12475960
Rigorous Enclosures of a Slow Manifold
Slow-fast dynamical systems have two time scales and an explicit parameter representing the ratio of these time scales. Locally invariant slow manifolds along which motion occurs on the slow time scale are a prominent feature of slow-fast systems. This paper introduces a rigorous numerical method to compute enclosures of the slow manifold of a slow-fast system with one fast and two slow variables. A triangulated first order approximation to the two dimensional invariant manifold is computed “algebraically.” Two translations of the computed manifold in the fast direction that are transverse to the vector field are computed as the boundaries of an initial enclosure. The enclosures are refined to bring them closer to each other by moving vertices of the enclosure boundaries one at a time. As an application we use the method to prove the existence of tangencies of invariant manifolds in the problem of singular Hopf bifurcation and to give bounds on the location of one such tangency.
14397302
Computing Two-Dimensional Global Invariant Manifolds in Slow-fast Systems
We present the GLOBALIZEBVP algorithm for the computation of two-dimensional stable and unstable manifolds of a vector field. Specifically, we use the collocation routines of AUTO to solve boundary problems that are used during the computation to find the next approximate geodesic level set on the manifold. The resulting implementation is numerically very stable and well suited for systems with multiple time scales. This is illustrated with the test-case examples of the Lorenz and Chua systems, and with a slow–fast model of a somatotroph cell.
207063885
Finite Resolution Dynamics
We develop a new mathematical model for describing a dynamical system at limited resolution (or finite scale), and we give precise meaning to the notion of a dynamical system having some property at all resolutions coarser than a given number. Open covers are used to approximate the topology of the phase space in a finite way, and the dynamical system is represented by means of a combinatorial multivalued map. We formulate notions of transitivity and mixing in the finite resolution setting in a computable and consistent way. Moreover, we formulate equivalent conditions for these properties in terms of graphs, and provide effective algorithms for their verification. As an application we show that the Hénon attractor is mixing at all resolutions coarser than 10−5.
13590790
Rigorous Computations of Homoclinic Tangencies
In this paper, we propose a rigorous computational method for detecting homoclinic tangencies and structurally unstable connecting orbits. It is a combination of several tools and algorithms, including the interval arithmetic, the subdivision algorithm, the Conley index theory, and the computational homology theory. As an example, we prove the existence of generic homoclinic tangencies in the Henon family.
7967640
Computer Assisted Proof of the Existence of Homoclinic Tangency for the Hénon Map and for the Forced Damped Pendulum
We present a topological method for the efficient computer assisted verification of the existence of the homoclinic tangency which unfolds generically in a one-parameter family of planar maps. The method has been applied to the Henon map and the forced damped pendulum ODE.
51849934
Cone Conditions and Covering Relations for Topologically Normally Hyperbolic Invariant Manifolds
We present a topological proof of the existence of invariant manifolds for maps with normally hyperbolic-like properties. The proof is conducted in the phase space of the system. In our approach we do not require that the map is a perturbation of some other map for which we already have an invariant manifold. We provide conditions which imply the existence of the manifold within an investigated region of the phase space. The required assumptions are formulated in a way which allows for rigorous computer assisted verification. We apply our method to obtain an invariant manifold within an explicit range of parameters for the rotating H\'enon map.
