Contents Part 4 Bifurcation Behavior 443 Bifurcation Hall of Fame 445 16. Origins of Bifurcation Concepts 449 16.1 The battle of the bulge 451 16.2 The figure of the Earth 458 16.3 The stirring machine 467 16.4 The big picture 482viii Contents 17. Subtle Bifurcations 489 17.1 First excitation 491 17.2 Second excitation 497 17.3 Octave jump in 2D 501 17.4 Octave jump in 3D 506 18. Fold Catastrophes 511 18.1 Static fold in 1D 513 18.2 Static fold in 2D 519 18.3 Periodic fold in 2D 525 18.4 Periodic fold in 3D 534 19. Pinch Catastraphes 541 19.1 Spiral pinch in 2D 543 19.2 Vortical pinch in 3D 547 19.3 Octave pinch in 2D 553 19.4 Octave pinch in 3D 557 20. Saddle Connection Catastrophe 563 20.1 Basin bifurcation in 2D 565 20.2 Periodic blue sky in 2D 574 20.3 Chaotic blue sky in 3D 578 20.4 Rossler's blue sky in 3D 585 21. Explosive Bifurcations 591 21.1 Blue loop in 2D 593 21.2 Blue loop in 3D 597 21.3 Zeeman's blue tangle in 3D 601 21.4 Ueda's chaotic explosion in 3D 604 22. Fractal Bifurcations 611 22.1 The octave cascade 613 22.2 The noisy cascade 617 22.3 Braid bifurcations 619 22.4 Tangle bifurcations 623 Appendix Symbolic Expressions 625 Notes 632 Bibliography 635 Index 639