Inspirational quotes with bifurcation.
Bifurcation embraces the weapon of success and failure.
Random thought fills the streambed, cascading with care and spilling over the flood plain until bifurcation forces it to arc and fork into a single, determined stream.
Though blessed with the enviable properties of a mink coat—graceful, unreasonable, and impractical no matter what she was draped over—she was nevertheless one of those people whose personality proved to be the bane of modern mathematicians. She was neither a flat nor solid shape. She showed no symmetry at all. Trigonometry, Calculus and Statistics all proved useless. Her Pie Chart was a muddle of arbitrary wedges, her Line Graph, the silhouette of the Alps. And just when one listed her under Chaos Theory—Butterfly Effects, Weather Predictions, Fractals, Bifurcation diagrams and whatnot—she showed up as an equilateral triangle, sometimes even a square.
If we increase r [in a logistic map] even more, we will eventually force the system into a period-8 limit cycle, then a period-16 cycle, and so on. The amount that we have to increase r to get another period doubling gets smaller and smaller for each new bifurcation. This cascade of period doublings is reminiscent of the race between Achilles and the tortoise, in that an infinite number of bifurcations (or time steps in the race) can be confined to a local region of finite size. At a very special critical value, the dynamical system will fall into what is essentially an infinite-period limit cycle. This is chaos.
Feel free to write to us if you have any questions. But before you do so, please take a look on our page with Frequently Asked Questions (FAQ) and even our sitemap to get a full overview of the content on our site.