I am seeking a help for a nonlinear instability problem. This is a pure mathematical problem, but it is very useful for turbulence.

"Nonlinear Stability of Tangent Function"

As is well known, the tangent function is defined within [-0.5pai, 0.5pai], and it is periodic with pai as the period. This function is singular at the two terminals of the zone, -0.5pai and o.5pai. Thus, the convergence region of this function is [-0.5pai, 0.5pai], except at the two terminals.

If subjected a nonlinear disturbance, this function (tan x) may be unstable and tends to diverge, when x is near to 0.5pai or -0.5pai.

We have guessed that this function (tan x) may be unstable when it is subjected a disturbance if x is beyond (-0.49917pai, 0.49917pai) or (-89.85 degree, 89.85 degree).

Now, the problem is to prove this stability region from mathematics with nonlinear instability analysis or computation, or any other methods.

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With a way to be easily understood, tan x is infinite, if x=90 degree. When x is near to 90 degree,for example, 89.9 degree, tan x is definite. However, if x is subjected a disturbance, x may reach the threshold and tan x may be infinite.

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Thank you very much in advance.

JackMM

I'm not sure I understand what you are really trying to do. Nonlinear stability (as I understand it) refers to the stability of a system that has non-linear elements in it, and hence can be represented by a non-linear (differential/algebraic) equation, or by a linear equation subject to non-linear boundary conditions and/or source terms. All you are describing is the limiting behavior of a non-linear function, which becomes infinite in the limit whether that limit is approached in a linear or non-linear fashion. Can you provide some clarification?

Thanks for your response.

The basic principle for this problem is as follows.

As is well known, tan x is infinite (divergent), if x=90 degree.

When x is near to 90 degree, for example, 89.9 degree, tan x is definite (tan 89.9 degree=572.9). However, if x is subjected a disturbance which is enough large, tan x may tend to infinite (divergent).

When x is not near to 90 degree, for example, 45 degree, tan x is definite. Then, no matter how large is the disturbance of x, tan x always is definite (tan45 degree=0.7071).

Now, the problem is how to determine the critical value of x, at which tan x could be diverge to infinite, at a disturbance. Now, I estimated this critical value is 89.85 degree, from other physical approach (when x<89.85 degree, tan x is stable, no matter of the disturbance; when x>89.85 degree, tan x is unstable and can diverge to infinite at a enough large disturbance). We need to prove this value from mathematics rigorously.

We can use any mathematical methods to solve this problem, if we can.

Assume: the disturbance of x is relatively small compared with the base value of x.

(tan45 degree=0.7071).

Above line is mistaken. Correct one is:

(tan45 degree=1.0).

As to the mathematical methods, we can employ any methods, e.g., linear, nonlinear, asymptotic analysis, or numerical simulation, or other methods not mentioned.

In the original message, I wrote "nonlinear instability."

"Nonlinear instability" is observed from the observation of the physical phenomenon, since the disturbance to trige the divergence needs a finite amplitude disturbance rather an infinitesmall disturbance.

This is not nonlinear instability it's a property of nonuniform convergence on the interval - look at any book on mathematical analysis (i.e. Weir's book on Lebesgue integration). Any function which contains a singularity within or at a limit point of a set has this property.

Thank you very much. I have no knowledge about this. I will check this book.

JackMM

The book (Alan J. Weir)looks like too mathematical. To prove the critical value (89.85 degree) quantitatively seems very difficult.

Look for any body inmathematics for collaboration for this very important work.

Thanks.