Equation

Rayleigh Basin of Capture

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

Target set: $x_{1}^{2}+x_{2}^{2}\leq (0.4)^{2}$

Search space: $[-2, 2]\times[-2,2]$

Figure

Code

from pyinvariant import *

# Define the search space
space = IntervalVector([[-2.0, 2.0],[-2.0,2.0]])

# Create the grpah structure
smartSubPaving = SmartSubPaving(space)

# Create the Domain
dom_outer = Domain(smartSubPaving, FULL_WALL)
dom_outer.set_border_path_in(False)
dom_outer.set_border_path_out(False)
f_sep = Function("x[2]", "(x[0])^2+(x[1])^2-(0.4)^2")
s_outer = SepFwdBwd(f_sep, LEQ) # possible options : LT, LEQ, EQ, GEQ, GT
dom_outer.set_sep_output(s_outer);

dom_inner = Domain(smartSubPaving, FULL_DOOR)
dom_inner.set_border_path_in(True)
dom_inner.set_border_path_out(True)
s_inner = SepFwdBwd(f_sep, GEQ) # possible options : LT, LEQ, EQ, GEQ, GT
dom_inner.set_sep_input(s_inner);

# Create the Dynamics
f = Function("x[2]", "(-x[1], (x[0]+x[1]^3-x[1]))") # Repulsive cycle
dyn = DynamicsFunction(f, BWD)

# Create the two Maze associated with the Domain and the dynamics
maze_inner = Maze(dom_inner, dyn)
maze_outer = Maze(dom_outer, dyn)

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

# Visualization
visu = VibesMaze("Rayleigh Basin of Capture", maze_outer, maze_inner)
visu.setProperties(0,0,512,512)
visu.show()
visu.drawCircle(0.0, 0.0, 0.4, "black[red]");