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 ) &gt; 0, such that x 3 &gt; 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 βThis 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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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  )  &amp;gt;  0, such that       x    3    &amp;gt;  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  βThis 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.

pampatsha · Accepted Answer

One can easily confirm that       x    3    &amp;gt;  0    ⟹    w  (  t  )  =  W  (  x  (  t  )  )  &amp;gt;  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  )  &amp;lt;  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    ¯  .

Comparison function for the Rossler system <mtable columnalign="right left right left right left

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