All Questions

Consider the sequence in the unit disk $D=\{(x,y)\,|\,x^2+y^2\leq 1\}$ iteratively defined by the quadratic map $$\begin{aligned} x_{n+1}&=2x_ny_n\\y_{n+1}&=1-2x_n^2\end{aligned},$$ starting ...

The following problem is presented in the paper Recent development of chaos theory in topological dynamics - by Jian Li and Xiangdong Ye : "If a dynamical system [$(X,f)$, $X$ metric space, $f$ ...

It is known that for $r=-2,2,4$ the logistic map $x_{n+1}=r x_n (1-x_n)$ has exact solutions of the form $$ x_n=\frac12 \left\{ 1- f\left(r^n f^{-1}(1-2x_0)\right)\right\} \qquad \qquad{(*)} $$ for ...

Suppose you have the following coupled discrete dynamical system: \begin{align*} e_{k+1}&=e_k - 2~\alpha~e_k~\lambda^2~\alpha_k^2 + \alpha^2~e_k^2~\lambda^3 \alpha_k^3\\ \alpha_{k+1}&= \...

Is the following statement true? How it can be proved/rejected? Initial conditions that correspond to unbounded orbits in polynomial symplectic mappings, which exhibit chaotic behavior (exponential ...