Sachin Bhalekar, Janardhan Chevala, Prashant M. Gade
In this work, we propose a generalization to the classical logistic map. The generalized map preserves most properties of the classical map and has richer dynamics as it contains the fractional order and one more parameter. We propose the stability bounds for each equilibrium point. The detailed bifurcation analysis with respect to both parameters is presented using the bifurcation diagrams in one and two dimensions. The chaos in this system is controlled using delayed feedback. We provide some non-linear feedback controllers to synchronize the system. The multistability in the proposed system is also discussed.
{"title":"Dynamical Analysis Of Fractional Order Generalized Logistic Map","authors":"Sachin Bhalekar, Janardhan Chevala, Prashant M. Gade","doi":"arxiv-2409.07174","DOIUrl":"https://doi.org/arxiv-2409.07174","url":null,"abstract":"In this work, we propose a generalization to the classical logistic map. The\u0000generalized map preserves most properties of the classical map and has richer\u0000dynamics as it contains the fractional order and one more parameter. We propose\u0000the stability bounds for each equilibrium point. The detailed bifurcation\u0000analysis with respect to both parameters is presented using the bifurcation\u0000diagrams in one and two dimensions. The chaos in this system is controlled\u0000using delayed feedback. We provide some non-linear feedback controllers to\u0000synchronize the system. The multistability in the proposed system is also\u0000discussed.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195863","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}
Attila Karsai, Tobias Breiten, Justus Ramme, Philipp Schulze
Port-Hamiltonian (pH) systems provide a powerful tool for modeling physical systems. Their energy-based perspective allows for the coupling of various subsystems through energy exchange. Another important class of systems, passive systems, are characterized by their inability to generate energy internally. In this paper, we explore first steps towards understanding the equivalence between passivity and the feasibility of port-Hamiltonian realizations in nonlinear systems. Based on our findings, we present a method to construct port-Hamiltonian representations of a passive system if the dynamics and the Hamiltonian are known.
{"title":"Nonlinear port-Hamiltonian systems and their connection to passivity","authors":"Attila Karsai, Tobias Breiten, Justus Ramme, Philipp Schulze","doi":"arxiv-2409.06256","DOIUrl":"https://doi.org/arxiv-2409.06256","url":null,"abstract":"Port-Hamiltonian (pH) systems provide a powerful tool for modeling physical\u0000systems. Their energy-based perspective allows for the coupling of various\u0000subsystems through energy exchange. Another important class of systems, passive\u0000systems, are characterized by their inability to generate energy internally. In\u0000this paper, we explore first steps towards understanding the equivalence\u0000between passivity and the feasibility of port-Hamiltonian realizations in\u0000nonlinear systems. Based on our findings, we present a method to construct\u0000port-Hamiltonian representations of a passive system if the dynamics and the\u0000Hamiltonian are known.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"403 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195865","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}
In this paper we prove that positive weak solutions for quasilinear parabolic equations on bounded domains subject to homogenous Neumann boundary conditions becme classical and global under the unique condition that the reaction doesn't change sign after certain positive time. We apply this result to reaction diffusion systems and prove global existence of theirs positive weak solutions under the same condition on theirs reactions. The nonlinearities growth isn't taken in consideration. The proof is based on the maximum principle.
{"title":"A surprising regularizing effect of the nonlinear semigroup associated to the semilinear heat equation and applications to reaction diffusion systems","authors":"Said Kouachi","doi":"arxiv-2409.06606","DOIUrl":"https://doi.org/arxiv-2409.06606","url":null,"abstract":"In this paper we prove that positive weak solutions for quasilinear parabolic\u0000equations on bounded domains subject to homogenous Neumann boundary conditions\u0000becme classical and global under the unique condition that the reaction doesn't\u0000change sign after certain positive time. We apply this result to reaction\u0000diffusion systems and prove global existence of theirs positive weak solutions\u0000under the same condition on theirs reactions. The nonlinearities growth isn't\u0000taken in consideration. The proof is based on the maximum principle.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195862","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}
Deep learning is revolutionizing weather forecasting, with new data-driven models achieving accuracy on par with operational physical models for medium-term predictions. However, these models often lack interpretability, making their underlying dynamics difficult to understand and explain. This paper proposes methodologies to estimate the Koopman operator, providing a linear representation of complex nonlinear dynamics to enhance the transparency of data-driven models. Despite its potential, applying the Koopman operator to large-scale problems, such as atmospheric modeling, remains challenging. This study aims to identify the limitations of existing methods, refine these models to overcome various bottlenecks, and introduce novel convolutional neural network architectures that capture simplified dynamics.
