chaos-doubling

Transition to Chaos through Period-Doubling Bifurcations in Maps, ODEs and PDEs

Logistic Map

$$ \begin{align} x_{n+1} = r x_n (1-x_n) \end{align} $$

Bifurcation diagram

Ordinary Differential Equations - Feigenbaum System

$$ \begin{align} &\dot{x} =a y \\ &\dot{y} = x + z^2 \\ &\dot{z} = b + cx + dz \end{align} $$

Period-doubling

Bifurcation diagram for component y

Partial Differential Equations - Kuramoto-Sivashinsky Equation

$$ \begin{align} \frac{\partial u}{\partial t} = -u\frac{\partial u}{\partial x} - \frac{\partial^2 u}{\partial x^2} - \nu \frac{\partial^4 u}{\partial x^r} \end{align} $$

Periodic

Chaotic

Transition from Periodic to Chaotic

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Article describing the work is found as conference proceeding: waiting to be published