# 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: 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] Формат: djvu Размер: 5687148 байт Язык: ENG Рейтинг: 181 Открыть: Нет поддержки JS :(