{"title":"Deep Learning for Koopman Operator Estimation in Idealized Atmospheric Dynamics","authors":"David Millard, Arielle Carr, Stéphane Gaudreault","doi":"arxiv-2409.06522","DOIUrl":"https://doi.org/arxiv-2409.06522","url":null,"abstract":"Deep learning is revolutionizing weather forecasting, with new data-driven\u0000models achieving accuracy on par with operational physical models for\u0000medium-term predictions. However, these models often lack interpretability,\u0000making their underlying dynamics difficult to understand and explain. This\u0000paper proposes methodologies to estimate the Koopman operator, providing a\u0000linear representation of complex nonlinear dynamics to enhance the transparency\u0000of data-driven models. Despite its potential, applying the Koopman operator to\u0000large-scale problems, such as atmospheric modeling, remains challenging. This\u0000study aims to identify the limitations of existing methods, refine these models\u0000to overcome various bottlenecks, and introduce novel convolutional neural\u0000network architectures that capture simplified dynamics.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195869","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}
In this article we study orbits of proximal pairs in almost automorphic subshifts. The corresponding orbits in the maximal equicontinuous factor are precisely those orbits that intersect the boundary of the subshift's separating cover. We impose certain finiteness conditions on this boundary and investigate the resulting consequences for the subshift, for instance in terms of complexity or the relations between proximal and asymptotic pairs. The last part of our article deals with Toeplitz subshifts without a finite boundary. There we treat the question of necessary conditions and sufficient conditions for the existence of a factor subshift with a finite boundary. Throughout the whole article, we provide numerous Toeplitz subshifts as examples and counterexamples to illustrate our findings and the necessity of our assumptions.
{"title":"Almost automorphic subshifts with finiteness conditions for the boundary of the separating cover","authors":"Daniel Sell, Franziska Sieron","doi":"arxiv-2409.06005","DOIUrl":"https://doi.org/arxiv-2409.06005","url":null,"abstract":"In this article we study orbits of proximal pairs in almost automorphic\u0000subshifts. The corresponding orbits in the maximal equicontinuous factor are\u0000precisely those orbits that intersect the boundary of the subshift's separating\u0000cover. We impose certain finiteness conditions on this boundary and investigate\u0000the resulting consequences for the subshift, for instance in terms of\u0000complexity or the relations between proximal and asymptotic pairs. The last\u0000part of our article deals with Toeplitz subshifts without a finite boundary.\u0000There we treat the question of necessary conditions and sufficient conditions\u0000for the existence of a factor subshift with a finite boundary. Throughout the\u0000whole article, we provide numerous Toeplitz subshifts as examples and\u0000counterexamples to illustrate our findings and the necessity of our\u0000assumptions.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"66 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195867","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}
Let $gcolon Lrightarrow L$ be an atoroidal, endperiodic map on an infinite type surface $L$ with no boundary and finitely many ends, each of which is accumulated by genus. By work of Landry, Minsky, and Taylor, $g$ is isotopic to a spun pseudo-Anosov map $f$. We show that spun pseudo-Anosov maps minimize the number of periodic points of period $n$ for sufficiently high $n$ over all maps in their homotopy class, strengthening a theorem of Landry, Minsky, and Taylor. We also show that the same theorem holds for atoroidal Handel--Miller maps when you only consider periodic points that lie in the intersection of the stable and unstable laminations.
