The Experts below are selected from a list of 1179 Experts worldwide ranked by ideXlab platform
Marian Mrozek - One of the best experts on this subject based on the ideXlab platform.
-
Chapter 9 – Conley Index
Handbook of Dynamical Systems, 2020Co-Authors: Konstantin Mischaikow, Marian MrozekAbstract:This chapter discusses the Conley Index theory. The Conley Index is an Index of isolating neighborhoods. The applicability of the Conley Index depends essentially on three properties. The first gives great freedom in the choice of regions in phase space on which one will perform the analysis. The second allows for passage from the isolating neighborhood to an understanding of the dynamics of the isolated invariant set. The Wazewski property, while the most fundamental, is the simplest result of this type. It contains a variety of more sophisticated theorems that can be used to prove the existence of connecting orbits, periodic orbits, and even chaotic dynamics in the sense of symbolic dynamics. The third property is important for the following reason. The Conley Index is a purely topological Index, and it is a very coarse measure of the dynamics. Typically, if it can be computed directly at a particular parameter value, then knowledge of the dynamics at that parameter value is reasonably complete. The power of the Index (as in degree theory) comes from being able to continue it to a parameter value where understanding of the dynamics is much less complete.
-
persistence of the Conley Index in combinatorial dynamical systems
arXiv: Algebraic Topology, 2020Co-Authors: Marian Mrozek, Ryan SlechtaAbstract:A combinatorial framework for dynamical systems provides an avenue for connecting classical dynamics with data-oriented, algorithmic methods. Combinatorial vector fields introduced by Forman and their recent generalization to multivector fields have provided a starting point for building such a connection. In this work, we strengthen this relationship by placing the Conley Index in the persistent homology setting. Conley indices are homological features associated with so-called isolated invariant sets, so a change in the Conley Index is a response to perturbation in an underlying multivector field. We show how one can use zigzag persistence to summarize changes to the Conley Index, and we develop techniques to capture such changes in the presence of noise. We conclude by developing an algorithm to track features in a changing multivector field.
-
Conley Index approach to sampled dynamics
arXiv: Dynamical Systems, 2019Co-Authors: Bogdan Batko, Konstantin Mischaikow, Marian Mrozek, Mateusz PrzybylskiAbstract:The topological method for the reconstruction of dynamics from time series [K. Mischaikow, M. Mrozek, J. Reiss, A. Szymczak. Construction of Symbolic Dynamics from Experimental Time Series, Physical Review Letters, 82 (1999), 1144-1147] is reshaped to improve its range of applicability, particularly in the presence of sparse data and strong expansion. The improvement is based on a multivalued map representation of the data. However, unlike the previous approach, it is not required that the representation has a continuous selector. Instead of a selector, a recently developed new version of Conley Index theory for multivalued maps [B. Batko and M. Mrozek. Weak Index pairs and the Conley Index for discrete multivalued dynamical systems, SIAM J. Applied Dynamical Systems 15 (2016), 1143-1162], [B.Batko. Weak Index pairs and the Conley Index for discrete multivalued dynamical systems. Part II: properties of the Index, SIAM J. Applied Dynamical Systems 16 (2017), 1587-1617] is used in computations. The existence of a continuous, single-valued generator of the relevant dynamics is guaranteed in the vicinity of the graph of the multivalued map constructed from data. Some numerical examples based on time series derived from the iteration of H\'enon type maps are presented.
-
weak Index pairs and the Conley Index for discrete multivalued dynamical systems
Siam Journal on Applied Dynamical Systems, 2016Co-Authors: Bogdan Batko, Marian MrozekAbstract:Motivated by the problem of reconstructing dynamics from samples, we revisit the Conley Index theory for discrete multivalued dynamical systems [T. Kaczynski and M. Mrozek, Topology Appl., 65 (1995), pp. 83--96]. We introduce a new, less restrictive definition of the isolating neighborhood. It turns out that then the main tool for the construction of the Index, i.e., the Index pair, is no longer useful. In order to overcome this obstacle we use the concept of weak Index pairs.
-
weak Index pairs and the Conley Index for discrete multivalued dynamical systems
arXiv: Dynamical Systems, 2015Co-Authors: Bogdan Batko, Marian MrozekAbstract:Motivated by the problem of reconstructing dynamics from samples we revisit the Conley Index theory for discrete multivalued dynamical systems. We introduce a new, less restrictive definition of the isolating neighbourhood. It turns out that then the main tool for the construction of the Index, i.e. the Index pair, is no longer useful. In order to overcome this obstacle we use the concept of weak Index pairs.
