe the effect of such a temporal heterogeneity on coupled oscillator networks, through the lens of the Kuramoto model. To do this, we shuffle repeatedly the intrinsic frequencies among the oscillators at either random or regular time intervals. What emerges is the remarkable effect that frequent shuffling induces earlier onset (i.e., at a lower coupling) of synchrony among the oscillator phases. Our study provides a novel strategy to induce and control synchrony under resource constraints. We demonstrate our results analytically and in experiments with a network of Wien Bridge oscillators with internal frequencies being shuffled in time.
- [9] arXiv:2405.13598 (cross-list from math.RA) [pdf, ps, html, other]
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Title: A classification of automorphic Lie algebras on complex toriComments: 36 pages. To appear in the Proceedings of the Edinburgh Mathematical SocietySubjects: Rings and Algebras (math.RA); Exactly Solvable and Integrable Systems (nlin.SI)
We classify the automorphic Lie algebras of equivariant maps from a complex torus to $\mathfrak{sl}_2(\mathbb{C})$. For each case we compute a basis in a normal form. The automorphic Lie algebras correspond precisely to two disjoint families of Lie algebras parametrised by the modular curve of $\mathrm{PSL}_2(\mathbb{Z})$, apart from four cases, which are all isomorphic to Onsager's algebra.
- [10] arXiv:2405.13716 (cross-list from physics.optics) [pdf, ps, html, other]
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Title: Bose-Einstein condensation of an optical thermodynamic system into a solitonic stateComments: 23 pages, 12 figuresSubjects: Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)
Recent years have seen a resurgence of interest in multimode fibers due to their intriguing physics and applications, with spatial beam self-cleaning (BSC) having received special attention. In BSC light condenses into the fundamental fiber mode at elevated intensities. Despite extensive efforts utilizing optical thermodynamics to explain such counterintuitive beam reshaping process, several challenges still remain in fully understanding underlying physics. Here we provide compelling experimental evidence that BSC in a dissipative dual-core fiber can be understood in full analogy to Bose-Einstein condensation (BEC) in dilute gases. Being ruled by the identical Gross-Pitaevskii Equation, both systems feature a Townes soliton solution, for which we find further evidence by modal decomposition of our experimental data. Specifically, we observe that efficient BSC only sets in after an initial thermalization phase, causing converge towards a Townes beam profile once a threshold intensity has been surpassed. This process is akin to a transition from classical to quantum-mechanical thermodynamics in BEC. Furthermore, our analysis also identifies dissipative processes as a crucial, yet previously unidentified component for efficient BSC in multimode fiber. This discovery paves the way for unprecedented applications of multimode-fiber based systems in ultrafast lasers, communications, and fiber-based delivery of high-power laser beams.
- [11] arXiv:2405.13809 (cross-list from cond-mat.quant-gas) [pdf, ps, html, other]
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Title: Self-trapping phenomenon, multistability and chaos in open anisotropic Dicke dimerSubjects: Quantum Gases (cond-mat.quant-gas); Statistical Mechanics (cond-mat.stat-mech); Chaotic Dynamics (nlin.CD); Optics (physics.optics)
We investigate semiclassical dynamics of coupled atom-photon interacting system described by a dimer of anisotropic Dicke model in the presence of photon loss, exhibiting a rich variety of non-linear dynamics. Based on symmetries and dynamical classification, we characterize and chart out various dynamical phases in a phase diagram. A key feature of this system is the multistability of different dynamical states, particularly the coexistence of various superradiant phases as well as limit cycles. Remarkably, this dimer system manifests self-trapping phenomena, resulting in a photon population imbalance between the cavities. Such a self-trapped state arises from saddle-node bifurcation, which can be understood from an equivalent Landau-Ginzburg description. Additionally, we identify a unique class of oscillatory dynamics self-trapped limit cycle hosting self-trapping of photons. The absence of stable dynamical phases leads to the onset of chaos, which is diagnosed using the saturation value of the decorrelator dynamics. Moreover, in a narrow region, the self-trapped states can coexist with chaotic attractor, which may have intriguing consequences in quantum dynamics. Finally, we discuss the experimental relevance of our findings, which can be tested in cavity and circuit quantum electrodynamics setups.
