Evolution (math)

In mathematics, the evolution of a differential equation is defined as a two-parameter mapping given by: ${\displaystyle \Phi }$ ${\displaystyle x'(t)=f(t,x(t))}$

${\displaystyle \Phi ^{t,t_{0}}x_{0}:=x(t)}$

in which

• ${\displaystyle x(t)}$the solution to the initial value problem is obtained from the above. Like and the initial condition exists ${\displaystyle x(t_{0})=x_{0}}$
• ${\displaystyle |t-t_{0}|}$ should be sufficiently small.

In words: Evolution maps the value of any solution curve at the point in time to the value of the solution curve at the point in time . So it describes the further development of the solution starting from the starting point . ${\displaystyle x_{0}}$ ${\displaystyle x}$ ${\displaystyle t_{0}}$ ${\displaystyle x(t)}$ ${\displaystyle t}$ ${\displaystyle x_{0}}$

The evolution of the differential equation has the following properties:

• ${\displaystyle \Phi ^{t_{0},t_{0}}x_{0}=x_{0}}$
• ${\displaystyle {\frac {d}{d\tau }}\Phi ^{t+\tau ,t}x|_{\tau =0}=f(t,x(t))}$
• ${\displaystyle \Phi ^{t_{2},t_{1}}\Phi ^{t_{1},t}x_{0}=\Phi ^{t_{2},t}x_{0}}$for ( transitivity ). ${\displaystyle t\leq t_{1}\leq t_{2}}$

In the case of autonomous differential equations , the start time is arbitrary. It then writes instead of simply and designated as phase flow . ${\displaystyle x'=f(x)}$ ${\displaystyle t_{0}}$ ${\displaystyle \Phi ^{t,t_{0}}}$ ${\displaystyle \Phi ^{t}}$ ${\displaystyle \Phi ^{t}}$