Nonlinear Sciences
- [1] arXiv:2405.08844 [pdf, ps, other]
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Title: Scalable synchronization cluster in networked chaotic oscillatorsComments: 12 pages, 8 figuresSubjects: Adaptation and Self-Organizing Systems (nlin.AO); Chaotic Dynamics (nlin.CD); Pattern Formation and Solitons (nlin.PS)
Cluster synchronization in synthetic networks of coupled chaotic oscillators is investigated. It is found that despite the asymmetric nature of the network structure, a subset of the oscillators can be synchronized as a cluster while the other oscillators remain desynchronized. Interestingly, with the increase of the coupling strength, the cluster is expanding gradually by recruiting the desynchronized oscillators one by one. This new synchronization phenomenon, which is named ``scalable synchronization cluster", is explored theoretically by the method of eigenvector-based analysis, and it is revealed that the scalability of the cluster is attributed to the unique feature of the eigenvectors of the network coupling matrix. The transient dynamics of the cluster in response to random perturbations are also studied, and it is shown that in restoring to the synchronization state, oscillators inside the cluster are stabilized in sequence, illustrating again the hierarchy of the oscillators. The findings shed new light on the collective behaviors of networked chaotic oscillators, and are helpful for the design of real-world networks where scalable synchronization clusters are concerned.
- [2] arXiv:2405.09147 [pdf, ps, html, other]
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Title: Synchronization of E. coli bacteria moving in coupled wellsAleksandre Japaridze, Victor Struijk, Kushal Swamy, Irek Roslon, Oriel Shoshani, Cees Dekker, Farbod AlijaniSubjects: Adaptation and Self-Organizing Systems (nlin.AO); Other Condensed Matter (cond-mat.other); Biological Physics (physics.bio-ph)
Synchronization plays a crucial role in the dynamics of living organisms, from fireflies flashing in unison to pacemaker cells that jointly generate heartbeats. Uncovering the mechanism behind these phenomena requires an understanding of individual biological oscillators and the coupling forces between them. Here, we develop a single-cell assay that studies rhythmic behavior in the motility of individual E.coli cells that can be mutually synchronized. Circular microcavities are used to isolate E.coli cells that swim along the cavity wall, resulting in self-sustained oscillations. Upon connecting these cavities by microchannels the bacterial motions can be coupled, yielding nonlinear dynamic synchronization patterns with phase slips. We demonstrate that the coordinated movement observed in coupled E. coli oscillators follows mathematical rules of synchronization which we use to quantify the coupling strength. These findings advance our understanding of motility in confinement, and lay the foundation for engineering desired dynamics in microbial active matter.
New submissions for Thursday, 16 May 2024 (showing 2 of 2 entries )
- [3] arXiv:2405.08825 (cross-list from stat.ML) [pdf, ps, html, other]
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Title: Thermodynamic limit in learning period threeComments: 26 pages, 19 figuresSubjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Adaptation and Self-Organizing Systems (nlin.AO); Chaotic Dynamics (nlin.CD)
A continuous one-dimensional map with period three includes all periods. This raises the following question: Can we obtain any types of periodic orbits solely by learning three data points? We consider learning period three with random neural networks and report the universal property associated with it. We first show that the trained networks have a thermodynamic limit that depends on the choice of target data and network settings. Our analysis reveals that almost all learned periods are unstable and each network has its characteristic attractors (which can even be untrained ones). Here, we propose the concept of characteristic bifurcation expressing embeddable attractors intrinsic to the network, in which the target data points and the scale of the network weights function as bifurcation parameters. In conclusion, learning period three generates various attractors through characteristic bifurcation due to the stability change in latently existing numerous unstable periods of the system.