Konstantin Mischaikow - One of the best experts on this subject based on the ideXlab platform.
-
Chapter 9 – Conley Index
Handbook of Dynamical Systems, 2020Co-Authors: Konstantin Mischaikow, Marian MrozekAbstract:This chapter discusses the Conley Index theory. The Conley Index is an Index of isolating neighborhoods. The applicability of the Conley Index depends essentially on three properties. The first gives great freedom in the choice of regions in phase space on which one will perform the analysis. The second allows for passage from the isolating neighborhood to an understanding of the dynamics of the isolated invariant set. The Wazewski property, while the most fundamental, is the simplest result of this type. It contains a variety of more sophisticated theorems that can be used to prove the existence of connecting orbits, periodic orbits, and even chaotic dynamics in the sense of symbolic dynamics. The third property is important for the following reason. The Conley Index is a purely topological Index, and it is a very coarse measure of the dynamics. Typically, if it can be computed directly at a particular parameter value, then knowledge of the dynamics at that parameter value is reasonably complete. The power of the Index (as in degree theory) comes from being able to continue it to a parameter value where understanding of the dynamics is much less complete.
-
Conley Index approach to sampled dynamics
arXiv: Dynamical Systems, 2019Co-Authors: Bogdan Batko, Konstantin Mischaikow, Marian Mrozek, Mateusz PrzybylskiAbstract:The topological method for the reconstruction of dynamics from time series [K. Mischaikow, M. Mrozek, J. Reiss, A. Szymczak. Construction of Symbolic Dynamics from Experimental Time Series, Physical Review Letters, 82 (1999), 1144-1147] is reshaped to improve its range of applicability, particularly in the presence of sparse data and strong expansion. The improvement is based on a multivalued map representation of the data. However, unlike the previous approach, it is not required that the representation has a continuous selector. Instead of a selector, a recently developed new version of Conley Index theory for multivalued maps [B. Batko and M. Mrozek. Weak Index pairs and the Conley Index for discrete multivalued dynamical systems, SIAM J. Applied Dynamical Systems 15 (2016), 1143-1162], [B.Batko. Weak Index pairs and the Conley Index for discrete multivalued dynamical systems. Part II: properties of the Index, SIAM J. Applied Dynamical Systems 16 (2017), 1587-1617] is used in computations. The existence of a continuous, single-valued generator of the relevant dynamics is guaranteed in the vicinity of the graph of the multivalued map constructed from data. Some numerical examples based on time series derived from the iteration of H\'enon type maps are presented.
-
Singular boundary value problems via the Conley Index
Topological Methods in Nonlinear Analysis, 2006Co-Authors: Tomáš Gedeon, Konstantin MischaikowAbstract:We use Conley Index theory to solve the singular boundary value problem $\varepsilon^2D u_{xx} + f(u,\varepsilon u_x,x) = 0$ on an interval $[-1,1]$, where $u \in \mathbb R^n$ and $D$ is a diagonal matrix, with separated boundary conditions. Since we use topological methods the assumptions we need are weaker then the standard set of assumptions. The Conley Index theory is used here not for detection of an invariant set, but for tracking certain cohomological information, which guarantees existence of a solution to the boundary value problem.
-
the Conley Index for fast slow systems ii multidimensional slow variable
Journal of Differential Equations, 2006Co-Authors: Tomáš Gedeon, Hiroshi Kokubu, Konstantin MischaikowAbstract:We develop a qualitative theory for fast-slow systems with a one-dimensional slow variable. Using Conley Index theory for singularity perturbed systems, conditions are given which imply that if one can construct heteroclinic connections and periodic orbits in systems with the derivative of the slow variable set to 0, these orbits persist when the derivative of the slow variable is small and nonzero.
-
The Conley Index for fast–slow systems II: Multidimensional slow variable☆
Journal of Differential Equations, 2006Co-Authors: Tomáš Gedeon, Hiroshi Kokubu, Konstantin MischaikowAbstract:We develop a qualitative theory for fast-slow systems with a one-dimensional slow variable. Using Conley Index theory for singularity perturbed systems, conditions are given which imply that if one can construct heteroclinic connections and periodic orbits in systems with the derivative of the slow variable set to 0, these orbits persist when the derivative of the slow variable is small and nonzero.
