Modeling interactions 1

Pendulum m – mass R – rod length x – angle of elevation Small angles x

What is characteristic for the model of pendulum Second order Linear, time invariant, homogeneous (for small x) Conservative

Two – compartmental model of drug turnover x’ = -k yx x + k xy y y’ = -k xy y + k yx y – k 0y y x,y - concentrations of a drug in two compartments

Linear systems x’’+ x = 0 y’’’ y’’ + y’ + 0.1y = u x’ = x + y y’ = x - y

Linear systems General solution (matrix exponential) Initial conditions Free motion (component) and forced component Characteristic equation State – space representation

Phase plane –second order systems f(x’’,x’,x)=0 Find solution: x(t), x’(t) Represent it as a parametric curve on the plane x x’

Classification of equilibria on a plane Neutral center Stable focus Unstable focus Stable node Unstable node Saddle point

Neutral center x x’

x

Stable focus x x’

x

Unstable focus x x’

x

Stable node x x’

x

Unstable node x x’

x

Saddle point x x’

x

Linearization x’=f(x) Two solutions: x 1 (t) – starting from x 1 (0) x 2 (t) – starting from x 2 (0) Difference:  x(t)= x 2 (t)- x 1 (t)

We take: x 1 = equilibrium Then equation for  x becomes linear, time invariant

Stability

First integrals

Lyapunov functions