- [4] arXiv:2405.08866 (cross-list from quant-ph) [pdf, ps, html, other]
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Title: On the quantum origin of limit cycles, fixed points, and critical slowing downComments: 4 pages, 6 figures + SupplementSubjects: Quantum Physics (quant-ph); Quantum Gases (cond-mat.quant-gas); Statistical Mechanics (cond-mat.stat-mech); Adaptation and Self-Organizing Systems (nlin.AO)
Among the most iconic features of classical dissipative dynamics are persistent limit-cycle oscillations and critical slowing down at the onset of such oscillations, where the system relaxes purely algebraically in time. On the other hand, quantum systems subject to generic Markovian dissipation decohere exponentially in time, approaching a unique steady state. Here we show how coherent limit-cycle oscillations and algebraic decay can emerge in a quantum system governed by a Markovian master equation as one approaches the classical limit, illustrating general mechanisms using a single-spin model and a two-site lossy Bose-Hubbard model. In particular, we demonstrate that the fingerprint of a limit cycle is a slow-decaying branch with vanishing decoherence rates in the Liouville spectrum, while a power-law decay is realized by a spectral collapse at the bifurcation point. We also show how these are distinct from the case of a classical fixed point, for which the quantum spectrum is gapped and can be generated from the linearized classical dynamics.
- [5] arXiv:2405.08922 (cross-list from math.DS) [pdf, ps, html, other]
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Title: Is every triangle a trajectory of an elliptical billiard?Comments: 18 pages, 16 figuresSubjects: Dynamical Systems (math.DS); Mathematical Physics (math-ph); Complex Variables (math.CV); Metric Geometry (math.MG); Exactly Solvable and Integrable Systems (nlin.SI)
Using Marden's Theorem from geometric theory of polynomials, we show that for every triangle there is a unique ellipse such that the triangle is a billiard trajectory within that ellipse. Since $3$-periodic trajectories of billiards within ellipses are examples of the Poncelet polygons, our considerations provide a new insight into the relationship between Marden's Theorem and the Poncelet Porism, two gems of exceptional classical beauty. We also show that every parallelogram is a billiard trajectory within a unique ellipse. We prove a similar result for the self-intersecting polygonal lines consisting of two pairs of congruent sides, named ``Darboux butterflies". In each of three considered cases, we effectively calculate the foci of the boundary ellipses.
- [6] arXiv:2405.09453 (cross-list from cs.LG) [pdf, ps, html, other]
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Title: Kuramoto Oscillators and Swarms on Manifolds for Geometry Informed Machine LearningSubjects: Machine Learning (cs.LG); Mathematical Physics (math-ph); Adaptation and Self-Organizing Systems (nlin.AO)
We propose the idea of using Kuramoto models (including their higher-dimensional generalizations) for machine learning over non-Euclidean data sets. These models are systems of matrix ODE's describing collective motions (swarming dynamics) of abstract particles (generalized oscillators) on spheres, homogeneous spaces and Lie groups. Such models have been extensively studied from the beginning of XXI century both in statistical physics and control theory. They provide a suitable framework for encoding maps between various manifolds and are capable of learning over spherical and hyperbolic geometries. In addition, they can learn coupled actions of transformation groups (such as special orthogonal, unitary and Lorentz groups). Furthermore, we overview families of probability distributions that provide appropriate statistical models for probabilistic modeling and inference in Geometric Deep Learning. We argue in favor of using statistical models which arise in different Kuramoto models in the continuum limit of particles. The most convenient families of probability distributions are those which are invariant with respect to actions of certain symmetry groups.
Cross submissions for Thursday, 16 May 2024 (showing 4 of 4 entries )
- [7] arXiv:2210.09812 (replaced) [pdf, ps, other]
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Title: Characterizing far from equilibrium states of the one-dimensional nonlinear Schr{\"o}dinger equationAbhik Kumar Saha, Romain Dubessy (LPL)Comments: Submission to SciPostSubjects: Pattern Formation and Solitons (nlin.PS)
We use the mathematical toolbox of the inverse scattering transform to study quantitatively the number of solitons in far from equilibrium one-dimensional systems described by the defocusing nonlinear Schr{ö}dinger equation. We present a simple method to identify the discrete eigenvalues in the Lax spectrum and provide a extensive benchmark of its efficiency. Our method can be applied in principle to all physical systems described by the defocusing nonlinear Schr{ö}dinger equation and allows to identify the solitons velocity distribution in numerical simulations and possibly experiments.