- [12] arXiv:2405.13892 (cross-list from cond-mat.stat-mech) [pdf, ps, html, other]
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Title: Emergence of Navier-Stokes hydrodynamics in chaotic quantum circuitsComments: 4+epsilon pages, 3 figures; 8 pages supplemental materialSubjects: Statistical Mechanics (cond-mat.stat-mech); Chaotic Dynamics (nlin.CD); Quantum Physics (quant-ph)
We construct an ensemble of two-dimensional nonintegrable quantum circuits that are chaotic but have a conserved particle current, and thus a finite Drude weight. The long-wavelength hydrodynamics of such systems is given by the incompressible Navier-Stokes equations. By analyzing circuit-to-circuit fluctuations in the ensemble we argue that these are negligible, so the circuit-averaged value of transport coefficients like the viscosity is also (in the long-time limit) the value in a typical circuit. The circuit-averaged transport coefficients can be mapped onto a classical irreversible Markov process. Therefore, remarkably, our construction allows us to efficiently compute the viscosity of a family of strongly interacting chaotic two-dimensional quantum systems.
- [13] arXiv:2405.13988 (cross-list from physics.ao-ph) [pdf, ps, html, other]
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Title: Melancholia States of the Atlantic Meridional Overturning CirculationSubjects: Atmospheric and Oceanic Physics (physics.ao-ph); Chaotic Dynamics (nlin.CD)
The Atlantic Meridional Overturning Circulation (AMOC) is a much studied component of the climate system, because its suspected multistability leads to tipping behaviour with large regional and global climatic impacts. In this paper we investigate the global stability properties of the system using an ocean general circulation model. We construct an unstable AMOC state, i.e., an unstable solution of the flow that resides between the stable regimes of a vigorous and collapsed AMOC. Such a solution, also known as a Melancholia or edge state, is a dynamical saddle embedded in the boundary separating the competing basins of attraction. It is physically relevant since it lies on the most probable path of a noise-induced transition between the two stable regimes, and because tipping occurs when one of the attractors and the Melancholia state collide. Its properties may thus give hints towards physical mechanisms and predictability of the critical transition. We find that while the AMOC Melancholia state as viewed from its upper ocean properties lies between the vigorous and collapsed regimes, it is characterized by an Atlantic deep ocean that is fresher and colder compared to both stable regimes. The Melancholia state has higher dynamic enthalpy than either stable state, representing a state of higher potential energy that is in good agreement with the dynamical landscape view on metastability.
- [14] arXiv:2405.14320 (cross-list from cond-mat.stat-mech) [pdf, ps, html, other]
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Title: Active Magnetic Matter: Propelling Ferrimagnetic Domain Walls by Dynamical FrustrationSubjects: Statistical Mechanics (cond-mat.stat-mech); Soft Condensed Matter (cond-mat.soft); Strongly Correlated Electrons (cond-mat.str-el); Pattern Formation and Solitons (nlin.PS)
Active matter encompasses many-particle systems with self-propelling units, such as flocks of birds or schools of fish. Here, we show how self-propelling domain walls can be realised in a solid-state system when a ferrimagnet is weakly driven out of thermal equilibrium by an oscillating field. This activates the Goldstone mode, inducing a rotation of the antiferromagnetic xy-order in a clockwise or anticlockwise direction, determined by the sign of the ferromagnetic component. Two opposite directions of rotation meet at a ferromagnetic domain wall, resulting in 'dynamical frustration', with three main consequences. (i) Domain walls move actively in a direction chosen by spontaneous symmetry breaking. Their speed is proportional to the square root of the driving power across large parameter regimes. (ii) In one dimension (1D), after a quench into the ferrimagnetic phase, this motion and strong hydrodynamic interactions lead to a linear growth of the magnetic correlation length over time, much faster than in equilibrium. (iii) The dynamical frustration makes the system highly resilient to noise. The correlation length of the weakly driven 1D system can be orders of magnitude larger than in the corresponding equilibrium system with the same noise level.
