Ebook: Shadowing in Dynamical Systems: Theory and Applications

In this book the theory of hyperbolic sets is developed, both for diffeomorphisms and flows, with an emphasis on shadowing. We show that hyperbolic sets are expansive and have the shadowing property. Then we use shadowing to prove that hyperbolic sets are robust under perturbation, that they have an asymptotic phase property and also that the dynamics near a transversal homoclinic orbit is chaotic.
It turns out that chaotic dynamical systems arising in practice are not quite hyperbolic. However, they possess enough hyperbolicity to enable us to use shadowing ideas to give computer-assisted proofs that computed orbits of such systems can be shadowed by true orbits for long periods of time, that they possess periodic orbits of long periods and that it is really true that they are chaotic.
Audience: This book is intended primarily for research workers in dynamical systems but could also be used in an advanced graduate course taken by students familiar with calculus in Banach spaces and with the basic existence theory for ordinary differential equations.

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In this book the theory of hyperbolic sets is developed, both for diffeomorphisms and flows, with an emphasis on shadowing. We show that hyperbolic sets are expansive and have the shadowing property. Then we use shadowing to prove that hyperbolic sets are robust under perturbation, that they have an asymptotic phase property and also that the dynamics near a transversal homoclinic orbit is chaotic.
It turns out that chaotic dynamical systems arising in practice are not quite hyperbolic. However, they possess enough hyperbolicity to enable us to use shadowing ideas to give computer-assisted proofs that computed orbits of such systems can be shadowed by true orbits for long periods of time, that they possess periodic orbits of long periods and that it is really true that they are chaotic.
Audience: This book is intended primarily for research workers in dynamical systems but could also be used in an advanced graduate course taken by students familiar with calculus in Banach spaces and with the basic existence theory for ordinary differential equations.

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In this book the theory of hyperbolic sets is developed, both for diffeomorphisms and flows, with an emphasis on shadowing. We show that hyperbolic sets are expansive and have the shadowing property. Then we use shadowing to prove that hyperbolic sets are robust under perturbation, that they have an asymptotic phase property and also that the dynamics near a transversal homoclinic orbit is chaotic.
It turns out that chaotic dynamical systems arising in practice are not quite hyperbolic. However, they possess enough hyperbolicity to enable us to use shadowing ideas to give computer-assisted proofs that computed orbits of such systems can be shadowed by true orbits for long periods of time, that they possess periodic orbits of long periods and that it is really true that they are chaotic.
Audience: This book is intended primarily for research workers in dynamical systems but could also be used in an advanced graduate course taken by students familiar with calculus in Banach spaces and with the basic existence theory for ordinary differential equations.
Content:
Front Matter....Pages i-xiv
Hyperbolic Fixed Points of Diffeomorphisms and Their Stable and Unstable Manifolds....Pages 1-20
Hyperbolic Sets of Diffeomorphisms....Pages 21-55
Transversal Homoclinic Points of Diffeomorphisms and Hyperbolic Sets....Pages 57-76
The Shadowing Theorem for Hyperbolic Sets of Diffeomorphisms....Pages 77-90
Symbolic Dynamics Near a Transversal Homoclinic Point of a Diffeomorphism....Pages 91-97
Hyperbolic Periodic Orbits of Ordinary Differential Equations, Stable and Unstable Manifolds and Asymptotic Phase....Pages 99-114
Hyperbolic Sets of Ordinary Differential Equations....Pages 115-169
Transversal Homoclinic Orbits and Hyperbolic Sets in Differential Equations....Pages 171-185
Shadowing Theorems for Hyperbolic Sets of Differential Equations....Pages 187-223
Symbolic Dynamics Near a Transversal Homoclinic Orbit of a System of Ordinary Differential Equations....Pages 225-239
Numerical Shadowing....Pages 241-284
Back Matter....Pages 285-299

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In this book the theory of hyperbolic sets is developed, both for diffeomorphisms and flows, with an emphasis on shadowing. We show that hyperbolic sets are expansive and have the shadowing property. Then we use shadowing to prove that hyperbolic sets are robust under perturbation, that they have an asymptotic phase property and also that the dynamics near a transversal homoclinic orbit is chaotic.
It turns out that chaotic dynamical systems arising in practice are not quite hyperbolic. However, they possess enough hyperbolicity to enable us to use shadowing ideas to give computer-assisted proofs that computed orbits of such systems can be shadowed by true orbits for long periods of time, that they possess periodic orbits of long periods and that it is really true that they are chaotic.
Audience: This book is intended primarily for research workers in dynamical systems but could also be used in an advanced graduate course taken by students familiar with calculus in Banach spaces and with the basic existence theory for ordinary differential equations.
Content:
Front Matter....Pages i-xiv
Hyperbolic Fixed Points of Diffeomorphisms and Their Stable and Unstable Manifolds....Pages 1-20
Hyperbolic Sets of Diffeomorphisms....Pages 21-55
Transversal Homoclinic Points of Diffeomorphisms and Hyperbolic Sets....Pages 57-76
The Shadowing Theorem for Hyperbolic Sets of Diffeomorphisms....Pages 77-90
Symbolic Dynamics Near a Transversal Homoclinic Point of a Diffeomorphism....Pages 91-97
Hyperbolic Periodic Orbits of Ordinary Differential Equations, Stable and Unstable Manifolds and Asymptotic Phase....Pages 99-114
Hyperbolic Sets of Ordinary Differential Equations....Pages 115-169
Transversal Homoclinic Orbits and Hyperbolic Sets in Differential Equations....Pages 171-185
Shadowing Theorems for Hyperbolic Sets of Differential Equations....Pages 187-223
Symbolic Dynamics Near a Transversal Homoclinic Orbit of a System of Ordinary Differential Equations....Pages 225-239
Numerical Shadowing....Pages 241-284
Back Matter....Pages 285-299
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