symmetry of a solution of an ordinary differential equation
Let γ be a symmetry of the ordinary differential equation (http://planetmath.org/SymmetryOfAnOrdinaryDifferentialEquation) and x0 be a steady state solution of ˙x=f(x). If
γx0=x0 |
then γ is called a symmetry of the solution of x0.
Let γ be a symmetry of the ordinary differential equation and x0(t) be a periodic solution of ˙x=f(x). If
γx0(t-t0)=x0(t) |
for a certain t0 then (γ,t0) is called a symmetry of the periodic solution of x0(t).
lemma: If γ is a symmetry of the ordinary differential equation and let x0(t) be a solution(either steady state or periodic) of ˙x=f(x). Then γx0(t) is a solution of ˙x=f(x).
proof: If x0(t) is a solution of dxdt=f(x) implies dx0(t)dt=f(x0(t)). Let’s now verify that γx0(t) is a solution, with a substitution into dxdt=f(x). The left hand side of the equation becomes dγx0(t)dt=γdx0(t)dt and the right hand side of the equation becomes f(γx0(t))=γf(x0(t)) since γ is a symmetry of the differential equation. Therefore we have that the left hand side equals the right hand side since dx0(t)dt=f(x0(t)).
qed
References
- GSS Golubitsky, Martin. Stewart, Ian. Schaeffer, G. David: Singularities and Groups in Bifurcation Theory (Volume II). Springer-Verlag, New York, 1988.
Title | symmetry of a solution of an ordinary differential equation |
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Canonical name | SymmetryOfASolutionOfAnOrdinaryDifferentialEquation |
Date of creation | 2013-03-22 13:42:26 |
Last modified on | 2013-03-22 13:42:26 |
Owner | Daume (40) |
Last modified by | Daume (40) |
Numerical id | 11 |
Author | Daume (40) |
Entry type | Definition |
Classification | msc 34-00 |
Synonym | symmetry of a periodic solution solution of an ordinary differential equation |