- [15] arXiv:2405.14376 (cross-list from cond-mat.dis-nn) [pdf, ps, html, other]
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Title: Quantum Chaos in Random Ising NetworksAndrás Grabarits, Kasturi Ranjan Swain, Mahsa Seyed Heydari, Pranav Chandarana, Fernando J. Gómez-Ruiz, Adolfo del CampoSubjects: Disordered Systems and Neural Networks (cond-mat.dis-nn); Statistical Mechanics (cond-mat.stat-mech); Chaotic Dynamics (nlin.CD)
We report a systematic investigation of universal quantum chaotic signatures in the transverse field Ising model on an Erdős-Rényi network. This is achieved by studying local spectral measures such as the level spacing and the level velocity statistics. A spectral form factor analysis is also performed as a global measure, probing energy level correlations at arbitrary spectral distances. Our findings show that these measures capture the breakdown of chaotic behavior upon varying the connectivity and strength of the transverse field in various regimes. We demonstrate that the level spacing statistics and the spectral form factor signal this breakdown for sparsely and densely connected networks. The velocity statistics capture the surviving chaotic signatures in the sparse limit. However, these integrable-like regimes extend over a vanishingly small segment in the full range of connectivity.
- [16] arXiv:2405.14442 (cross-list from cs.ET) [pdf, ps, html, other]
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Title: Fully parallel implementation of digital memcomputing on FPGASubjects: Emerging Technologies (cs.ET); Chaotic Dynamics (nlin.CD)
We present a fully parallel digital memcomputing solver implemented on a field-programmable gate array (FPGA) board. For this purpose, we have designed an FPGA code that solves the ordinary differential equations associated with digital memcomputing in parallel. A feature of the code is the use of only integer-type variables and integer constants to enhance optimization. Consequently, each integration step in our solver is executed in 96~ns. This method was utilized for difficult instances of the Boolean satisfiability (SAT) problem close to a phase transition, involving up to about 150 variables. Our results demonstrate that the parallel implementation reduces the scaling exponent by about 1 compared to a sequential C++ code on a standard computer. Additionally, compared to C++ code, we observed a time-to-solution advantage of about three orders of magnitude. Given the limitations of FPGA resources, the current implementation of digital memcomputing will be especially useful for solving compact but challenging problems.
- [17] arXiv:2405.14542 (cross-list from physics.bio-ph) [pdf, ps, html, other]
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Title: Emergence of metastability in frustrated oscillatory networks: the key role of hierarchical modularitySubjects: Biological Physics (physics.bio-ph); Adaptation and Self-Organizing Systems (nlin.AO)
Oscillatory complex networks in the metastable regime have been used to study the emergence of integrated and segregated activity in the brain, which are hypothesised to be fundamental for cognition. Yet, the parameters and the underlying mechanisms necessary to achieve the metastable regime are hard to identify, often relying on maximising the correlation with empirical functional connectivity dynamics. Here, we propose and show that the brain's hierarchically modular mesoscale structure alone can give rise to robust metastable dynamics and (metastable) chimera states in the presence of phase frustration. We construct unweighted $3$-layer hierarchical networks of identical Kuramoto-Sakaguchi oscillators, parameterized by the average degree of the network and a structural parameter determining the ratio of connections between and within blocks in the upper two layers. Together, these parameters affect the characteristic timescales of the system. Away from the critical synchronization point, we detect the emergence of metastable states in the lowest hierarchical layer coexisting with chimera and metastable states in the upper layers. Using the Laplacian renormalization group flow approach, we uncover two distinct pathways towards achieving the metastable regimes detected in these distinct layers. In the upper layers, we show how the symmetry-breaking states depend on the slow eigenmodes of the system. In the lowest layer instead, metastable dynamics can be achieved as the separation of timescales between layers reaches a critical threshold. Our results show an explicit relationship between metastability, chimera states, and the eigenmodes of the system, bridging the gap between harmonic based studies of empirical data and oscillatory models.