58427338
Validated Numerics: A Short Introduction to Rigorous Computations
This textbook provides a comprehensive introduction to the theory and practice of validated numerics, an emerging new field that combines the strengths of scientific computing and pure mathematics. In numerous fields ranging from pharmaceutics and engineering to weather prediction and robotics, fast and precise computations are essential. Based on the theory of set-valued analysis, a new suite of numerical methods is developed, producing efficient and reliable solvers for numerous problems in nonlinear analysis. Validated numerics yields rigorous computations that can find all possible solutions to a problem while taking into account all possible sources of error--fast, and with guaranteed accuracy.Validated Numerics offers a self-contained primer on the subject, guiding readers from the basics to more advanced concepts and techniques. This book is an essential resource for those entering this fast-developing field, and it is also the ideal textbook for graduate students and advanced undergraduates needing an accessible introduction to the subject. Validated Numerics features many examples, exercises, and computer labs using MATLAB/C++, as well as detailed appendixes and an extensive bibliography for further reading.Provides a comprehensive, self-contained introduction to validated numerics Requires no advanced mathematics or programming skills Features many examples, exercises, and computer labs Includes code snippets that illustrate implementation Suitable as a textbook for graduate students and advanced undergraduates
210707079
On a Fast and Accurate Method to Enclose All Zeros of an Analytic Function on a Triangulated Domain
We present a fast hybrid method designed to enclose all zeros of an analytic function on a triangulated domain. The method consists of three parts: first the zeros are isolated (up to some resolution) using a combination of winding number computations and bisections; in the second step we approximate the location of each zero using a floating point complex Newton method; in the final step we validate the existence of an exact zero near each approximate zero by recalculating the winding number on a small enclosing triangle. The output of the program is a subset of a refinement of the original triangulation of the domain, together with the number of zeros on each remaining triangle. To demonstrate the method, we apply it on a few non–trivial examples.
334995
The parameterization method for invariant manifolds III: overview and applications
We describe a method to establish existence and regularity of 
invariant manifolds and, at the same time to find simple maps which are 
conjugated to the dynamics on them. The method establishes several invariant 
manifold theorems. For instance, it reduces the proof of the usual 
stable manifold theorem near hyperbolic points to an application of the 
implicit function theorem in Banach spaces. We also present several other 
applications of the method.
51838125
Covering relations and the existence of topologicallynormally hyperbolic invariant sets
We present a topological method for the detection of normally hyperbolic 
type invariant sets for maps. The invariant set covers a sub-manifold without a boundary in $\mathbb{R}^k$. For the method to hold we 
only need to assume that the movement of the system transversal to the 
manifold has directions of topological expansion and contraction. The 
movement in the direction of the manifold can be arbitrary. The result is 
based on the method of covering relations and local Brouwer degree theory.
446712
Computing Slow Manifolds of Saddle Type
Slow manifolds are important geometric structures in the state spaces of dynamical systems with multiple time scales. This paper introduces an algorithm for computing trajectories on slow manifolds that are normally hyperbolic with both stable and unstable fast manifolds. We present two examples of bifurcation problems where these manifolds play a key role and a third example in which saddle-type slow manifolds are part of a traveling wave profile of a partial differential equation. Initial value solvers are incapable of computing trajectories on saddle-type slow manifolds, so the slow manifold of saddle type (SMST) algorithm presented here is formulated as a boundary value method. We take an empirical approach here to assessing the accuracy and effectiveness of the algorithm.
120998331
Covering relations, cone conditions and the stable manifold theorem
Abstract We show how to effectively link covering relations with cone conditions. We give a new, ‘geometric,’ proof of the stable manifold theorem for hyperbolic fixed point of a map.
7841084
Phase Portraits of Planar Vector Fields: Computer Proofs
We present an algorithm for computer verification of the global structure of structurally stable planar vector fields. Constructing analytical proofs for the qualitative properties of phase portraits has been difficult. We try to avoid this barrier by augmenting numerical computations of trajectories of dynamical systems with error estimates that yield rigorous proofs. Our approach lends itself to high-precision estimates, because the proofs are broken into independent calculations whose length in f1oatingpoint operations does not increase with increasing precision. The algorithm is tested on a system that arises in the study of Hopf bifurcation of periodic orbits with 1:4 resonance.
115170200
On a computer-aided approach to the computation of Abelian integrals
An accurate method to compute enclosures of Abelian integrals is developed. This allows for an accurate description of the phase portraits of planar polynomial systems that are perturbations of Hamiltonian systems. As an example, it is applied to the study of bifurcations of limit cycles arising from a cubic perturbation of an elliptic Hamiltonian of degree four.
