Sunday, 3 May 2015

Constants or integrals of motion: Invariants of a dynamical flow

In this post we will shortly review a concept in dynamical systems namely of invariants of a dynamical flow with a simple derivation using famous Lotka-Volterra system as an example, due to Lotka (1925) and Volterra (1927).

Concept

A dynamical flow associated with an observation vector ${\bf y}(t)$ may have functions, $I({\bf y})$ that are time independent, being $dI/dt=0$. The number of invariants and the length of the observation vector have an effect on overall dynamics.

Lotka-Volterra (LV) System

The LV dynamics explains the behaviour between population of the prey $v$  and population of predators $u$, a case of predator-prey model. We will use a special case of the LV dynamics, remember the dot notation, meaning time derivatives, for predators,
$$ \dot{u}  =  u (v-2) $$
and for prays,
$$ \dot{v}  = v (1-u) $$
Observation vector will consist of $y=(u,v)$.

If we divide these equations, hoping that we can collect $u$ and $v$ in separate terms,

$$
\begin{eqnarray}
\frac{\dot{u}}{\dot{v}}            & = & \frac{u (v-2)}{v(1-u)} \\
\dot{u} v (1-u)                            & = & \dot{v} u (v-2) \\
\dot{u} v (1-u) - \dot{v} u (v-2)  & = & 0 \\
\dot{u} (1-u) - \dot{v} u/v (v -2) & = & 0\\
\dot{u} (1-u)/u - \dot{v}(v-2)/v   & = & 0
\end{eqnarray}
$$

If we integrate both sides over time $dt$,
$$
\begin{eqnarray}
\int \frac{1-u}{u} \frac{du}{dt} dt - \int \frac{v-2}{v} \frac{dv}{dt} dt & = &0 \\
\int \frac{1-u}{u}du - \int \frac{v-2}{v} dv & = &0 \\
\end{eqnarray}
$$

Solution of these indefinite integrals yields to a an invariant of the LV dynamics
$$ d I({\bf y})/dt = ln u - u + 2 ln v - v  $$
We have shown one invariant of the system. This is important to determine the structure of the system, such as volume preserving dynamics, i.e., Hamiltonian Dynamics.

Further Reading
(c) Copyright 2008-2015 Mehmet Suzen (suzen at acm dot org)

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