MHB Could this nondimensionalized ODE reveal hysteresis through its steady states?

AI Thread Summary
The discussion centers on a nondimensionalized ordinary differential equation (ODE) that seeks to demonstrate three nonzero steady states under the condition that the product of parameters r and q exceeds 4. The model's structure suggests potential hysteresis, which requires specific substitutions to analyze. Participants are exploring the implications of these substitutions and how to derive the necessary conditions for hysteresis, particularly identifying a critical point at R = 0.638. The conversation highlights challenges in solving for steady states and understanding the relationships between parameters in the r-q space. Overall, the focus remains on clarifying the conditions for hysteresis and the mathematical implications of the ODE.
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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?
 
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dwsmith said:
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?
Can anyone offer any guidance? I tried using Mathematica but the solution is unmanageable.

What should I do since it says rq > 4? I don't really understand how that affects. Does epsilon being really small change anything?
 
In order to determine if the model has hysteresis, I have to make the substitution

$r=\frac{R}{\sqrt{\varepsilon}}$ and $u=U\sqrt{\varepsilon}$.

And show that there is a nose at $R = 0.638$.

After I make the substitutions, what do I do to show the nose at $R = 0.638$?
 
dwsmith said:
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?

Can you explain your notation, in particular what are constants and what are functions of time and state?

Also, when you have:

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

you do not proceed by looking for solutions of:

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

CB
 
CaptainBlack said:
Can you explain your notation, in particular what are constants and what are functions of time and state?

Also, when you have:

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

you do not proceed by looking for solutions of:

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

CB

u is a parameter and everything else is a constant.

I know one steady state is u = 0 but the others are rather difficult to solve for which is what I need some aid with.

As well as parameterizing the r and q.
 
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dwsmith said:
In order to determine if the model has hysteresis, I have to make the substitution

$r=\frac{R}{\sqrt{\varepsilon}}$ and $u=U\sqrt{\varepsilon}$.

And show that there is a nose at $R = 0.638$.

After I make the substitutions, what do I do to show the nose at $R = 0.638$?
I have everything solved now except the hysteresis. If you decide to respond to this question, ignore everything else.

Thanks.
 
How would I come up with the r-q space for this model?
 
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