Contents 

Part I     Periodic Behavior        1

	Dynamics Hall of Fame     3

	1.  Basic Concepts of Dynamics     13
	    1.1  State spaces     15
	    1.2  Dynamical systems     21
	    1.3  Special trajectories     30
	    1.4  Asymptotic approach to limit sets     36
	    1.5  Attractors, basins, and separatrices     42
	    1.6  Gradient systems     47

	2.  Classical Applications: Limit Points in 2D from Newton to Rayleigh     53
	    2.1  Pendula     55
	    2.2  Buckling columns     65
	    2.3  Percussion instruments     71
	    2.4  Predators and prey     82

	3.  Vibrations: Limit Cycles in 2D from Rayleigh to Rashevsky     87
	    3.1  Wind instruments     89
	    3.2  Bowed instruments     94
	    3.3  Radio transmitters     103
	    3.4  Biological morphogenesis     106

	4.  Forced Vibrations: Limit Cycles in 3D from Rayleigh to Duffing     113
	    4.1  The ring model for forced springs     115
	    4.2  Forced linear springs     126
	    4.3  Forced hard springs     139
	    4.4  Harmonics     148vi  Contents

	5.  Compound Oscillations: Invariant Tori in 3D from Huyghens to Hayashi     159
	    5.1  The torus model for two oscillators     161
	    5.2  The torus model for coupled oscillators     167
	    5.3  The ring model for forced oscillators     172
	    5.4  Braids: the dynamics of entrainment     176
	    5.5  Response curves for frequency changes     185
	    5.6  Forced electric oscillators     192

	Conclusion     199