Equation

Van Der Pol Cycle Invariant

$$\left\{ \begin{array}{lll}\dot{x}_{1} & = & x_{2}\\\dot{x}_{2} & = & \left(1-x_{1}^{2}\right)\cdot x_{2}-x_{1}.\end{array}\right. $$

Search space: $[-3, 3]\times[-3, 3]\setminus{\left\{\mathbf{x}\mid x_{1}^{2}+x_{2}^{2}\leq 1\right\}}$

Figure

Code

from pyibex import *
from pyinvariant import *

# Define the search space
space = IntervalVector([[-3, 3],[-3,3]])

# Create the grpah structure
smartSubPaving = SmartSubPaving(space)

# Create the Domain
dom = Domain(smartSubPaving, FULL_DOOR)
dom.set_border_path_in(False)
dom.set_border_path_out(False)

f_sep = Function("x[2]", "(x[0])^2+(x[1])^2-(1.0)^2")
s = SepFwdBwd(f_sep, GEQ) # possible options : LT, LEQ, EQ, GEQ, GT
dom.set_sep(s); # Reduce the domain with a separator

# Create the Dynamics
f = Function("x[2]", "(x[1],(1.0*(1.0-x[0]^2))*x[1]-x[0])")
dyn = DynamicsFunction(f, FWD_BWD)

# Create the Maze associated with the Domain and the dynamics
maze = Maze(dom, dyn)

# Contract the system
for i in range(15):
	print(i)
	smartSubPaving.bisect()
	maze.contract()

# Visualization
visu = VibesMaze("smartSubPaving", maze)
visu.setProperties(0,0,512,512)
visu.show()