The Lorenz Equations: Bifurcations, Chaos, and Strange AttractorsSpringer Science & Business Media, 6 dec 2012 - 270 pagina's The equations which we are going to study in these notes were first presented in 1963 by E. N. Lorenz. They define a three-dimensional system of ordinary differential equations that depends on three real positive parameters. As we vary the parameters, we change the behaviour of the flow determined by the equations. For some parameter values, numerically computed solutions of the equations oscillate, apparently forever, in the pseudo-random way we now call "chaotic"; this is the main reason for the immense amount of interest generated by the equations in the eighteen years since Lorenz first presented them. In addition, there are some parameter values for which we see "preturbulence", a phenomenon in which trajectories oscillate chaotically for long periods of time before finally settling down to stable stationary or stable periodic behaviour, others in which we see "intermittent chaos", where trajectories alternate be tween chaotic and apparently stable periodic behaviours, and yet others in which we see "noisy periodicity", where trajectories appear chaotic though they stay very close to a non-stable periodic orbit. Though the Lorenz equations were not much studied in the years be tween 1963 and 1975, the number of man, woman, and computer hours spent on them in recent years - since they came to the general attention of mathematicians and other researchers - must be truly immense. |
Inhoudsopgave
1 | |
Chapter | 7 |
PRETURBULENCE STRANGE ATTRACTORS AND GEOMETRIC MODELS | 26 |
Chapter 4 | 37 |
PERIOD DOUBLING AND STABLE ORBITS | 51 |
3 | 59 |
5 | 66 |
7 | 74 |
Chapter 8 | 151 |
2 | 184 |
Appendix | 192 |
Appendix D | 199 |
Appendix | 211 |
Appendix | 217 |
GEOMETRIC MODELS OF THE LORENZ EQUATIONS | 223 |
Appendix | 234 |
Overige edities - Alles bekijken
The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors Colin Sparrow Fragmentweergave - 1982 |
The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors Colin Sparrow Geen voorbeeld beschikbaar - 2005 |
Veelvoorkomende woorden en zinsdelen
A₁ analysis Appendix averaged equations Bifurcation diagram C₁ and C₂ calculated chaotic behaviour Chapter compute Conjecture curve described differential equations eigenvalues exist expect extra twisting Figure final xy finite global 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 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 bifurcation period doubling windows preturbulent rectangles region return map return plane return surface right-hand branch saddle-node bifurcation schematic sequence of homoclinic shown in Fig shows stable orbit stable periodic orbits stable symmetric xy 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 unstable manifold x²y xy period z-axis ху