I have recently been playing around with the discrete map $$z_{n+1} = z_n – \frac{1}{z_n}$$ That is, repeatedly mapping each number to the difference between itself and its reciprocal. It shows some interesting behaviour. This map seems so simple/obvious, I highly doubt this has never been analysed before. However, I (and several people I asked) […]

I’m new to this “integrable system” stuff, but from what I’ve read, if there are as many linearly independent constants of motion that are compatible with respect to the poisson brackets as degrees of freedom, then the system is solvable in terms of elementary functions. Is this correct? I get that for each linearly independent […]

In the ODE where $y’=f(y(t))$ and $y(0)=yo$. The omega limit set $w(yo)$ is positively invariant and also negatively invariant. I want to prove first that its positively invariant and then prove its negatively invariant. But how do I show that using a flow function ($phi(y,t)$) given that I know only the definition and the identity […]

I have a (possibly very) limited math background, but I would like to find a book, website, or other type of work that will teach almost everything about strange attractors. As a motivating example, I found a recurrence of the following form: $$x_{n+1} = a x_n + y_n$$ $$y_{n+1} = b + (x_n)^2$$ …at this […]

Assuming we have an ordinary differential equation (ODE) such as Lorenz system: $$ \dot x=\sigma(y-x)\\ \dot y=\gamma x-y-xz\\ \dot z=xy-bz $$ where $$ \sigma = 10\\ \gamma = 28\\ b = \frac{8}{3}\\ x(0)=10\\ y(0)=1\\ z(0)=1 $$ This system is known to be chaotic because of its behavior [1], [2]. However, we normally judge about a […]

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