- [18] arXiv:2405.14546 (cross-list from cs.GT) [pdf, ps, html, other]
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Title: Global Behavior of Learning Dynamics in Zero-Sum Games with Memory AsymmetryComments: 11 pages, 4 figures (main); 4 pages (appendix)Subjects: Computer Science and Game Theory (cs.GT); Multiagent Systems (cs.MA); Optimization and Control (math.OC); Chaotic Dynamics (nlin.CD)
This study examines the global behavior of dynamics in learning in games between two players, X and Y. We consider the simplest situation for memory asymmetry between two players: X memorizes the other Y's previous action and uses reactive strategies, while Y has no memory. Although this memory complicates the learning dynamics, we discover two novel quantities that characterize the global behavior of such complex dynamics. One is an extended Kullback-Leibler divergence from the Nash equilibrium, a well-known conserved quantity from previous studies. The other is a family of Lyapunov functions of X's reactive strategy. These two quantities capture the global behavior in which X's strategy becomes more exploitative, and the exploited Y's strategy converges to the Nash equilibrium. Indeed, we theoretically prove that Y's strategy globally converges to the Nash equilibrium in the simplest game equipped with an equilibrium in the interior of strategy spaces. Furthermore, our experiments also suggest that this global convergence is universal for more advanced zero-sum games than the simplest game. This study provides a novel characterization of the global behavior of learning in games through a couple of indicators.
- [19] arXiv:2405.14733 (cross-list from physics.flu-dyn) [pdf, ps, html, other]
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Title: Space-time statistics of 2D soliton gas in shallow water studied by stereoscopic surface mappingComments: Accepted for publication in Experiments in FluidsSubjects: Fluid Dynamics (physics.flu-dyn); Pattern Formation and Solitons (nlin.PS)
We describe laboratory experiments in a 2D wave tank that aim at building up and monitor 2D shallow water soliton gas. The water surface elevation is obtained over a large ($\sim 100\,\text{m}^2$) domain, with centimetre-resolution, by stereoscopic vision using two cameras. Floating particles are seeded to get surface texture and determine the wave field by image correlation. With this set-up, soliton propagation and multiple interactions can be measured with a previously unreachable level of detail. The propagation of an oblique soliton is analysed, the amplitude decay and local incidence are compared to analytical predictions. We further present two cases of 2D soliton gas, emerging from multiple line solitons with random incidence ($|\theta|<30^\circ$) and from irregular random waves forced with a {\sc jonswap} spectrum ($|\theta|<45^\circ$). To our knowledge, those are the first observations of random 2D soliton gas for gravity waves. In both cases Mach reflections and Mach expansions result in solitons that mainly propagate in directions perpendicular to the wave-makers.
Cross submissions for Friday, 24 May 2024 (showing 14 of 14 entries )
- [20] arXiv:2404.00721 (replaced) [pdf, ps, html, other]
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Title: Designing robust trajectories by lobe dynamics in low-dimensional Hamiltonian systemsComments: 5 pages, 4 figures + Supplemental Material (5 pages, 6 figures)Subjects: Chaotic Dynamics (nlin.CD); Dynamical Systems (math.DS); Optimization and Control (math.OC); Classical Physics (physics.class-ph)
Modern space missions with uncrewed spacecraft require robust trajectory design to connect multiple chaotic orbits by small controls. To address this issue, we propose a control scheme to design robust trajectories by leveraging a geometrical structure in chaotic zones, known as a {\it lobe}. Our scheme shows that appropriately selected lobes reveal possible paths to traverse chaotic zones in a short time. The effectiveness of our method is demonstrated through trajectory design in both the standard map and Hill's equation.
- [21] arXiv:2201.05789 (replaced) [pdf, ps, html, other]
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Title: Out-of-Time Ordered Correlators in Kicked Coupled Tops: Information Scrambling in Mixed Phase Space and the Role of Conserved QuantitiesComments: 17 pages, 13 figures, 1 table. Close to the accepted version in Chaos: An Interdisciplinary Journal of Nonlinear ScienceSubjects: Quantum Physics (quant-ph); Chaotic Dynamics (nlin.CD)
We study operator growth in a bipartite kicked coupled tops (KCT) system using out-of-time ordered correlators (OTOCs), which quantify ``information scrambling" due to chaotic dynamics and serve as a quantum analog of classical Lyapunov exponents. In the KCT system, chaos arises from the hyper-fine coupling between the spins. Due to a conservation law, the system's dynamics decompose into distinct invariant subspaces. Focusing initially on the largest subspace, we numerically verify that the OTOC growth rate aligns well with the classical Lyapunov exponent for fully chaotic dynamics. While previous studies have largely focused on scrambling in fully chaotic dynamics, works on mixed-phase space scrambling are sparse. We explore scrambling behavior in both mixed-phase space and globally chaotic dynamics. In the mixed phase space, we use Percival's conjecture to partition the eigenstates of the Floquet map into ``regular" and ``chaotic." Using these states as the initial states, we examine how their mean phase space locations affect the growth and saturation of the OTOCs. Beyond the largest subspace, we study the OTOCs across the entire system, including all other smaller subspaces. For certain initial operators, we analytically derive the OTOC saturation using random matrix theory (RMT). When the initial operators are chosen randomly from the unitarily invariant random matrix ensembles, the averaged OTOC relates to the linear entanglement entropy of the Floquet operator, as found in earlier works. For the diagonal Gaussian initial operators, we provide a simple expression for the OTOC.