Krzysztof P Rybakowski - One of the best experts on this subject based on the ideXlab platform.
-
An Introduction to the Conley Index Theory in Noncompact Spaces
Dynamics in Infinite Dimensions, 2020Co-Authors: Krzysztof P RybakowskiAbstract:This appendix serves to introduce the reader to the main aspects of the Conley Index theory.
-
Conley Index continuation for a singularly perturbed periodic boundary value problem
Topological Methods in Nonlinear Analysis, 2019Co-Authors: Maria C. Carbinatto, Krzysztof P RybakowskiAbstract:We establish spectral convergence and Conley Index continuation results for a class of singularly perturbed periodic boundary value problems.
-
A note on Conley Index and some parabolic problems with locally large diffusion
Topological Methods in Nonlinear Analysis, 2017Co-Authors: Maria C. Carbinatto, Krzysztof P RybakowskiAbstract:We prove singular Conley Index continuation results for a class of scalar parabolic equations with locally large diffusion considered by Fusco \cite{F} and Carvalho and Pereira \cite{CP}.
-
localized singularities and the Conley Index
Topological Methods in Nonlinear Analysis, 2011Co-Authors: Maria C. Carbinatto, Krzysztof P RybakowskiAbstract:We establish some abstract convergence and Conley Index continuation principles for families of singularly perturbed semilinear parabolic equations and apply them to reaction-diffusion equations with nonlinear boundary conditions and localized large diffusion. This extends and refines previous results of [A. Rodriguez-Bernal, Localized spatial homogenization and large diffusion , SIAM J. Math. Anal. 29 (1998), 1361-1380] and [J.M. Arrieta, A.N. Carvalho and A. Rodr\'{\i}guez-Bernal, Parabolic problems with nonlinear boundary conditions and critical nonlinearities , J. Differential Equations 156 (1999), 376-406].
-
on curved squeezing and Conley Index
Topological Methods in Nonlinear Analysis, 2011Co-Authors: Krzysztof P RybakowskiAbstract:We consider reaction-diffusion equations on a family of domains depending on a parameter $\eps> 0$. As $\eps\to 0$, the domains degenerate to a lower dimensional manifold. Using some abstract results introduced in the recent paper \cite{\rfa{CR2}} we show that there is a limit equation as $\eps\to 0$ and obtain various convergence and admissibility results for the corresponding semiflows. As a consequence, we also establish singular Conley Index and homology Index continuation results. Under an additional dissipativeness assumption, we also prove existence and upper-semicontinuity of global attractors. The results of this paper extend and refine earlier results of \cite{\rfa{CR1}} and \cite{\rfa{PRR}}.
Tomáš Gedeon - One of the best experts on this subject based on the ideXlab platform.
-
Singular boundary value problems via the Conley Index
Topological Methods in Nonlinear Analysis, 2006Co-Authors: Tomáš Gedeon, Konstantin MischaikowAbstract:We use Conley Index theory to solve the singular boundary value problem $\varepsilon^2D u_{xx} + f(u,\varepsilon u_x,x) = 0$ on an interval $[-1,1]$, where $u \in \mathbb R^n$ and $D$ is a diagonal matrix, with separated boundary conditions. Since we use topological methods the assumptions we need are weaker then the standard set of assumptions. The Conley Index theory is used here not for detection of an invariant set, but for tracking certain cohomological information, which guarantees existence of a solution to the boundary value problem.
-
the Conley Index for fast slow systems ii multidimensional slow variable
Journal of Differential Equations, 2006Co-Authors: Tomáš Gedeon, Hiroshi Kokubu, Konstantin MischaikowAbstract:We develop a qualitative theory for fast-slow systems with a one-dimensional slow variable. Using Conley Index theory for singularity perturbed systems, conditions are given which imply that if one can construct heteroclinic connections and periodic orbits in systems with the derivative of the slow variable set to 0, these orbits persist when the derivative of the slow variable is small and nonzero.
-
The Conley Index for fast–slow systems II: Multidimensional slow variable☆
Journal of Differential Equations, 2006Co-Authors: Tomáš Gedeon, Hiroshi Kokubu, Konstantin MischaikowAbstract:We develop a qualitative theory for fast-slow systems with a one-dimensional slow variable. Using Conley Index theory for singularity perturbed systems, conditions are given which imply that if one can construct heteroclinic connections and periodic orbits in systems with the derivative of the slow variable set to 0, these orbits persist when the derivative of the slow variable is small and nonzero.
