Hysteresis Model with two real parameters

Question

I would like to ask the following:

I am trying to make a throughout analysis of a Hysteresis model in one dimension, with two real parameters:

• $\frac{dx}{dt}=f(x,ν,μ)=νx-x^3+μ$, where $ν,μ\in{\mathbb{R}}$.

The problem is that I am not able to understand how to find the equilibrium points of this particular system, in order to use them for my bifurcation analysis.

When I try to set $f$ equal to zero I will have to solve this equation: $νx-x^3+μ=0$. What I suspect is that I have to take the values where the derivative of $f$ is zero, meaning $x_{1,2}=\pm \sqrt{\frac{ν}{3}}$, and then add these points to $f$ and solve with respect to $μ$.

Thank you for your time everyone!

Answer 1

You have a cubic on the right hand side. We can characterize the roots using the discriminant.

In this particular case, the discriminant is $\Delta = 4\nu^3-27\mu^2$. You will will have

• 3 real equilibria when $\Delta>0$,
• 1 real equilibium when $\Delta<0$, and
• 2 real equilibria with one repeated when $\Delta=0$.

Thus, the curve $4\nu^3-27\mu^2 = 0$ will certainly be relevant to your bifurcation analysis.

A more detailed response would certainly be possible with more information. For example, in what range of $\mu$ and $\nu$ parameters are you interested? And in what types of initial conditions are of interest?