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Stability analysis

After linearization of the system (6)-(7) in the vicinity of the equilibrium points, we get the Jacobi matrix:
equation204
Let us consider the point tex2html_wrap_inline1495. Then
equation224
and the corresponding eigenvalues are equal to:
equation235
i.e. this equilibrium is a stable node, if
 equation239
and it is an unstable saddle, if
 equation245
Since tex2html_wrap_inline1497 for any tex2html_wrap_inline1499, then tex2html_wrap_inline1501, and any equilibrium tex2html_wrap_inline1503 is stable.

If tex2html_wrap_inline1505 then
equation254
and the corresponding eigenvalues are equal to
equation269
Since tex2html_wrap_inline1507 for any tex2html_wrap_inline1509, then tex2html_wrap_inline1511, and the equilibrium with tex2html_wrap_inline1513 cannot be an unstable node.

This equilibrium is a saddle point if
equation285
If tex2html_wrap_inline1515, we have a stable node or stable focus, moreover for the latest
equation297


next up previous
Next: Parametrization Up: The model: Formulation and Previous: The cycles

Werner von Bloh (Data & Computation)
Thu Jul 13 11:24:58 MEST 2000