Perron - Frobenius operator

The Perron – Frobenius operator is an operator that describes the change in probability density over time in the phase space of states of a dynamical system . Named after the German mathematicians Ferdinand Frobenius and Oscar Perron .

Content

Definition

Consider the ensemble of trajectories of a dynamical system, the initial data for which are distributed in phase space with a certain probability density $P(x)$ ${\ displaystyle P (x)}$ . Let the state of a dynamic system in phase space change over time $x(t)=\varphi (x,t)$ ${\ displaystyle x (t) = \ varphi (x, t)}$ . In this case, the probability density changes: $P(x,t)=L(P(x),t)$ ${\ displaystyle P (x, t) = L (P (x), t)}$ . Operator $L$ ${\ displaystyle L}$ is called the Perron-Frobenius operator ^[1] .

Examples

Consider the mapping $x_{n+1}=f(x_{n})$ ${\ displaystyle x_ {n + 1} = f (x_ {n})}$ . Let on $n$ ${\ displaystyle n}$ -th step in the phase space, the probability density $p_{n}(x)$ ${\ displaystyle p_ {n} (x)}$ . Then for the probability density in the next step we get: $p_{n+1}(y)=\int \delta (f(x)-y)p_{n}(x)dx=L(p_{n})$ ${\ displaystyle p_ {n + 1} (y) = \ int \ delta (f (x) -y) p_ {n} (x) dx = L (p_ {n})}$ . Here is the operator $L$ ${\ displaystyle L}$ called the Perron-Frobenius operator for mapping $f(x)$ ${\ displaystyle f (x)}$ ^[2] . He is a linear non-self-adjoint operator .

Notes

↑ Nonlinear Dynamics and Chaos, 2011 , p. 159.
↑ Nonlinear Dynamics and Chaos, 2011 , p. 168.

Literature

Malinetskii G. G. , Potapov A. B. Non-linear dynamics and chaos: basic concepts. - M .: Librocom, 2011 .-- 240 p. - ISBN 978-5-397-01583-7 .

[_424afbdc16ac7a9e-1] Nonlinear Dynamics and Chaos, 2011 , p. 159.

[_424afcdc16ac7c62-2] Nonlinear Dynamics and Chaos, 2011 , p. 168.