In 1966, American mathematician Steve Smale was awarded a Fields Medal, a kind of Nobel Prize for mathematics. At a press conference at the International Congress of Mathematicians in Moscow, Smale attacked both US and Russian foreign policy. He was vehemently opposed to military aggression by the two super-powers. His pacifist position resulted in serious … Continue reading Finding a Horseshoe on the Beaches of Rio
Tag: Chaos
The Logistic Map is hiding in the Mandelbrot Set
The logistic map is a simple second-order function on the unit interval: $latex \displaystyle x_{n+1} = r x_n (1-x_n) \,, &fg=000000$ where $latex {x_n}&fg=000000$ is the variable value at stage $latex {n}&fg=000000$ and $latex {r}&fg=000000$ is the ``growth rate''. For $latex {1 \le r \le 4}&fg=000000$, the map sends the unit interval [0,1] into itself. … Continue reading The Logistic Map is hiding in the Mandelbrot Set
Sharkovsky’s Theorem
This post is an extension and elaboration of two recent posts, with more technical details ``The reasonable man adapts himself to the world: the unreasonable one persists in trying to adapt the world to himself. Therefore all progress depends on the unreasonable man.'' … Continue reading Sharkovsky’s Theorem
Oleksandr Sharkovsky and Chaos Theory
We all know how a simple action at a critical moment can change our lives. Over the past half-century, with the growing evidence of how small changes can lead to dramatic developments, there has been a paradigm shift in science. Earlier attempts to predict the future as if it were determined with certainty have given … Continue reading Oleksandr Sharkovsky and Chaos Theory
The Logistic Map: a Simple Model with Rich Dynamics
Suppose the population of the world $latex {P(t)}&fg=000000$ is described by the equation $latex \displaystyle \frac{\mathrm{d}P} {\mathrm{d}t} = a P \,. &fg=000000$ Then $latex {P(t)}&fg=000000$ grows exponentially: $latex {P(t) = P_0 \exp(at)}&fg=000000$. This was the nightmare prediction of Thomas Robert Malthus. Taking a value $latex {a=0.02\ \mathrm{yr}^{-1}}&fg=000000$ for the growth rate, we get a doubling … Continue reading The Logistic Map: a Simple Model with Rich Dynamics
Can We Control the Weather?
Atmospheric motions are chaotic: a minute perturbation can lead to major changes in the subsequent evolution of the flow. How do we know this? There is just one atmosphere and, if we perturb it, we can never know how it might have evolved if left alone. We know, from simple nonlinear models that exhibit chaos, … Continue reading Can We Control the Weather?
ThatsMaths 