Boundary Value Problems

• Boundary Value Problems
• • Basic Usage
  • Important notes
  • • 1. Relation with periodic orbits
    • 2. Accessing the solution
    • 3. Record from solution
    • 4. Plot solution
  • Methods
  • • Trapeze method
    • Collocation method
    • Shooting method

Consider the Boundary Value Problem (BVP)

\[\frac{du}{dt}=F(u,p),\quad t\in[t_0,t_f]\]

with boundary conditions

\[g(u(t_0), u(t_f), p)=0.\]

A periodic orbit is a special case where $g(u_0,u_1)=u_0-u_1$ and $t_f-t_0=T$ is the period.

The BVP module provides a unified interface for solving such problems using the same discretization methods as periodic orbits:

• BVPModel: mathematical formulation (vector field + boundary conditions)
• Trapeze, Collocation, Shooting: discretization methods
• discretize: combine model + method into a DiscretizedBVP
• BVPBifProblem: wrap for continuation

Basic Usage

using BifurcationKit
const BK = BifurcationKit

# 1. Define the vector field
F(u, p) = [u[2], -p.ω^2 * u[1]]

# 2. Define boundary conditions
g(u0, u1, p) = [u0[1], u1[1]]

# 3. Create a BVP model
model = BK.BVP.BVPModel(F, g; n=2)

# 4. Choose discretization
disc = BK.BVP.Trapeze(M=100)

# 5. Discretize
bvp = BK.BVP.discretize(model, disc)

# 6. Generate initial guess
x0 = BK.BVP.generate_solution(bvp, t -> [cos(t), sin(t)])

# 7. Create BVP bifurcation problem
prob = BK.BVP.BVPBifProblem(bvp, x0, (ω=1.0,), (@optic _.ω))

# 8. Continue
br = BK.continuation(prob, PALC(), ContinuationPar())

Important notes

1. Relation with periodic orbits

The BVP module is a generalization of the periodic orbit solvers. It uses the same underlying discretizations (Trapeze, Collocation, Shooting) but exposes a cleaner interface where the boundary conditions are user-defined rather than hardcoded to $u(0)=u(T)$.

2. Accessing the solution

After a computation, you can obtain the time-domain solution using get_solution_bvp:

sol = BK.BVP.get_solution_bvp(bvp, x, p)
# sol.t contains the time mesh
# sol.u contains the solution at each time slice

For periodic orbits, you can also use get_periodic_orbit.

3. Record from solution

You can pass your own function to continuation. It is called as record_from_solution(x, opt; k...) where opt.p is the current continuation parameter.

4. Plot solution

Similarly, the method is called like plot_solution(x, opt; k...) where opt.p is the current value of the continuation parameter.

Methods

We provide three discretization methods:

Trapeze method

The Trapezoid method discretizes the BVP using a finite difference scheme based on the trapezoidal rule. It is usually fast and works well for small to medium sized problems. See BVP Trapeze.

Collocation method

The Collocation method uses orthogonal collocation on time intervals. It is the most precise method and supports mesh adaptation during continuation. See BVP Collocation.

Shooting method

The Shooting method solves the BVP by integrating the ODE between shooting points and matching conditions. It is well suited for large-scale problems and supports parallel integration. See BVP Shooting.