I will give it a try:
Let us take first order homogenous delay differential equation without dependence of derivative of solution \( y' (t) \) on \( y(t) \) at the same moment of time \( t \):
\( y'(t) +\alpha y(t-T)=0 \)
Here T is delay, which shows how values of solution \( y(t) \) T moments in past ( or future if T<0) impact derivative of solution \( y'(t) \) at current moment \( t \).
Let us introduce new function \( z(t) \) such that :
\( \ln z(t) = -y(t) \) then \( z(t) = e^{-y(t)} \)
As is known from literature, solution of such delay equation is given by :
\( y(t) = \sum_{k=-\infty}^{\infty} C_k *e^{{1/T}*W_k(-\alpha T) t} \)
So \( z(t) = e^{-y(t)}= e^{\sum_{k=-\infty}^{\infty} -C_k *e^{{1/T}*W_k(-\alpha T) t}} \)
Where \( W_k \) is a k-th branch of Lambert W function.
To reach the solution I am looking for, of form:
\( e^{-iwt*e^{iwt}} \)
Coefficients \( C_k \) have to be:
\( C_k= {1/T}*W_k(-\alpha T) t \) so
\( z(t) = e^{\sum_{k=-\infty}^{\infty} -{1/T}*W_k(-\alpha T) t *e^{{1/T}*W_k(-\alpha T) t}} \)
This can be expressed as infinite product, with each term of sum as exponent of one multiplier \( e \) in this infinite product.
From here, obviously:
\( -iwt = -{1/T}*W_k(-\alpha T) t \)
so
\( {1/T}= I \)
\( T=1/I=-I \)
\( w= W_k (-\alpha/ I) \)
Now if we look back to original delay equation, this means that solution of a form:
\( e^{-Iwt*e^{Iwt} \) is one of particular solutions corresponding to particular branch of W function of a homogenous logarithmic first order differential equation with IMAGINARY DELAY \( T=-I \).
\( e^{Iwt*e^{-Iwt}} \)
corresponds to Imaginary delay \( T=1/I \).
We can generalize this to imaginary delay \( T= +-1/(I\tau) \). (Not sure if + sign is OK).
Then we have particular solutions of :
\( - (\ln z(t))' - \alpha \ln z(t+-1/(I\tau) )=0 \)
In form:
\( z_k(t)= e^{+-w_k t/(I\tau) )*e^{-+w_k t/I\tau } \)
Where
\( w_k = W_k (-+\alpha/ (I \tau) ) \)
I hope there are not too many mistakes
I think I could not choose coefficients \( C_k \) so arbitrary, it must have an impact on initial conditions or so called preshape function \( \phi(t) \) for times \( ln z(t) =\phi (t) \) \( t [0; +- I\tau] \) which I do not know how to calculate.
Next question is how these logarithmic differential delay equations (and thus processes they represent) with imaginary delay have to be nested to produce tetration and , perhaps, turbulence. In case of turbulence, the statistical character of processes will have to be added, somewhere, so that we deal with mean values , distributions and structure functions. (probably, In projective space as that is the space of turbulent time).
Ivars
Let us take first order homogenous delay differential equation without dependence of derivative of solution \( y' (t) \) on \( y(t) \) at the same moment of time \( t \):
\( y'(t) +\alpha y(t-T)=0 \)
Here T is delay, which shows how values of solution \( y(t) \) T moments in past ( or future if T<0) impact derivative of solution \( y'(t) \) at current moment \( t \).
Let us introduce new function \( z(t) \) such that :
\( \ln z(t) = -y(t) \) then \( z(t) = e^{-y(t)} \)
As is known from literature, solution of such delay equation is given by :
\( y(t) = \sum_{k=-\infty}^{\infty} C_k *e^{{1/T}*W_k(-\alpha T) t} \)
So \( z(t) = e^{-y(t)}= e^{\sum_{k=-\infty}^{\infty} -C_k *e^{{1/T}*W_k(-\alpha T) t}} \)
Where \( W_k \) is a k-th branch of Lambert W function.
To reach the solution I am looking for, of form:
\( e^{-iwt*e^{iwt}} \)
Coefficients \( C_k \) have to be:
\( C_k= {1/T}*W_k(-\alpha T) t \) so
\( z(t) = e^{\sum_{k=-\infty}^{\infty} -{1/T}*W_k(-\alpha T) t *e^{{1/T}*W_k(-\alpha T) t}} \)
This can be expressed as infinite product, with each term of sum as exponent of one multiplier \( e \) in this infinite product.
From here, obviously:
\( -iwt = -{1/T}*W_k(-\alpha T) t \)
so
\( {1/T}= I \)
\( T=1/I=-I \)
\( w= W_k (-\alpha/ I) \)
Now if we look back to original delay equation, this means that solution of a form:
\( e^{-Iwt*e^{Iwt} \) is one of particular solutions corresponding to particular branch of W function of a homogenous logarithmic first order differential equation with IMAGINARY DELAY \( T=-I \).
\( e^{Iwt*e^{-Iwt}} \)
corresponds to Imaginary delay \( T=1/I \).
We can generalize this to imaginary delay \( T= +-1/(I\tau) \). (Not sure if + sign is OK).
Then we have particular solutions of :
\( - (\ln z(t))' - \alpha \ln z(t+-1/(I\tau) )=0 \)
In form:
\( z_k(t)= e^{+-w_k t/(I\tau) )*e^{-+w_k t/I\tau } \)
Where
\( w_k = W_k (-+\alpha/ (I \tau) ) \)
I hope there are not too many mistakes
I think I could not choose coefficients \( C_k \) so arbitrary, it must have an impact on initial conditions or so called preshape function \( \phi(t) \) for times \( ln z(t) =\phi (t) \) \( t [0; +- I\tau] \) which I do not know how to calculate.
Next question is how these logarithmic differential delay equations (and thus processes they represent) with imaginary delay have to be nested to produce tetration and , perhaps, turbulence. In case of turbulence, the statistical character of processes will have to be added, somewhere, so that we deal with mean values , distributions and structure functions. (probably, In projective space as that is the space of turbulent time).
Ivars