Hello all,
I have developed a model for a 9 dimensional dynamical system. I intend to perform bifurcation analysis on the aforementioned system in MatCont. I should add that, the system is highly nonlinear with very large equations in its Jacobian and Hessian matrices. The continuation on the stationary branch of solutions is easily performed and occurrence of several bifurcations can be observed. However, when I try to start the continuation study in the periodic orbits emerged from a Hopf bifurcation, MatCont returns the error of "Current step size is too small (point 1)". It means that the continuation algorithm does not reach convergence in calculating the second point of continuation on the periodic branch of solutions. Could anyone suggest me how to deal with this issue?
P.S.
I have tried to manipulate the amplitude, initial step size, Min step size, Max step size and tolerance values of the numerical algorithm. Can it be the case that I have not tried enough?
Hello
By chance, do you know what method is being used to solve your equations? Have you tried an implicit solver?
Hi,
Actually I don't. I have used the default solver in MatCont. I guess it is Newton's method
From what i have been digging up, I am assuming you are using MatCont from Sourceforge. From their documentation, they make use of Matlab's ODE solvers along with some new Stiff solvers. If your are using the default, I would assume you would be using Matlab's go to ode solver ode45.
Ode45 is an explicit ode solver and can not handle Stiff problems. Somewhere in MatCont (I have not looked deeply into the package to know where), you should be able to use a different ode solver. Matlab recommends ode15s if ode45 fails. If your system needs to preserve energy, try ode23tb.
I hope this was somewhat helpful.
(edit: The package peeked my interest so I have downloaded the package. Please disregard the statement on them adding 2 new Stiff solvers... They are not Stiff but high order explicit runge kutta. Both are of order 7(8). From what I have seen, you should be able to choose a solver from the drop down box in the integration window)
Dear Matt,
I want to thank you for your attention and response. The problem is that the integration window you mentioned is open to manipulation for time simulations, i.e. finding the equilibrium of the system. For bifurcation analysis, which is a continuation process, there are only two windows, Starter window (for initial conditions) and Continuer window (for options regarding the continuation), in none of which there is an option for choosing the solver. I somehow need to change the solver that MatCont uses by default.
If steps are small, then the motion is fast. If steps are moderate, then the motion be slow. Separate the fast and slow orbits. Try a problem of linear damped oscillator with small or infinitesimal mass on your solver instead yours huge equation. If the problem remains, then we should return to the theory, since the solver cannot do nothing.
Dear Dr. Rano,
Thank you for your attention. Unfortunately system is highly stiff. However, I believe numerical calculation of Hessian Matrix might solve the problem, since using analytical expressions for Hessian Matrix and higher order derivatives, as MATCONT suggests, could be problematic in continuation process due to induced singularities in system.
It is a speculation, nevertheless useful in my case.
Hello Pouya, I'm having the same problem. Did you solve this issue?
Hi Pouya, I am loosing my mind over this. Could tell me if you solved this problem?
Thank you
- Badges
- Science topic
- Similar topics
- Methodology
- Research Methodology
- Research Papers