The "lumped" suite of MATLAB code is designed to analyze the dynamical system 

  dS_R/dt = c_1*Q_R*Psi + nu*uh*(1-S_R/S_0_R) - c_2*S_R*|N|^(n_Glen-1)*N
  dS_K/dt = c_1*Q_K/T*Psi + uh*(1-S_K/S_0_K) - c_2*S_K*|N|^(n_Glen-1)*N
  -V_p_0*dN/dt = Q_tot - Q_R - (n_c-1)*Q_K
with
  Q_R = c_3*S_R^alpha*Psi^(beta-2);
  Q_K = c_3*S_K^alpha*T^(1-beta)*Psi^(beta-2);
  Psi = Psi_0 - N/L;

Available functionality:

1. Initialize a parameter structure (parallel_parameter_generator) required to run all other codes. The parameter structure can also be created manually by specifying the fields required of the parameter structure as described in the file headers for each of the routines listed below. The default parameter structures output by parallel_parameter_generator are dimensionless, and the corresponding scales to convert output back to dimensional are included as additional output in parallel_parameter_generator
2. Forward solution of the initial value problem (parallel_evolve_v2)
3. Compute a steady state solution for a given set of parameters (parallel_steady, which calls the Newton solver Newton; parallel_initguess may be used to compute an initial guess for the steady state based on the asymptotic form for a fully channel- or cavity-like conduit, in which there is no conduit opening due to sliding over bed roughness, or due to dissipation-driven melting, respectively)
4. Find the stability boundary in any parameter plane and compute the Landau coefficients for the Hopf bifurcation that occurs at the stability boundary (parallel_stability_boundary_v5)
5. Compute the eigenvalues for the linearized problem at a gridded set of steady states in parameter space (parallel_bifurcation_plane_v2, which calls parallel_initguess, parallel_steady and parallel_spectrum). The eigenvalues for the same steady states can also be re-computed for changed storage capacity V_p using parallel_stability_plane_v2; the steady states are independent of V_p and do not need to be recomputed.
6. Find the boundary in any parameter plane marking where periodic solutions that have N < 0 occur (parallel_zero_boundary, which calls ArcLength_single_symmetric and parallel_shoot_par_zero_v2)
7. Compute a bifurcation diagram showing maximum and minimum effective pressure along a closed orbit as a function of a single parameter to visualize the Hopf bifurcation (parallel_periodic_arclength_nullcline, whichcalls parallel_shoot_nullcline to find an initial point, and parallel_shoot_arclength_nullcline in conjunction with parallel_shoot_par_nullcline to compute the boundary using arc length continuation)
8. Compute periodic orbits provided initial conditions on the N-nullcline for the closed orbit have been computed. Using a Newton solver on parallel_shoot_nullcline, or run parallel_periodic_arclength_nullcline for a single point, npts =1, will find the initial conditions, parallel_shoot_closed_orbit_nullcline can then be used output the orbit itself
9. Solvers for asymptotic versions of the problem with n_c = 1 as described in (floodinitialization, which calls Newton_single and shoot_v1, and flood)

(c) 2020 Christian Schoof

Distributed under the BSD-3 license, see https://opensource.org/licenses/BSD-3-Clause
