Lie point symmetry

In this paragraph, we consider precisely expanded Lie point symmetries i.e. we work in an expanded space meaning that the distinction between independent variable, state variables and parameters are avoided as much as possible.

A symmetry group of a system is a continuous dynamical system defined on a local Lie group ${\displaystyle G}$ acting on a manifold ${\displaystyle M}$. For the sake of clarity, we restrict ourselves to n-dimensional real manifolds ${\displaystyle M=\mathbb {R} ^{n}}$ where ${\displaystyle n}$ is the number of system coordinates.

Let us define algebraic systems used in the forthcoming symmetry definition.

Let ${\displaystyle F=(f_{1},\dots ,f_{k})=(p_{1}/q_{1},\dots ,p_{k}/q_{k})}$ be a finite set of rational functions over the field ${\displaystyle \mathbb {R} }$ where ${\displaystyle p_{i}}$ and ${\displaystyle q_{i}}$ are polynomials in ${\displaystyle \mathbb {R} [Z]}$ i.e. in variables ${\displaystyle Z=(z_{1},\dots ,z_{n})}$ with coefficients in ${\displaystyle \mathbb {R} }$. An algebraic system associated to ${\displaystyle F}$ is defined by the following equalities and inequalities:

${\displaystyle {\begin{array}{ccc}\left\{{\begin{array}{l}p_{1}(Z)=0,\\\vdots \\p_{k}(Z)=0\end{array}}\right.&{\mbox{and}}&\left\{{\begin{array}{l}q_{1}(Z)\neq 0,\\\vdots \\q_{k}(Z)\neq 0.\end{array}}\right.\end{array}}}$

An algebraic system defined by ${\displaystyle F=(f_{1},\dots ,f_{k})}$ is regular (a.k.a. smooth) if the system ${\displaystyle F}$ is of maximal rank ${\displaystyle k}$, meaning that the Jacobian matrix ${\displaystyle (\partial f_{i}/\partial z_{j})}$ is of rank ${\displaystyle k}$ at every solution ${\displaystyle Z}$ of the associated semi-algebraic variety.

The following theorem (see th. 2.8 in ch.2 of ^[5]) gives necessary and sufficient conditions so that a local Lie group ${\displaystyle G}$ is a symmetry group of an algebraic system.

Theorem. Let ${\displaystyle G}$ be a connected local Lie group of a continuous dynamical system acting in the n-dimensional space ${\displaystyle \mathbb {R} ^{n}}$. Let ${\displaystyle F:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{k}}$ with ${\displaystyle k\leq n}$ define a regular system of algebraic equations:

${\displaystyle f_{i}(Z)=0\quad \forall i\in \{1,\dots ,k\}.}$

Then ${\displaystyle G}$ is a symmetry group of this algebraic system if, and only if,

${\displaystyle \delta f_{i}(Z)=0\quad \forall i\in \{1,\dots ,k\}{\mbox{ whenever }}f_{1}(Z)=\dots =f_{k}(Z)=0}$

for every infinitesimal generator ${\displaystyle \delta }$ in the Lie algebra ${\displaystyle {\mathfrak {g}}}$ of ${\displaystyle G}$.

Consider the algebraic system defined on a space of 6 variables, namely ${\displaystyle Z=(P,Q,a,b,c,l)}$ with:

${\displaystyle \left\{{\begin{array}{l}f_{1}(Z)=(1-cP)+bQ+1,\\f_{2}(Z)=aP-lQ+1.\end{array}}\right.}$

The infinitesimal generator

${\displaystyle \delta =a(a-1){\dfrac {\partial }{\partial a}}+(l+b){\dfrac {\partial }{\partial b}}+(2ac-c){\dfrac {\partial }{\partial c}}+(-aP+P){\dfrac {\partial }{\partial P}}}$

is associated to one of the one-parameter symmetry groups. It acts on 4 variables, namely ${\displaystyle a,b,c}$ and ${\displaystyle P}$. One can easily verify that ${\displaystyle \delta f_{1}=f_{1}-f_{2}}$ and ${\displaystyle \delta f_{2}=0}$. Thus the relations ${\displaystyle \delta f_{1}=\delta f_{2}=0}$ are satisfied for any ${\displaystyle Z}$ in ${\displaystyle \mathbb {R} ^{6}}$ that vanishes the algebraic system.

Let us define systems of first-order ODEs used in the forthcoming symmetry definition.