- [22] arXiv:2305.05328 (replaced) [pdf, ps, html, other]
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Title: Dynamical properties and mechanisms of metastability: a perspective in neuroscienceKalel L. Rossi, Roberto C. Budzinski, Everton S. Medeiros, Bruno R. R. Boaretto, Lyle Muller, Ulrike FeudelComments: 4 figuresSubjects: Neurons and Cognition (q-bio.NC); Dynamical Systems (math.DS); Chaotic Dynamics (nlin.CD)
Metastability, characterized by a variability of regimes in time, is a ubiquitous type of neural dynamics. It has been formulated in many different ways in the neuroscience literature, however, which may cause some confusion. In this Perspective, we discuss metastability from the point of view of dynamical systems theory. We extract from the literature a very simple but general definition through the concept of metastable regimes as long-lived but transient epochs of activity with unique dynamical properties. This definition serves as an umbrella term that encompasses formulations from other works, and readily connects to concepts from dynamical systems theory. This allows us to examine general dynamical properties of metastable regimes, propose in a didactic manner several dynamics-based mechanisms that generate them, and discuss a theoretical tool to characterize them quantitatively. This perspective leads to insights that help to address issues debated in the literature and also suggest pathways for future research.
- [23] arXiv:2308.11279 (replaced) [pdf, ps, html, other]
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Title: Thermocapillary Thin Films: Periodic Steady States and Film RuptureComments: 31 pages, 8 figures; we added a remark regarding the instability of positive solutions with additional referencesSubjects: Analysis of PDEs (math.AP); Pattern Formation and Solitons (nlin.PS); Fluid Dynamics (physics.flu-dyn)
We study stationary, periodic solutions to the thermocapillary thin-film model
\begin{equation*}
\partial_t h + \partial_x \Bigl(h^3(\partial_x^3 h - g\partial_x h) + M\frac{h^2}{(1+h)^2}\partial_xh\Bigr) = 0,\quad t>0,\ x\in \mathbb{R},
\end{equation*} which can be derived from the Bénard-Marangoni problem via a lubrication approximation. When the Marangoni number $M$ increases beyond a critical value $M^*$, the constant solution becomes spectrally unstable via a (conserved) long-wave instability and periodic stationary solutions bifurcate. For a fixed period, we find that these solutions lie on a global bifurcation curve of stationary, periodic solutions with a fixed wave number and mass. Furthermore, we show that the stationary periodic solutions on the global bifurcation branch converge to a weak stationary periodic solution which exhibits film rupture. The proofs rely on a Hamiltonian formulation of the stationary problem and the use of analytic global bifurcation theory. Finally, we show the instability of the bifurcating solutions close to the bifurcation point and give a formal derivation of the amplitude equation governing the dynamics close to the onset of instability. - [24] arXiv:2402.02994 (replaced) [pdf, ps, html, other]
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Title: Extreme statistics and extreme events in dynamical models of turbulenceXander M. de Wit, Giulio Ortali, Alessandro Corbetta, Alexei A. Mailybaev, Luca Biferale, Federico ToschiSubjects: Fluid Dynamics (physics.flu-dyn); Chaotic Dynamics (nlin.CD)
We present a study of the intermittent properties of a shell model of turbulence with unprecedented statistics, about $\sim 10^7$ eddy turn over time, achieved thanks to an implementation on a large-scale parallel GPU factory. This allows us to quantify the inertial range anomalous scaling properties of the velocity fluctuations up to the 24th order moment. Through a careful assessment of the statistical and systematic uncertainties, we show that none of the phenomenological and theoretical models previously proposed in the literature to predict the anomalous power-law exponents in the inertial range is in agreement with our high-precision numerical measurements. We find that at asymptotically high order moments, the anomalous exponents tend towards a linear scaling, suggesting that extreme turbulent events are dominated by one leading singularity. We found that systematic corrections to scaling induced by the infrared and ultraviolet (viscous) cut-offs are the main limitations to precision for low-order moments, while high orders are mainly affected by the finite statistical samples. The unprecedentedly high fidelity numerical results reported in this work offer an ideal benchmark for the development of future theoretical models of intermittency in dynamical systems for either extreme events (high-order moments) or typical fluctuations (low-order moments). For the latter, we show that we achieve a precision in the determination of the inertial range scaling exponents of the order of one part over ten thousand (5th significant digit), which must be considered a record for out-of-equilibrium fluid-mechanics systems and models.