{"title":"Periodic points of endperiodic maps","authors":"Ellis Buckminster","doi":"arxiv-2409.05963","DOIUrl":"https://doi.org/arxiv-2409.05963","url":null,"abstract":"Let $gcolon Lrightarrow L$ be an atoroidal, endperiodic map on an infinite\u0000type surface $L$ with no boundary and finitely many ends, each of which is\u0000accumulated by genus. By work of Landry, Minsky, and Taylor, $g$ is isotopic to\u0000a spun pseudo-Anosov map $f$. We show that spun pseudo-Anosov maps minimize the\u0000number of periodic points of period $n$ for sufficiently high $n$ over all maps\u0000in their homotopy class, strengthening a theorem of Landry, Minsky, and Taylor.\u0000We also show that the same theorem holds for atoroidal Handel--Miller maps when\u0000you only consider periodic points that lie in the intersection of the stable\u0000and unstable laminations.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"188 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195868","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}
Let $f,g$ be $C^2$ area-preserving Anosov diffeomorphisms on $mathbb{T}^2$ which are topologically conjugate by a homeomorphism $h$ ($hf=gh$). We assume that the Jacobian periodic data of $f$ and $g$ are matched by $h$ for all points of some large period $Ninmathbb{N}$. We show that $f$ and $g$ are ``approximately smoothly conjugate." That is, there exists a $C^{1+alpha}$ diffeomorphism $overline{h}_N$ such that $h$ and $overline{h}_N$ are $C^0$ exponentially close in $N$, and $f$ and $f_N:=overline{h}_N^{-1}goverline{h}_N$ are $C^1$ exponentially close in $N$. Moreover, the rates of convergence are uniform among different $f,g$ in a $C^2$ bounded set of Anosov diffeomorphisms. The main idea in constructing $overline{h}_N$ is to do a ``weighted holonomy" construction, and the main technical tool in obtaining our estimates is a uniform effective version of Bowen's equidistribution theorem of weighted discrete orbits to the SRB measure.
{"title":"Finite Periodic Data Rigidity For Two-Dimensional Area-Preserving Anosov Diffeomorphisms","authors":"Thomas Aloysius O'Hare","doi":"arxiv-2409.05857","DOIUrl":"https://doi.org/arxiv-2409.05857","url":null,"abstract":"Let $f,g$ be $C^2$ area-preserving Anosov diffeomorphisms on $mathbb{T}^2$\u0000which are topologically conjugate by a homeomorphism $h$ ($hf=gh$). We assume\u0000that the Jacobian periodic data of $f$ and $g$ are matched by $h$ for all\u0000points of some large period $Ninmathbb{N}$. We show that $f$ and $g$ are\u0000``approximately smoothly conjugate.\" That is, there exists a $C^{1+alpha}$\u0000diffeomorphism $overline{h}_N$ such that $h$ and $overline{h}_N$ are $C^0$\u0000exponentially close in $N$, and $f$ and\u0000$f_N:=overline{h}_N^{-1}goverline{h}_N$ are $C^1$ exponentially close in $N$.\u0000Moreover, the rates of convergence are uniform among different $f,g$ in a $C^2$\u0000bounded set of Anosov diffeomorphisms. The main idea in constructing\u0000$overline{h}_N$ is to do a ``weighted holonomy\" construction, and the main\u0000technical tool in obtaining our estimates is a uniform effective version of\u0000Bowen's equidistribution theorem of weighted discrete orbits to the SRB\u0000measure.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195871","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}
This paper investigates the secular motion of a massless asteroid within the framework of the double-averaged elliptic restricted three-body problem. By employing Poincar'e variables, we analyze the stability properties of asteroid orbits in the presence of planetary perturbations. Our study reveals that periodic orbits identified in the planar configuration maintain stability in the spatial perturbed problem across a wide range of parameter values. These findings, supported by numerical simulations, contribute to a deeper understanding of asteroid dynamics and have implications for studying exoplanetary systems with highly eccentric host stars.