117014104
Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems
The subject of this book is the solution of stiff differential equations and of differential-algebraic systems (differential equations with constraints). The book is divided into four chapters. The beginning of each chapter is of introductory nature, followed by practical applications, the discussion of numerical results, theoretical investigations on the order and accuracy, linear and nonlinear stability, convergence and asymptotic expansions. Stiff and differential-algebraic problems arise everywhere in scientific computations (e.g., in physics, chemistry, biology, control engineering, electrical network analysis, mechanical systems). Many applications as well as computer programs are presented.
8843351
Mixed-mode oscillations and slow manifolds in the self-coupled FitzHugh-Nagumo system.
We investigate the organization of mixed-mode oscillations in the self-coupled FitzHugh-Nagumo system. These types of oscillations can be explained as a combination of relaxation oscillations and small-amplitude oscillations controlled by canard solutions that are associated with a folded singularity on a critical manifold. The self-coupled FitzHugh-Nagumo system has a cubic critical manifold for a range of parameters, and an associated folded singularity of node-type. Hence, there exist corresponding attracting and repelling slow manifolds that intersect in canard solutions. We present a general technique for the computation of two-dimensional slow manifolds (smooth surfaces). It is based on a boundary value problem approach where the manifolds are computed as one-parameter families of orbit segments. Visualization of the computed surfaces gives unprecedented insight into the geometry of the system. In particular, our techniques allow us to find and visualize canard solutions as the intersection curves of the attracting and repelling slow manifolds.
119590527
Bifurcations of mixed-mode oscillations in a three-variable autonomous Van der Pol-Duffing model with a cross-shaped phase diagram
Abstract The bifurcation structure of a three-variable Van der Pol-Duffing-type model is studied in some detail, with special attention to the mixed-mode solutions, a type of complex periodic behavior frequently encountered in oscillating chemical reactions. The mixed-mode oscillations in the model occur close to two Hopf bifurcations, which are arranged with the saddle-node bifurcations in a so-called cross-shaped phase diagram, a bifurcation diagram also typical for chemical reactions. The mixed-mode oscillations are shown to lie on isolated bifurcation curves, which are all born in a single codimension-two bifurcation known as the neutrally twisted homoclinic orbit or inclination switch. With the introduction of an additional slow time scale, the same model can exhibit more complex mixed-mode oscillations and torus bifurcations.
1245682
Mixed-Mode Oscillations in Three Time-Scale Systems: A Prototypical Example
Mixed-mode dynamics is a complex type of dynamical behavior that is characterized by a combination of small-amplitude oscillations and large-amplitude excursions. Mixed-mode oscillations (MMOs) hav...
121853294
Global return maps for mixed-mode oscillations with one fast and two slow variables
Alternating patterns of small and large amplitude oscillations occur in a wide variety of physical, chemical, biological and engineering systems. These mixed-mode oscillations (MMOs) are often found in systems with multiple time scales. Previous differential equation modeling and analysis of MMOs has mainly focused on local mechanisms to explain the small oscillations. The goal of this paper is to provide a systematic starting point for the study of global return maps that generate the large oscillations. This paper contributes to the understanding of global return maps in the following ways: (a) We provide a detailed numerical study of the singular return maps for the Koper model which is a prototypical example for MMOs that also relates to local normal form theory, (b) we decompose the global return map to provide first steps towards towards analytical proofs of complicated MMO patterns, (c) we develop affine and quadratic return map models based on the previous analysis and (d) we demonstrate how to combine the global map models with small oscillations generated by two types of local normal forms. We find that the local-global decomposition is an efficient way to model, simulate and analyze MMOs.
14877154
Mixed-Mode Oscillations with Multiple Time Scales
Mixed-mode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied for over thirty years in chemical, physical, and biological systems. Few attempts have been made thus far to classify different patterns of MMOs, in contrast to the classification of the related phenomena of bursting oscillations. This paper gives a survey of different types of MMOs, concentrating its analysis on MMOs whose small-amplitude oscillations are produced by a local, multiple-time-scale “mechanism.” Recent work gives substantially improved insight into the mathematical properties of these mechanisms. In this survey, we unify diverse observations about MMOs and establish a systematic framework for studying their properties. Numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper.