-
The Conley Index for Fast-Slow Systems I. One-Dimensional Slow Variable
Journal of Dynamics and Differential Equations, 1999Co-Authors: Tomáš Gedeon, Konstantin Mischaikow, Hiroshi Kokubu, James F. ReineckAbstract:We develop a qualitative theory for fast-slow systems with a one-dimensional slow variable. Using Conley Index theory for singularity perturbed systems, conditions are given which imply that if one can construct heteroclinic connections and periodic orbits in systems with the derivative of the slow variable set to 0, these orbits persist when the derivative of the slow variable is small and nonzero.
Roman Srzednicki - One of the best experts on this subject based on the ideXlab platform.
-
a topological approach to the algorithmic computation of the Conley Index for poincare maps
Siam Journal on Applied Dynamical Systems, 2015Co-Authors: Marian Mrozek, Roman Srzednicki, Frank WeilandtAbstract:A new algorithm for computing the Conley Index of the Poincare map of a time-periodic nonautonomous ordinary differential equation is presented. The algorithm is based on a theorem which reduces the computation of the Index to the study of certain singular chains on an Index pair for some small-step translation operator of the equation. In particular, no numerical enclosures of the Poincare map are required. Concrete numerical examples for planar systems are provided.
-
Conley Index of poincare maps in isolating segments
Nonlinear Analysis-theory Methods & Applications, 2009Co-Authors: Marian Mrozek, Roman SrzednickiAbstract:Abstract We calculate the discrete-time Conley Index of the Poincare map of a time-periodic ordinary differential equation in an isolated invariant set generated by a periodic isolating segment. As an application, we present results on the existence of bounded solutions of some planar equations.
-
chapter 7 wazewski method and Conley Index
Handbook of Differential Equations: Ordinary Differential Equations, 2000Co-Authors: Roman SrzednickiAbstract:This chapter discusses the retract method introduced by Tadeusz Wazwski. It is a method of proving the existence of solutions that remain in a given set and refers to differential equations describing some evolution in time. The sets under consideration should satisfy the condition that all egress points are strict or its less restrictive variant. Now they are called Wazewski sets. The method is based on theorems that roughly assert that there is a solution contained a Wazewski set for all positive values of time if the subset of egress points is not a retract of the whole set. If, moreover, the set is compact, then its invariant part is nonempty. For isolating blocks , that is, compact Wazewski sets that do not contain any full solutions intersecting their boundaries, Charles Conley discovered homotopical invariant, which provides quantitative information on their invariant parts. It is called the Conley Index . The chapter describes the Wazewski method in detail and presents information on foundations of the Conley Index theory, which directly relates to the method and essentially does not overlap with the exposition of Mischaikow and Mrozek.
-
CHAPTER 7 – Ważewski Method and Conley Index
Handbook of Differential Equations: Ordinary Differential Equations, 2000Co-Authors: Roman SrzednickiAbstract:This chapter discusses the retract method introduced by Tadeusz Wazwski. It is a method of proving the existence of solutions that remain in a given set and refers to differential equations describing some evolution in time. The sets under consideration should satisfy the condition that all egress points are strict or its less restrictive variant. Now they are called Wazewski sets. The method is based on theorems that roughly assert that there is a solution contained a Wazewski set for all positive values of time if the subset of egress points is not a retract of the whole set. If, moreover, the set is compact, then its invariant part is nonempty. For isolating blocks , that is, compact Wazewski sets that do not contain any full solutions intersecting their boundaries, Charles Conley discovered homotopical invariant, which provides quantitative information on their invariant parts. It is called the Conley Index . The chapter describes the Wazewski method in detail and presents information on foundations of the Conley Index theory, which directly relates to the method and essentially does not overlap with the exposition of Mischaikow and Mrozek.
-
the Conley Index over a base
Transactions of the American Mathematical Society, 2000Co-Authors: Marian Mrozek, James F. Reineck, Roman SrzednickiAbstract:We construct a generalization of the Conley Index for flows. The new Index preserves information which in the classical case is lost in the process of collapsing the exit set to a point. The new Index has most of the properties of the classical Index. As examples, we study a flow with a knotted orbit in R3, and the problem of continuing two periodic orbits which are not homotopic as loops.