Dynamical systems. Differential equations, maps and chaotic behaviour

Автор(ы):Arrowsmith D. K., Place C. M.
06.10.2007
Год изд.:1992
Описание: В книге описаны различные динамические системы, приведены упражнения по всем имеющимся разделам.
Оглавление:
Dynamical systems. Differential equations, maps and chaotic behaviour — обложка книги.
1. Introduction [1]
  1.1 Preliminary ideas [1]
    1.1.1 Existence and uniqueness [1]
    1.1.2 Geometrical representation [3]
  1.2 Autonomous equations [6]
    1.2.1 Solution curves and the phase portrait [6]
    1.2.2 Phase portraits and dynamics [11]
  1.3 Autonomous systems in the plane [12]
  1.4 Construction of phase portraits in the plane [17]
    1.4.1 Use of calculus [17]
    1.4.2 Isoclines [20]
  1.5 Flows and evolution [23]
      Exercises [27]
2. Linear systems [35]
  2.1 Linear changes of variable [35]
  2.2 Similarity types for 2 x 2 real matrices [38]
  2.3 Phase portraits for canonical systems in the plane [43]
    2.3.1 Simple canonical systems [43]
    2.3.2 Non-simple canonical systems [46]
  2.4 Classification of simple linear phase portraits in the plane [48]
    2.4.1 Phase portrait of a simple linear system [48]
    2.4.2 Types of canonical system and qualitative equivalence [50]
    2.4.3 Classification of linear systems [52]
  2.5 The evolution operator [52]
  2.6 Affine systems [55]
  2.7 Linear systems of dimension greater than two [57]
    2.7.1 Three-dimensional systems [57]
    2.7.2 Four-dimensional systems [61]
    2.7.3 n-Dimensional systems [62]
      Exercises [63]
3. Non-linear systems in the plane [71]
  3.1 Local and global behaviour [71]
  3.2 Linearization at a fixed point [74]
  3.3 The linearization theorem [77]
  3.4 Non-simple fixed points [81]
  3.5 Stability of fixed points [84]
  3.6 Ordinary points and global behaviour [93]
    3.6.1 Ordinary points [93]
    3.6.2 Global phase portraits [95]
  3.7 First integrals [96]
  3.8 Limit points and limit cycles [101]
  3.9 Poincare- Bendixson theory [105]
      Exercises [110]
4. Flows on поп-planar phase spaces [120]
  4.1 Fixed points [120]
    4.1.1 Hyperbolic fixed points [120]
    4.1.2 N on-hyperbolic fixed points [125]
  4.2 Closed orbits [129]
    4.2.1 Poincare maps and hyperbolic closed orbits [129]
    4.2.2 Topolofiical classification of hyperbolic closed orbits [132]
    4.2.3 Periodic orbits and quasi-periodic motion [136]
  4.3 Attracting sets and attractors [138]
    4.3.1 Trapping regions for Poincare maps [140]
    4.3.2 Saddle points in attracting sets [143]
  4.4 Further integrals [147]
    4.4.1 Hamilton's equations [148]
    4.4.2 Poincare maps of Hamiltonian flows [152]
      Exercises [155]
5 Applications I: planar phase spaces [162]
  5.1 Linear models [162]
    5.1.1 A mechanical oscillator [162]
    5.1.2 Electrical circuits [167]
    5.1.3 Economics [170]
    5.1.4 Coupled oscillators [172]
  5.2 Afilne models [175]
    5.2.1 The forced harmonic oscillator [176]
    5.2.2 Resonance [177]
  5.3 Non-linear models [179]
    5.3.1 Competing species [180]
    5.3.2 Volterra-Lotka equations [183]
    5.3.3 The Holling tanner model [185]
  5.4 Relaxation oscillations [188]
    5.4.1 Van der Pol oscillator [188]
    5.4.2 Jumps and regularization [192]
  5.5.Piecewise modelling [195]
    5.5.1 The jump assumption and piecewise models [196]
    5.5.2 A limit cycle from linear equations [198]
      Exercises [202]
6 Applications II rum-planar pliasc spaces, families of systems and bifurcations [212]
  6.1 The Zeeman models of heartbeat and nerve impulse [212]
  6.2 A model оГ animal conflict [218]
  6.3 Families of differential equations and bifurcations [223]
    6.3.1 introductory remarks [223]
    6.3.2 Saddle-node bifurcation [226]
    6.3.3 Hopf bifurcation [228]
  6.4 A mathematical model of tumour growth [232]
    6.4.1 Construction of the model [232]
    6.4.2 An analysis of the dynamics [233]
  6.5 Some bifurcations in families of one-dimensional maps [240]
    6.5.1 The fold bifurcation [240]
    6.5.2 The flip bifurcation [242]
    6.5.3 The logistic map [245]
  6.6 Some bifurcations in families of Iwo-dimensionaJ maps [251]
    6.6.1 The child on a swing [251]
    6.6.2 The Duffing equation [254]
  6.7 Area-preserving maps, homoclinic tangles and strange attractors [259]
    6.7.1 Introductory remarks [259]
    6.7.2 Periodic orbits and island chains [261]
    6.7.3 Chaotic orbits and homoclinic tangles [264]
    6.7.4 Strange attracting sets [267]
  6.8 Symbolic dynamics [271]
  6.9 New directions [279]
    6.9.1 Introductory remarks [279]
    6.9.2 Iterated function schemes [280]
    6.9.3 Cellular automata [284]
      Exercises [288]
Bibliography [303]
Hints to lixcrcises [306]
Index [326]
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Язык:ENG
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