18833142
Unfoldings of Singular Hopf Bifurcation
Singular Hopf bifurcation occurs in generic families of vector fields with two slow variables and one fast variable. Prototypes for this bifurcation depend upon several parameters, and the dynamics displayed by these systems is intricate. The core of this paper is a numerical investigation of the bifurcations in one prototypical family. It presents extensive diagrams of bifurcations of equilibrium points and periodic orbits that are close to singular Hopf bifurcation. In addition, parameters are determined where there is a tangency between invariant manifolds that are important in the appearance of mixed-mode oscillations in systems near singular Hopf bifurcation. One parameter of the prototype is identified as the primary bifurcation parameter, and the paper presents a catalogue of bifurcation sequences that occur as the primary bifurcation parameter is varied. These results are applied to estimating the parameters for the onset of mixed-mode oscillations in a model of chemical oscillations.
115764682
Introduction to the Modern Theory of Dynamical Systems: EQUILIBRIUM STATES AND SMOOTH INVARIANT MEASURES
Part I. Examples and Fundamental Concepts Introduction 1. First examples 2. Equivalence, classification, and invariants 3. Principle classes of asymptotic invariants 4. Statistical behavior of the orbits and introduction to ergodic theory 5. Smooth invariant measures and more examples Part II. Local Analysis and Orbit Growth 6. Local hyperbolic theory and its applications 7. Transversality and genericity 8. Orbit growth arising from topology 9. Variational aspects of dynamics Part III. Low-Dimensional Phenomena 10. Introduction: What is low dimensional dynamics 11. Homeomorphisms of the circle 12. Circle diffeomorphisms 13. Twist maps 14. Flows on surfaces and related dynamical systems 15. Continuous maps of the interval 16. Smooth maps of the interval Part IV. Hyperbolic Dynamical Systems 17. Survey of examples 18. Topological properties of hyperbolic sets 19. Metric structure of hyperbolic sets 20. Equilibrium states and smooth invariant measures Part V. Sopplement and Appendix 21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.
15670491
{22 () an Interval Hermite-obreschkoo Method for Computing Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Diierential Equation *
To date, the only eeective approach for computing guaranteed bounds on the solution of an initial value problem (IVP) for an ordinary diierential equation (ODE) has been interval methods based on Taylor series. This paper derives a new approach, an interval Hermite-Obreschkoo (IHO) method, for computing such enclosures. Compared to interval Taylor series (ITS) methods, for the same stepsize and order, our IHO scheme has a smaller truncation error, better stability, and requires fewer Taylor coeecients and high-order Jacobians. The stability properties of the ITS and IHO methods are investigated. We show as an important by-product of this analysis that the stability of an interval method is determined not only by the stability function of the underlying formula, as in a standard method for an IVP for an ODE, but also by the associated formula for the truncation error.
16327912
Validated solutions of initial value problems for ordinary differential equations
Compared to standard numerical methods for initial-value problems (IVPs) for ordinary differential equations (ODEs), validated methods for IVPs for ODEs have two important advantages: if they return a solution to a problem, then (1) the problem is guaranteed to have a unique solution, and (2) an enclosure of the true solution is produced. The authors survey Taylor series methods for validated solutions of IVPs for ODEs, describe several such methods in a common framework, and identify areas for future research.
16069082
An Effective High-Order Interval Method for Validating Existence and Uniqueness of the Solution of an IVP for an ODE
Validated methods for initial value problems for ordinary differential equations produce bounds that are guaranteed to contain the true solution of a problem. When computing such bounds, these methods verify that a unique solution to the problem exists in the interval of integration and compute a priori bounds for the solution in this interval. A major difficulty in this verification phase is how to take as large a stepsize as possible, subject to some tolerance requirement. We propose a high-order enclosure method for proving existence and uniqueness of the solution and computing a priori bounds.
19143030
Covering relations for multidimensional dynamical systems
Abstract We present a topological technique for analyzing dynamical systems with complex behavior, based on the general notion of covering relations. Our method can be used to study multidimensional dynamical systems with an arbitrary number of ‘topologically’ expanding directions.