{"title":"A curious dynamical system in the plane","authors":"Stefan Steinerberger, Tony Zeng","doi":"arxiv-2409.08961","DOIUrl":"https://doi.org/arxiv-2409.08961","url":null,"abstract":"For any irrational $alpha > 0$ and any initial value $z_{-1} in\u0000mathbb{C}$, we define a sequence of complex numbers $(z_n)_{n=0}^{infty}$ as\u0000follows: $z_n$ is $z_{n-1} + e^{2 pi i alpha n}$ or $z_{n-1} - e^{2 pi i\u0000alpha n}$, whichever has the smaller absolute value. If both numbers have the\u0000same absolute value, the sequence terminates at $z_{n-1}$ but this happens\u0000rarely. This dynamical system has astonishingly intricate behavior: the choice\u0000of signs in $z_{n-1} pm e^{2 pi i alpha n}$ appears to eventually become\u0000periodic (though the period can be large). We prove that if one observes\u0000periodic signs for a sufficiently long time (depending on $z_{-1}, alpha$),\u0000the signs remain periodic for all time. The surprising complexity of the system\u0000is illustrated through examples.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"188 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250413","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Computing Lyapunov Exponents using Weighted Birkhoff Averages","authors":"E. Sander, J. D. Meiss","doi":"arxiv-2409.08496","DOIUrl":"https://doi.org/arxiv-2409.08496","url":null,"abstract":"The Lyapunov exponents of a dynamical system measure the average rate of\u0000exponential stretching along an orbit. Positive exponents are often taken as a\u0000defining characteristic of chaotic dynamics. However, the standard\u0000orthogonalization-based method for computing Lyapunov exponents converges\u0000slowly -- if at all. Many alternatively techniques have been developed to\u0000distinguish between regular and chaotic orbits, though most do not compute the\u0000exponents. We compute the Lyapunov spectrum in three ways: the standard method,\u0000the weighted Birkhoff average (WBA), and the ``mean exponential growth rate for\u0000nearby orbits'' (MEGNO). The latter two improve convergence for nonchaotic\u0000orbits, but the WBA is fastest. However, for chaotic orbits the three methods\u0000convergence at similar, slow rates. Though the original MEGNO method does not\u0000compute Lyapunov exponents, we show how to reformulate it as a weighted average\u0000that does.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250420","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jonah Botvinick-Greenhouse, Maria Oprea, Romit Maulik, Yunan Yang
{"title":"Measure-Theoretic Time-Delay Embedding","authors":"Jonah Botvinick-Greenhouse, Maria Oprea, Romit Maulik, Yunan Yang","doi":"arxiv-2409.08768","DOIUrl":"https://doi.org/arxiv-2409.08768","url":null,"abstract":"The celebrated Takens' embedding theorem provides a theoretical foundation\u0000for reconstructing the full state of a dynamical system from partial\u0000observations. However, the classical theorem assumes that the underlying system\u0000is deterministic and that observations are noise-free, limiting its\u0000applicability in real-world scenarios. Motivated by these limitations, we\u0000rigorously establish a measure-theoretic generalization that adopts an Eulerian\u0000description of the dynamics and recasts the embedding as a pushforward map\u0000between probability spaces. Our mathematical results leverage recent advances\u0000in optimal transportation theory. Building on our novel measure-theoretic\u0000time-delay embedding theory, we have developed a new computational framework\u0000that forecasts the full state of a dynamical system from time-lagged partial\u0000observations, engineered with better robustness to handle sparse and noisy\u0000data. We showcase the efficacy and versatility of our approach through several\u0000numerical examples, ranging from the classic Lorenz-63 system to large-scale,\u0000real-world applications such as NOAA sea surface temperature forecasting and\u0000ERA5 wind field reconstruction.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250414","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Fixed point indices of iterates of orientation-reversing homeomorphisms","authors":"Grzegorz Graff, Patryk Topór","doi":"arxiv-2409.08753","DOIUrl":"https://doi.org/arxiv-2409.08753","url":null,"abstract":"We show that any sequence of integers satisfying necessary Dold's congruences\u0000is realized as the sequence of fixed point indices of the iterates of an\u0000orientation-reversing homeomorphism of $mathbb{R}^{m}$ for $mgeq 3$. As an\u0000element of the construction of the above homeomorphism, we consider the class\u0000of boundary-preserving homeomorphisms of $mathbb{R}^{m}_{+}$ and give the\u0000answer to [Problem 10.2, Topol. Methods Nonlinear Anal. 50 (2017), 643 - 667]\u0000providing a complete description of the forms of fixed point indices for this\u0000class of maps.