- [25] arXiv:2404.00754 (replaced) [pdf, ps, html, other]
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Title: Imitation dynamics and the replicator equationSubjects: Physics and Society (physics.soc-ph); Adaptation and Self-Organizing Systems (nlin.AO); Populations and Evolution (q-bio.PE)
Evolutionary game theory has impacted many fields of research by providing a mathematical framework for studying the evolution and maintenance of social and moral behaviors. This success is owed in large part to the demonstration that the central equation of this theory - the replicator equation - is the deterministic limit of a stochastic imitation (social learning) dynamics. Here we offer an alternative elementary proof of this result, which holds for the scenario where players compare their instantaneous (not average) payoffs to decide whether to maintain or change their strategies, and only more successful individuals can be imitated.
- [26] arXiv:2404.05755 (replaced) [pdf, ps, html, other]
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Title: The Rosenzweig Porter model revisited for the three Wigner Dyson symmetry classesSubjects: Statistical Mechanics (cond-mat.stat-mech); Chaotic Dynamics (nlin.CD)
We present numerical results for the Rosenzweig Porter model for all symmetry classes of the Dyson threefold way. We analyzed the fluctuation properties in the eigenvalue spectra, and compared them with existing and new analytical results. Based on these results we propose characteristics of the spectral properties as measures to explore the transition from Poisson to Wigner Dyson WD statistics. Furthermore, we performed thorough studies of the properties of the eigenvectors in terms of the fractal dimensions, the Kullback Leibler KL divergences and the fidelity susceptibility. The ergodic and Anderson transitions take place at the same parameter values and a finite size scaling analysis of the KL divergences at the transitions yields the same critical exponents for all three WD classes, thus indicating superuniversality of these transitions.
- [27] arXiv:2405.02870 (replaced) [pdf, ps, html, other]
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Title: Tetrahedron dualityComments: 22 pages. v2: references addedSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI)
A certain two-dimensional supersymmetric gauge theory is argued to satisfy a relation that promotes the Zamolodchikov tetrahedron equation to an infrared duality between two quantum field theories. Solutions of the tetrahedron equation with continuous spin variables are obtained from partition functions of this theory and its variants.
- [28] arXiv:2405.10190 (replaced) [pdf, ps, html, other]
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Title: Comparative Analysis of Predicting Subsequent Steps in H\'enon MapComments: 19 pages, 9 figuresSubjects: Machine Learning (cs.LG); Chaotic Dynamics (nlin.CD)
This paper explores the prediction of subsequent steps in Hénon Map using various machine learning techniques. The Hénon map, well known for its chaotic behaviour, finds applications in various fields including cryptography, image encryption, and pattern recognition. Machine learning methods, particularly deep learning, are increasingly essential for understanding and predicting chaotic phenomena. This study evaluates the performance of different machine learning models including Random Forest, Recurrent Neural Network (RNN), Long Short-Term Memory (LSTM) networks, Support Vector Machines (SVM), and Feed Forward Neural Networks (FNN) in predicting the evolution of the Hénon map. Results indicate that LSTM network demonstrate superior predictive accuracy, particularly in extreme event prediction. Furthermore, a comparison between LSTM and FNN models reveals the LSTM's advantage, especially for longer prediction horizons and larger datasets. This research underscores the significance of machine learning in elucidating chaotic dynamics and highlights the importance of model selection and dataset size in forecasting subsequent steps in chaotic systems.