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I need to demonstrate that there are 3 possible nonzero steady states if r and q lie in a domain in r,q space given approximately by rq>4. Could this model exhibit hysteresis?

The below ODE is nondimensionalized.

$0<\varepsilon\ll 1$

$\displaystyle

\frac{du}{d\tau} = ru\left(1 - \frac{u}{q}\right) - \left(1 - \exp\left[-\frac{u^2}{\varepsilon}\right]\right)=0

$

From that we get these two equations,

$\displaystyle

h(u) = ru\left(1 - \frac{u}{q}\right), \quad f(u) = 1 - \exp\left[-\frac{u^2}{\varepsilon}\right].

$

The derivatives of $h$ and $f$ are

$\displaystyle

h'(u) = r - \frac{2ru}{q}, \quad f'(u) = \frac{2u}{\varepsilon}\exp\left[-\frac{u^2}{\varepsilon}\right].

$

Solving for $\displaystyle r = \frac{2uq}{\varepsilon(q - 2u)}\exp\left[-\frac{u^2}{\varepsilon}\right]$.

I tried substitution r into the original equation and solving for q but that is ridiculous. What can I do here?

The below ODE is nondimensionalized.

$0<\varepsilon\ll 1$

$\displaystyle

\frac{du}{d\tau} = ru\left(1 - \frac{u}{q}\right) - \left(1 - \exp\left[-\frac{u^2}{\varepsilon}\right]\right)=0

$

From that we get these two equations,

$\displaystyle

h(u) = ru\left(1 - \frac{u}{q}\right), \quad f(u) = 1 - \exp\left[-\frac{u^2}{\varepsilon}\right].

$

The derivatives of $h$ and $f$ are

$\displaystyle

h'(u) = r - \frac{2ru}{q}, \quad f'(u) = \frac{2u}{\varepsilon}\exp\left[-\frac{u^2}{\varepsilon}\right].

$

Solving for $\displaystyle r = \frac{2uq}{\varepsilon(q - 2u)}\exp\left[-\frac{u^2}{\varepsilon}\right]$.

I tried substitution r into the original equation and solving for q but that is ridiculous. What can I do here?

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