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Multivariate mean equicontinuity for finite-to-one topomorphic extensions","authors":"Jonas Breitenbücher, Lino Haupt, Tobias Jäger","doi":"arxiv-2409.08707","DOIUrl":"https://doi.org/arxiv-2409.08707","url":null,"abstract":"In this note, we generalise the concept of topo-isomorphic extensions and\u0000define finite topomorphic extensions as topological dynamical systems whose\u0000factor map to the maximal equicontinuous factor is measure-theoretically at\u0000most $m$-to-one for some $minmathbb{N}$. We further define multivariate\u0000versions of mean equicontinuity, complementing the notion of multivariate mean\u0000sensitivity introduced by Li, Ye and Yu, and then show that any $m$-to-one\u0000topomorphic extension is mean $(m+1)$-equicontinuous. This falls in line with\u0000the well-known result, due to Downarowicz and Glasner, that strictly ergodic\u0000systems are isomorphic extensions if and only if they are mean equicontinuous.\u0000While in the multivariate case we can only conjecture that the converse\u0000direction also holds, the result provides an indication that multivariate\u0000equicontinuity properties are strongly related to finite extension structures.\u0000For minimal systems, an Auslander-Yorke type dichotomy between multivariate\u0000mean equicontinuity and sensitivity is shown as well.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"295 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250416","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Multiple recurrence without commutativity","authors":"Wen Huang, Song Shao, Xiangdong Ye","doi":"arxiv-2409.07979","DOIUrl":"https://doi.org/arxiv-2409.07979","url":null,"abstract":"We study multiple recurrence without commutativity in this paper. We show\u0000that for any two homeomorphisms $T,S: Xrightarrow X$ with $(X,T)$ and $(X,S)$\u0000being minimal, there is a residual subset $X_0$ of $X$ such that for any $xin\u0000X_0$ and any nonlinear integral polynomials $p_1,ldots, p_d$ vanishing at $0$,\u0000there is some subsequence ${n_i}$ of $mathbb Z$ with $n_ito infty$\u0000satisfying $$ S^{n_i}xto x, T^{p_1(n_i)}xto x, ldots, T^{p_d(n_i)}xto x,\u0000itoinfty.$$","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"86 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195864","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Maren Bråthen Kristoffersen, Bjørn Fredrik Nielsen, Susanne Solem
{"title":"Estimating neural connection strengths from firing intervals","authors":"Maren Bråthen Kristoffersen, Bjørn Fredrik Nielsen, Susanne Solem","doi":"arxiv-2409.07241","DOIUrl":"https://doi.org/arxiv-2409.07241","url":null,"abstract":"We propose and analyze a procedure for using a standard activity-based neuron\u0000network model and firing data to compute the effective connection strengths\u0000between neurons in a network. We assume a Heaviside response function, that the\u0000external inputs are given and that the initial state of the neural activity is\u0000known. The associated forward operator for this problem, which maps given\u0000connection strengths to the time intervals of firing, is highly nonlinear.\u0000Nevertheless, it turns out that the inverse problem of determining the\u0000connection strengths can be solved in a rather transparent manner, only\u0000employing standard mathematical tools. In fact, it is sufficient to solve a\u0000system of decoupled ODEs, which yields a linear system of algebraic equations\u0000for determining the connection strengths. The nature of the inverse problem is\u0000investigated by studying some mathematical properties of the aforementioned\u0000linear system and by a series of numerical experiments. Finally, under an\u0000assumption preventing the effective contribution of the network to each neuron\u0000from staying at zero, we prove that the involved forward operator is\u0000continuous. Sufficient criteria on the external input ensuring that the needed\u0000assumption holds are also provided.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195861","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Erik Bergland, Jason J Bramburger, Bjorn Sandstede
