We can analyze stability of equilibrium of linear system X'=AX by finding characteristic polynomial of matrix A. If we deal with delay differential equation of the form x'(t)=x(t-r), r is constant, then we get characteristic equation s-Exp(rs)=0. The real part of root of this transcendental equation will decide the stability of equilibrium (i.e. zero) solution.
Now consider the equation x'(t)=x(rt), r is constant i.e. a differential equation with proportional delay. In this case, the equilibrium is x(t)=0 but it is difficult to find a characteristic equation and hence the stability of equilibrium. Please help.
Regards:
S. Bhalekar
I think that the Differential equation with proportional delay is reduced to the scaling equation of the type
s*F(s)=aF(rs). (a, b are some comstants). The solution of this equation can be expressed in the form of the product. Then you can investigate this product.
Another option - the usage of new logarithmic variables F(lns)=aF(lns+lnr). May be this transfromation will be helpful as well.
But in general your problem is not trivial one.
You can consider a solution of the equation x'(t)=x(rt) in the form of Taylor expansion. Let s=square root of r. Then the solution is:
x(t)=x(0)*sum_{n=0}^{infinity} s^{n^2-n}*t^n/n!
This expansion is convergent for any t if abs(s)
hi
just some intuition .. hope it is of some use..
1. x'(t)=x(rt) , if r >1 then we want the values of x(t) at a time greater than t . This seems not likely. If we do know it then we already have the solution!!! unless this is used in retrospect with data already collected.
2. if r< 1 then we need to show integral x(t) upto t is bounded.
3 if r
Hi,
this is called a pantograph equation, there are many papers on it. You may try this: https://www.researchgate.net/publication/242981179_Exact_and_discretized_stability_of_the_pantograph_equation
Article Exact and discretized stability of the pantograph equation
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