By Harry Dankowicz

This publication presents a finished advent to the mathematical technique of parameter continuation, the computational research of households of ideas to nonlinear mathematical equations. It develops a scientific formalism for developing summary representations of continuation difficulties and for imposing those in an current computational platform.

*Recipes for Continuation* lends equivalent significance to theoretical rigor, set of rules improvement, and software program engineering. The ebook demonstrates using totally constructed toolbox templates for unmarried- and multisegment boundary-value difficulties to the research of periodic orbits in delicate and hybrid dynamical structures, quasi-periodic invariant tori, and homoclinic and heteroclinic connecting orbits among equilibria and/or periodic orbits. It additionally exhibits using vectorization for optimum computational potency, an object-oriented paradigm for the modular building of continuation difficulties, and adaptive discretization algorithms for assured bounds on predicted errors.

The ebook includes broad and entirely labored examples that illustrate the appliance of the MATLAB-based Computational Continuation center (COCO) to difficulties from contemporary study literature which are correct to dynamical method versions from mechanics, electronics, biology, economics, and neuroscience. a good number of the routines on the finish of every bankruptcy can be utilized as self-study or for direction assignments that variety from reflections on theoretical content material to implementations in code of algorithms and toolboxes that generalize the dialogue within the booklet or the literature. Open-ended initiatives during the publication supply possibilities for summative assessments.

**Audience:** it's meant for college students and academics of nonlinear dynamics and engineering, in addition to engineers and scientists engaged in modeling and simulation, and is effective to strength builders of computational instruments for research of nonlinear dynamical platforms. It assumes a few familiarity with MATLAB programming and a theoretical sophistication anticipated of upper-level undergraduate or first-year graduate scholars in utilized arithmetic and/or computational technology and engineering.

**Contents:** half I: layout basics: bankruptcy 1: A Continuation Paradigm; bankruptcy 2: Encapsulation; bankruptcy three: building; bankruptcy four: Toolbox improvement; bankruptcy five: job Embedding; half II: Toolbox Templates: bankruptcy 6: Discretization; bankruptcy 7: The Collocation Continuation challenge; bankruptcy eight: Single-Segment Continuation difficulties; bankruptcy nine: Multisegment Continuation difficulties; bankruptcy 10: The Variational Collocation challenge; half III: Atlas Algorithms: bankruptcy eleven: protecting Manifolds; bankruptcy 12: Single-Dimensional Atlas Algorithms; bankruptcy thirteen: Multidimensional Manifolds; bankruptcy 14: Computational domain names; half IV: occasion dealing with: bankruptcy 15: unique issues and occasions; bankruptcy sixteen: Atlas occasions and Toolbox Integration; bankruptcy 17: occasion Handlers and department Switching; half V: variation: bankruptcy 18: Pointwise variation and Comoving Meshes; bankruptcy 19: A Spectral Toolbox; bankruptcy 20: integrating variation in Atlas Algorithms; half VI: Epilogue: bankruptcy 21: Toolbox initiatives; Index

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**Example text**

Hint: show that the derivative of Y+ (a) can never equal zero. Similarly, show that √ the derivative of Y− (a) has only one zero (which necessarily must occur for a > 2). 6. Explain how the observations regarding the behavior of Y± (a) are used to draw conclusions as to the number of solutions to Eqs. 13). 7. The results obtained in Sect. 2 show that stationary curves of the integral functional in Eq. 1) exist only for Y ≥ Y ∗ . Under the transformation f (x) → f (1 − x) /Y , however, the boundary conditions f (0) = 1, f (1) = Y become 1 .

Then, as suggested in Fig. , that T m ak t k x ∗ (t) − · f x ∗ (t), λ = 0. 7) k=0 q q for some sequence of quadrature weights w j j=1 and quadrature nodes β j T j=1 on the interval [0, T ] for some integer q. Either of these phase conditions now provides closure to the continuation problem. Assuming that the closed continuation problem may be solved for the n (m + 1) + 1 unknowns {ak }m k=0 and T , its constituent equations collectively define 22 Chapter 2. Encapsulation a family of polynomial approximants for viable periodic solutions implicitly in terms of the value of λ.

Show that ϕ± (a) (x) ≈ ± sinh x for a near 1. Use the result of the previous two exercises and the slope at x = 0 to explain the conclusion regarding a change in extremum from a local minimum to a saddle accompanying a change in sign of ϕ± (a)|x=1 . Derive the expression for ϕ± (a)|x=1 given in Eq. 30). Derive the expression for the value of the functional J given in Eq. 32). Explain the sense in which the curve consisting of a line segment from (0, 1) to (0, 0), followed by a line segment from (0, 0) to (1, 0), followed by a line segment from (1, 0) to (1, Y ) is a “boundary” of the solution space of smooth curves.