{"title":"Localized synchronous patterns in weakly coupled bistable oscillators","authors":"Erik Bergland, Jason J Bramburger, Bjorn Sandstede","doi":"arxiv-2409.07546","DOIUrl":"https://doi.org/arxiv-2409.07546","url":null,"abstract":"Motivated by numerical continuation studies of coupled mechanical\u0000oscillators, we investigate branches of localized time-periodic solutions of\u0000one-dimensional chains of coupled oscillators. We focus on Ginzburg-Landau\u0000equations with nonlinearities of lambda-omega type and establish the existence\u0000of on-site and off-site solutions in the case of weak coupling and\u0000weak-amplitude dependence of the oscillator periods. Utilizing a core-far-field\u0000decomposition, we demonstrate that in-phase solutions lie on branches that\u0000exhibit snaking behavior.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"2012 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195859","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yu Huang, Sebastian Bathiany, Peter Ashwin, Niklas Boers
{"title":"Deep Learning for predicting rate-induced tipping","authors":"Yu Huang, Sebastian Bathiany, Peter Ashwin, Niklas Boers","doi":"arxiv-2409.07590","DOIUrl":"https://doi.org/arxiv-2409.07590","url":null,"abstract":"Nonlinear dynamical systems exposed to changing forcing can exhibit\u0000catastrophic transitions between alternative and often markedly different\u0000states. The phenomenon of critical slowing down (CSD) can be used to anticipate\u0000such transitions if caused by a bifurcation and if the change in forcing is\u0000slow compared to the internal time scale of the system. However, in many\u0000real-world situations, these assumptions are not met and transitions can be\u0000triggered because the forcing exceeds a critical rate. For example, given the\u0000pace of anthropogenic climate change in comparison to the internal time scales\u0000of key Earth system components, such as the polar ice sheets or the Atlantic\u0000Meridional Overturning Circulation, such rate-induced tipping poses a severe\u0000risk. Moreover, depending on the realisation of random perturbations, some\u0000trajectories may transition across an unstable boundary, while others do not,\u0000even under the same forcing. CSD-based indicators generally cannot distinguish\u0000these cases of noise-induced tipping versus no tipping. This severely limits\u0000our ability to assess the risks of tipping, and to predict individual\u0000trajectories. To address this, we make a first attempt to develop a deep\u0000learning framework to predict transition probabilities of dynamical systems\u0000ahead of rate-induced transitions. Our method issues early warnings, as\u0000demonstrated on three prototypical systems for rate-induced tipping, subjected\u0000to time-varying equilibrium drift and noise perturbations. Exploiting\u0000explainable artificial intelligence methods, our framework captures the\u0000fingerprints necessary for early detection of rate-induced tipping, even in\u0000cases of long lead times. Our findings demonstrate the predictability of\u0000rate-induced and noise-induced tipping, advancing our ability to determine safe\u0000operating spaces for a broader class of dynamical systems than possible so far.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195860","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Benjamín A. Itzá-Ortiz, Mónica Moreno Rocha, Víctor Nopal-Coello
{"title":"Sturmian external angles of primitive components in the Mandelbrot set","authors":"Benjamín A. Itzá-Ortiz, Mónica Moreno Rocha, Víctor Nopal-Coello","doi":"arxiv-2409.07636","DOIUrl":"https://doi.org/arxiv-2409.07636","url":null,"abstract":"In this work we introduce the broken line construction, which is a geometric\u0000and combinatorial algorithm that computes periodic Sturmian angles of a given\u0000period, yielding the locations of their landing parameters in the Mandelbrot\u0000set. An easy to implement method to compute the conjugated angle of a periodic\u0000Sturmian angle is also provided. Furthermore, if $theta$ is a periodic\u0000Sturmian angle computed by the broken line construction, then we show the\u0000existence of a one-to-one correspondence between its binary expansion and its\u0000associated kneading sequence.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195858","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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