{"title":"Stability analysis of spatial perturbed elliptic restricted 3-body problem with double-averaging","authors":"Yan Luo, Kaicheng Sheng","doi":"arxiv-2409.05299","DOIUrl":"https://doi.org/arxiv-2409.05299","url":null,"abstract":"This paper investigates the secular motion of a massless asteroid within the\u0000framework of the double-averaged elliptic restricted three-body problem. By\u0000employing Poincar'e variables, we analyze the stability properties of asteroid\u0000orbits in the presence of planetary perturbations. Our study reveals that\u0000periodic orbits identified in the planar configuration maintain stability in\u0000the spatial perturbed problem across a wide range of parameter values. These\u0000findings, supported by numerical simulations, contribute to a deeper\u0000understanding of asteroid dynamics and have implications for studying\u0000exoplanetary systems with highly eccentric host stars.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"188 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195896","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}
In this paper, we establish Lasota-Yorke inequality for the Frobenius-Perron Operator of a piecewise expanding $C^{1+varepsilon}$ map of an interval. By adapting this inequality to satisfy the assumptions of the Ionescu-Tulcea and Marinescu ergodic theorem cite{ionescu1950}, we demonstrate the existence of an absolutely continuous invariant measure (ACIM) for the map. Furthermore, we prove the quasi-compactness of the Frobenius-Perron operator induced by the map. Additionally, we explore significant properties of the system, including weak mixing and exponential decay of correlations.
{"title":"Existence of ACIM for Piecewise Expanding $C^{1+varepsilon}$ maps","authors":"Aparna Rajput, Paweł Góra","doi":"arxiv-2409.06076","DOIUrl":"https://doi.org/arxiv-2409.06076","url":null,"abstract":"In this paper, we establish Lasota-Yorke inequality for the Frobenius-Perron\u0000Operator of a piecewise expanding $C^{1+varepsilon}$ map of an interval. By\u0000adapting this inequality to satisfy the assumptions of the Ionescu-Tulcea and\u0000Marinescu ergodic theorem cite{ionescu1950}, we demonstrate the existence of\u0000an absolutely continuous invariant measure (ACIM) for the map. Furthermore, we\u0000prove the quasi-compactness of the Frobenius-Perron operator induced by the\u0000map. Additionally, we explore significant properties of the system, including\u0000weak mixing and exponential decay of correlations.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195866","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}
In this work, we explore the dynamics of fractional differential equations (FDEs) through a rigorous topological analysis of strange attractors. By investigating systems with Caputo derivatives of order ( alpha in (0, 1) ), we identify conditions under which chaotic behavior emerges, characterized by positive topological entropy and the presence of homoclinic and heteroclinic structures. We introduce novel methods for computing the fractional Conley index and Lyapunov exponents, which allow us to distinguish between chaotic and non-chaotic attractors. Our results also provide new insights into the fractal and spectral properties of strange attractors in fractional systems, establishing a comprehensive framework for understanding chaos and stability in this context.
在这项工作中，我们通过对奇异吸引子进行严格的拓扑分析来探索分数微分方程（FDEs）的动力学。通过研究阶数为 ( alpha in (0, 1) )的卡普托导数系统，我们确定了出现混沌行为的条件，这些条件的特征是拓扑熵为正以及存在同线性和异线性结构。我们引入了计算分数康利指数和李亚普诺夫指数的新方法，这使我们能够区分混沌吸引子和非混沌吸引子。我们的研究结果还为分形系统中奇异吸引子的分形和谱特性提供了新的见解，为在此背景下理解混沌和稳定性建立了一个全面的框架。
{"title":"Strange Attractors in Fractional Differential Equations: A Topological Approach to Chaos and Stability","authors":"Ronald Katende","doi":"arxiv-2409.05053","DOIUrl":"https://doi.org/arxiv-2409.05053","url":null,"abstract":"In this work, we explore the dynamics of fractional differential equations\u0000(FDEs) through a rigorous topological analysis of strange attractors. By\u0000investigating systems with Caputo derivatives of order ( alpha in (0, 1) ),\u0000we identify conditions under which chaotic behavior emerges, characterized by\u0000positive topological entropy and the presence of homoclinic and heteroclinic\u0000structures. We introduce novel methods for computing the fractional Conley\u0000index and Lyapunov exponents, which allow us to distinguish between chaotic and\u0000non-chaotic attractors. Our results also provide new insights into the fractal\u0000and spectral properties of strange attractors in fractional systems,\u0000establishing a comprehensive framework for understanding chaos and stability in\u0000this context.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195872","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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