- [8] arXiv:2305.00539 (replaced) [pdf, ps, html, other]
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Title: Dynein-driven self-organization of microtubules: An entropy- and network-based analysisComments: 15 pages, 10 figures, 3 tables, 44 referencesSubjects: Adaptation and Self-Organizing Systems (nlin.AO); Pattern Formation and Solitons (nlin.PS); Subcellular Processes (q-bio.SC)
Microtubules self-organize to form part of the cellular cytoskeleton. They give cells their shape and play a crucial role in cell division and intracellular transport. Strikingly, microtubules driven by motor proteins reorganize into stable mitotic/meiotic spindles with high spatial and temporal precision during successive cell division cycles. Although the topic has been extensively studied, the question remains: What defines such microtubule networks' spatial order and robustness? Here, we aim to approach this problem by analyzing a simplified computational model of radial microtubule self-organization driven by a single type of motor protein -- dyneins. We establish that the spatial order of the steady-state pattern is likely associated with the dynein-driven microtubule motility. At the same time, the structure of the microtubule network is likely linked to its connectivity at the beginning of self-organization. Using the continuous variation of dynein concentration, we reveal hysteresis in microtubule self-organization, ensuring the stability of radial filament structures.
- [9] arXiv:2404.06394 (replaced) [pdf, ps, html, other]
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Title: On the minimal memory set of cellular automataComments: 10 pagesSubjects: Cellular Automata and Lattice Gases (nlin.CG); Formal Languages and Automata Theory (cs.FL); Dynamical Systems (math.DS)
For a group $G$ and a finite set $A$, a cellular automaton (CA) is a transformation $\tau : A^G \to A^G$ defined via a finite memory set $S \subseteq G$ and a local map $\mu : A^S \to A$. Although memory sets are not unique, every CA admits a unique minimal memory set, which consists on all the essential elements of $S$ that affect the behavior of the local map. In this paper, we study the links between the minimal memory set and the generating patterns $\mathcal{P}$ of $\mu$; these are the patterns in $A^S$ that are not fixed when the cellular automaton is applied. In particular, we show that when $\vert S \vert \geq 2$ and $\vert \mathcal{P} \vert$ is not a multiple of $\vert A \vert$, then the minimal memory set must be $S$ itself. Moreover, when $\vert \mathcal{P} \vert = \vert A \vert$, $\vert S \vert \geq 3$, and the restriction of $\mu$ to these patterns is well-behaved, then the minimal memory set must be $S$ or $S \setminus \{s\}$, for some $s \in S \setminus \{e\}$. These are some of the first general theoretical results on the minimal memory set of a cellular automaton.
- [10] arXiv:2402.14330 (replaced) [pdf, ps, other]
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Title: Collisions of Light Bullets with Different Circular PolarizationsComments: 7 pages, 15 figures, in English, to appear as [JETP Lett. 119(8), 585 (2024)]Subjects: Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)
Collisions of left- and right-polarized spatiotemporal optical solitons have been numerically simulated for a locally isotropic focusing Kerr medium with anomalous chromatic dispersion. The stable propagation of such ``light bullets'' in a moderate nonlinear regime is ensured by a transverse parabolic profile of the refraction index in a multimode waveguide. The transverse motion of centers of mass of wave packets in such systems occurs on classical trajectories of a harmonic oscillator, whereas the motion in the longitudinal direction is uniform. Therefore, collisions of two solitons can be not only head-on but also tangential. An inelastic collision of two solitons with opposite circular polarizations can result either in two binary light bullets combining the left and right polarization or in more complex bound systems.
DOI: https://doi.org/10.1134/S0021364024600691