Let ${\displaystyle d\cdot /dt}$ be a derivation w.r.t. the continuous independent variable ${\displaystyle t}$. We consider two sets ${\displaystyle X=(x_{1},\dots ,x_{k})}$ and ${\displaystyle \Theta =(\theta _{1},\dots ,\theta _{l})}$. The associated coordinate set is defined by ${\displaystyle Z=(z_{1},\dots ,z_{n})=(t,x_{1},\dots ,x_{k},\theta _{1},\dots ,\theta _{l})}$ and its cardinal is ${\displaystyle n=1+k+l}$. With these notations, a system of first-order ODEs is a system where:

${\displaystyle \left\{{\begin{array}{l}{\dfrac {dx_{i}}{dt}}=f_{i}(Z){\mbox{ with }}f_{i}\in \mathbb {R} (Z)\quad \forall i\in \{1,\dots ,k\},\\{\dfrac {d\theta _{j}}{dt}}=0\quad \forall j\in \{1,\dots ,l\}\end{array}}\right.}$

and the set ${\displaystyle F=(f_{1},\dots ,f_{k})}$ specifies the evolution of state variables of ODEs w.r.t. the independent variable. The elements of the set ${\displaystyle X}$ are called state variables, these of ${\displaystyle \Theta }$ parameters.

One can associate also a continuous dynamical system to a system of ODEs by resolving its equations.

An infinitesimal generator is a derivation that is closely related to systems of ODEs (more precisely to continuous dynamical systems). For the link between a system of ODEs, the associated vector field and the infinitesimal generator, see section 1.3 of.^[4] The infinitesimal generator ${\displaystyle \delta }$ associated to a system of ODEs, described as above, is defined with the same notations as follows:

${\displaystyle \delta ={\dfrac {\partial }{\partial t}}+\sum _{i=1}^{k}f_{i}(Z){\dfrac {\partial }{\partial x_{i}}}\cdot }$

Here is a geometrical definition of such symmetries. Let ${\displaystyle {\mathcal {D}}}$ be a continuous dynamical system and ${\displaystyle \delta _{\mathcal {D}}}$ its infinitesimal generator. A continuous dynamical system ${\displaystyle {\mathcal {S}}}$ is a Lie point symmetry of ${\displaystyle {\mathcal {D}}}$ if, and only if, ${\displaystyle {\mathcal {S}}}$ sends every orbit of ${\displaystyle {\mathcal {D}}}$ to an orbit. Hence, the infinitesimal generator ${\displaystyle \delta _{\mathcal {S}}}$ satisfies the following relation^[8] based on Lie bracket:

${\displaystyle [\delta _{\mathcal {D}},\delta _{\mathcal {S}}]=\lambda \delta _{\mathcal {D}}}$

where ${\displaystyle \lambda }$ is any constant of ${\displaystyle \delta _{\mathcal {D}}}$ and ${\displaystyle \delta _{\mathcal {S}}}$ i.e. ${\displaystyle \delta _{\mathcal {D}}\lambda =\delta _{\mathcal {S}}\lambda =0}$. These generators are linearly independent.

One does not need the explicit formulas of ${\displaystyle {\mathcal {D}}}$ in order to compute the infinitesimal generators of its symmetries.

Consider Pierre François Verhulst's logistic growth model with linear predation,^[14] where the state variable ${\displaystyle x}$ represents a population. The parameter ${\displaystyle a}$ is the difference between the growth and predation rate and the parameter ${\displaystyle b}$ corresponds to the receptive capacity of the environment:

${\displaystyle {\dfrac {dx}{dt}}=(a-bx)x,{\dfrac {da}{dt}}={\dfrac {db}{dt}}=0.}$

The continuous dynamical system associated to this system of ODEs is:

${\displaystyle {\begin{array}{rccc}{\mathcal {D}}:&(\mathbb {R} ,+)\times \mathbb {R} ^{4}&\rightarrow &\mathbb {R} ^{4}\\&({\hat {t}},(t,x,a,b))&\rightarrow &\left(t+{\hat {t}},{\frac {axe^{a{\hat {t}}}}{a-(1-e^{a{\hat {t}}})bx}},a,b\right).\end{array}}}$

The independent variable ${\displaystyle {\hat {t}}}$ varies continuously; thus the associated group can be identified with ${\displaystyle \mathbb {R} }$.

The infinitesimal generator associated to this system of ODEs is:

${\displaystyle \delta _{\mathcal {D}}={\dfrac {\partial }{\partial t}}+((a-bx)x){\dfrac {\partial }{\partial x}}\cdot }$

The following infinitesimal generators belong to the 2-dimensional symmetry group of ${\displaystyle {\mathcal {D}}}$:

${\displaystyle \delta _{{\mathcal {S}}_{1}}=-x{\dfrac {\partial }{\partial x}}+b{\dfrac {\partial }{\partial b}},\quad \delta _{{\mathcal {S}}_{2}}=t{\dfrac {\partial }{\partial t}}-x{\dfrac {\partial }{\partial x}}-a{\dfrac {\partial }{\partial a}}\cdot }$

There exist many software packages in this area.^[15][16][17] For example, the package liesymm of Maple provides some Lie symmetry methods for PDEs.^[18] It manipulates integration of determining systems and also differential forms. Despite its success on small systems, its integration capabilities for solving determining systems automatically are limited by complexity issues. The DETools package uses the prolongation of vector fields for searching Lie symmetries of ODEs. Finding Lie symmetries for ODEs, in the general case, may be as complicated as solving the original system.