Calculate Equilibrium points for continuous system DynamicalSystems.jl

Specific Domains Modelling & Simulations
1

I need calculate all equilibrium points for dynamical system Hindmarsh Rose (HR). Code for system:

function HR!(du, u, p, t)
    
    function sigma(x)
            return 1.0 / ( 1.0 + exp( -10.0 * ( x  - ( - 0.25 ) ) ) )
    end
    
    a, b, c, d, s, xr, r,  I, vs, k1, k2, el_link  = p
    x1, y1, z1, x2, y2, z2 = u
    
    du[1] = y1 + b * x1 ^ 2 - a * x1 ^3 - z1 + I - k1 * ( x1 - vs ) * sigma(x2) + el_link * ( x2 - x1 )
    du[2] = c - d * x1 ^2 - y1
    du[3] = r * ( s * ( x1 - xr ) - z1 )
    
    du[4] = y2 + b * x2 ^ 2 - a * x2 ^3 - z2 + I - k2 * ( x2 - vs ) * sigma(x1) + el_link * ( x1 - x2 )
    du[5] = c - d * x2 ^2 - y2
    du[6] = r * ( s * ( x2 - xr ) - z2 )
    
    return SVector(du[1], du[2], du[3],
                    du[4], du[5], du[6])
end

That is two coupled HR element. How can i get equilibrium points ? Are there any packages that look for equilibrium points for the system ?
For such a system it is too problematic to find equilibrium points analytically
Monodromy matrix i can not use because need find equilibrium points in regular and chaotic modes

2

Typing “equilibrium points” in the search bar of the docs gives this as first result :wink:

https://juliadynamics.github.io/DynamicalSystems.jl/dev/chaos/periodicity/#Fixed-points

2 Likes
3

I’m sorry, i’m too inattentive. Thank you.

4

No, this isn’t possible out of the box. You would need to combine the Poincare map integrator with a newton solver or NLSolve.jl. GitHub - JuliaNLSolvers/NLsolve.jl: Julia solvers for systems of nonlinear equations and mixed complementarity problems

if you do write this code, please consider contributing it back to DynamicalSystems.jl.

5

Okay, I’m sorting out the code for poincaré. I think it will take some time because some lines of code are not clear to me

6

What do you mean ? The source code of the poincare map should be completely irrelevant. You have want to find the zeros of pmal(u) - u using Newton method or whatever nlsolve jl gives you. This is really easy. See also chapter 4 section 2 of our book if this newton’s approach for fjnding fixed points isn’t clear to you.

7

I don’t understand yet how I can get the Poincare mapping itself in the form of equations

8

That’s why you need to use nlsolve.jl, because the poincare map doesn’t doesn’t equations.

10

how to find equilibrium points of a non linear closed delay differential system of equations?

the equations add up to give zero.(closed)

11

apart from the trivial 0 vector