Isochronous Systems

by Calogero, Francesco

• ISBN13: 9780199535286
• ISBN10: 0199535280
• Format: Hardcover
• Copyright: 2008-04-15
• Publisher: Oxford University Press

Summary

A dynamical system is called isochronous if it features in its phase space an open, fully-dimensional region where all its solutions are periodic in all its degrees of freedom with the same, fixed period. Recently a simple transformation has been introduced, applicable to quite a large classof dynamical systems, that yields autonomous systems which are isochronous. This justifies the notion that isochronous systems are not rare.In this book the procedure to manufacture isochronous systems is reviewed, and many examples of such systems are provided. Examples include many-body problems characterized by Newtonian equations of motion in spaces of one or more dimensions, Hamiltonian systems, and also nonlinear evolutionequations (PDEs).The book shall be of interest to students and researchers working on dynamical systems, including integrable and nonintegrable models, with a finite or infinite number of degrees of freedom. It might be used as a basic textbook, or as backup material for an undergraduate or graduate course.

Author Biography

Francesco Calogero
Professor of Theoretical Physics, University of Rome "La Sapienza"

Table of Contents

Introduction	p. 1
Notes to Chapter 1	p. 7
Isochronous systems are not rare	p. 9
The trick	p. 9
Examples	p. 13
Notes to Chapter 2	p. 21
A single ODE of arbitrary order	p. 23
A class of autonomous ODEs	p. 23
Examples	p. 31
First-order algebraic complex ODE	p. 32
Polynomial vector field in the plane	p. 33
Oscillator with additional inverse-cube force	p. 34
Isochronous versions of the first and second Painleve ODEs (complex and real versions)	p. 34
Autonomous second-order ODEs (complex and real versions)	p. 35
Autonomous third-order ODEs (complex and real versions)	p. 42
Isochronous version of a solvable second-order ODE due to Painleve	p. 44
Isochronous versions of five solvable ODEs due to Chazy	p. 46
Notes to Chapter 3	p. 50
Systems of ODEs: many-body problems, nonlinear harmonic oscillators	p. 51
A class of isochronous dynamical systems	p. 52
A lemma	p. 53
Examples	p. 60
One-dimensional systems	p. 69
Many-body problems with two-body velocity-independent forces	p. 69
Goldfishing	p. 69
Nonlinear oscillators	p. 114
Two Hamiltonian systems	p. 115
Two-dimensional systems	p. 120
Three-dimensional systems	p. 123
Multi-dimensional systems	p. 127
Notes to Chapter 4	p. 131
Isochronous Hamiltonian systems are not rare	p. 135
Another trick	p. 136
Partially isochronous Hamiltonian systems	p. 148
A simple variant	p. 149
A more general variant	p. 151
More general Hamiltonians	p. 153
Examples	p. 155
Yet another trick	p. 163
Main results	p. 164
Transient chaos	p. 169
A simple example	p. 170
Quantization: equispaced spectrum	p. 172
Notes to Chapter 5	p. 174
Asymptotically isochronous systems	p. 176
An asymptotically isochronous class of solvable many-body problems	p. 177
A specific example	p. 179
A (generally nonintegrable) class of asymptotically isochronous many-body models	p. 180
A theorem and its proof	p. 182
Some additional considerations	p. 185
Isochronous PDEs	p. 188
The trick for PDEs	p. 188
A list of isochronous PDEs	p. 190
PDEs with lots of solutions periodic in time and in space	p. 206
Notes to Chapter 7	p. 208
Outlook	p. 211
Notes to Chapter 8	p. 212
Some useful identities	p. 213
Two proofs	p. 218
Diophantine findings and conjectures	p. 229
References	p. 239
Index	p. 249
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