The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors

A₁ Appendix averaged equations Bifurcation diagram C₁ and C₂ calculated CALIFORNIA LIBRARY chaotic behaviour Chapter compute Conjecture curve described differential equations eigenvalues exist expect extra twisting Figure final xy finite heteroclinic homoclinic explosions homoclinic orbit Hopf bifurcation increases infinite number intersects the return interval larger r-values locate Lorenz attractors Lorenz equations Lorenz flow Lorenz list Lorenz system manifold of C₁ non-stable periodic orbit non-symmetric orbits non-wandering set numerical experiments occur one-dimensional maps orbits and trajectories parameter range parameter values period doubling windows rectangles region return map return plane return surface right-hand branch saddle-node bifurcation schematic sequence of homoclinic shown in Fig shows stable manifold stable orbit stable periodic orbits stationary point strange attractor strange invariant set symbolic descriptions symbolic sequences symmetric orbit symmetric xy orbit tion top face topological Trajectories started tubes two-dimensional UNIVERSITY OF CALIFORNIA unstable xy period z-axis ху