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 class of 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 evolution equations (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(s): Francesco Calogero
Publisher: Oxford University Press, USA
Year: 2008
Language: English
Pages: 256
Contents......Page 10
1 Introduction......Page 12
1.N Notes to Chapter 1......Page 18
2.1 The trick......Page 20
2.2 Examples......Page 24
2.N Notes to Chapter 2......Page 32
3.1 A class of autonomous ODEs......Page 34
3.2 Examples......Page 42
3.2.1 First-order algebraic complex ODE......Page 43
3.2.2 Polynomial vector field in the plane......Page 44
3.2.4 Isochronous versions of the first and second Painlevé ODEs (complex and real versions)......Page 45
3.2.5 Autonomous second-order ODEs (complex and real versions)......Page 46
3.2.6 Autonomous third-order ODEs (complex and real versions)......Page 53
3.2.7 Isochronous version of a solvable second-order ODE due to Painlevé......Page 55
3.2.8 Isochronous versions of five solvable ODEs due to Chazy......Page 57
3.N Notes to Chapter 3......Page 61
4 Systems of ODEs: many-body problems, nonlinear harmonic oscillators......Page 62
4.1 A class of isochronous dynamical systems......Page 63
4.1.1 A lemma......Page 64
4.1.2 Examples......Page 71
4.2.2 Goldfishing......Page 80
4.2.3 Nonlinear oscillators......Page 125
4.2.4 Two Hamiltonian systems......Page 126
4.3 Two-dimensional systems......Page 131
4.4 Three-dimensional systems......Page 134
4.5 Multi-dimensional systems......Page 138
4.N Notes to Chapter 4......Page 142
5 Isochronous Hamiltonian systems are not rare......Page 146
5.1 Another trick......Page 147
5.2 Partially isochronous Hamiltonian systems......Page 159
5.2.1 A simple variant......Page 160
5.2.2 A more general variant......Page 162
5.3 More general Hamiltonians......Page 164
5.4 Examples......Page 166
5.5 Yet another trick......Page 174
5.5.1 Main results......Page 175
5.5.2 Transient chaos......Page 180
5.5.3 A simple example......Page 181
5.5.4 Quantization: equispaced spectrum......Page 183
5.N Notes to Chapter 5......Page 185
6 Asymptotically isochronous systems......Page 187
6.1 An asymptotically isochronous class of solvable many-body problems......Page 188
6.1.1 A specific example......Page 190
6.2 A (generally nonintegrable) class of asymptotically isochronous many-body models......Page 191
6.2.1 A theorem and its proof......Page 193
6.3 Some additional considerations......Page 196
7.1 The trick for PDEs......Page 199
7.2 A list of isochronous PDEs......Page 201
7.3 PDEs with lots of solutions periodic in time and in space......Page 217
7.N Notes to Chapter 7......Page 219
8 Outlook......Page 222
8.N Notes to Chapter 8......Page 223
A Some useful identities......Page 224
B Two proofs......Page 229
C Diophantine findings and conjectures......Page 240
References......Page 250
P......Page 260
Z......Page 261
