Equation

Rayleigh Invariant

$$\left\{ \begin{array}{lll}\dot{x}_{1} & = & x_{2}\\\dot{x}_{2} & = & -(x_{1}+x_{2}^{3}-x_{2})\end{array}\right.$$

Search space: $[-2, 2]\times[-2, 2]\setminus{\left\{\mathbf{x}\mid x_{1}^{2}+x_{2}^{2}\leq 0.3\right\}}$

Figure

Code

from pyinvariant import *

# Define the search space
space = IntervalVector([[-2, 2],[-2,2]])

# 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-(0.3)^2")
s = SepFwdBwd(f_sep, GEQ) # possible options : LT, LEQ, EQ, GEQ, GT
dom.set_sep(s);

# Create the Dynamics
f = Function("x[2]", "(x[1],(-1.0*(x[0]+x[1]^3-x[1])))")
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): # Number of bisections
	print(i)
	smartSubPaving.bisect()
	maze.contract()

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