ScommaMaruj

ScommaMaruj

Answered

2022-07-04

Comparison function for the Rossler system
x ˙ 1 = x 2 x 3 x ˙ 2 = x 1 + α x 2 x ˙ 3 = β + x 3 ( x 1 γ )
With α = β = 0.1 and γ = 14, and x 3 ( 0 ) > 0, such that x 3 > 0 t 0.
we have found that for the comparison function W = x 1 2 + x 2 2 + 2 x 3 the following inequality holds:
W ˙ 2 α W + 2 β
This implies that the solutions of this system are well defined on the infinite time interval [ 0 , ), and implies that solutions can not escape to infinity in finite time.

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Answer & Explanation

pampatsha

pampatsha

Expert

2022-07-05Added 15 answers

One can easily confirm that x 3 > 0 w ( t ) = W ( x ( t ) ) > 0 and thus
α w ( t ) + β ( α w ( 0 ) + β ) e 2 α t w ( t ) w ¯ ( t ) = w ( 0 ) e 2 α t + β α ( e 2 α t 1 ) .
This means that over any finite time interval [ 0 , T ] the solution is bounded by
W ( x ( t ) ) w ¯ ( T ) .
So if one considers the region W ( x ) < 2 w ¯ ( T ), it is bounded, and has x ( T ) as inner point. Thus the ODE function has bound and a Lipschitz constant there, thus the IVP with the IC at x ( T ) can be solved locally in forwar direction and the solution x thus continued.
Note that the movement forward in time is essential for the claim that x 3 stays positive, backwards in time x 3 can become negative and thus unbounded by w ¯ .

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