Focus

A type of arrangement of the trajectories of an autonomous system of first-order ordinary differential equations

$$ \tag{* } \dot{x} = f ( x),\ \ x = ( x _ {1} , x _ {2} ),\ \ f: G \subset \mathbf R ^ {2} \rightarrow \mathbf R ^ {2} , $$

$ f \in C ( G) $, where $ G $ is the domain of uniqueness, in a neighbourhood of a singular point $ x _ {0} $( cf. Equilibrium position). This type is characterized as follows. There is a neighbourhood $ U $ of $ x _ {0} $ such that for all trajectories of the system starting in $ U \setminus \{ x _ {0} \} $, the negative semi-trajectories are escaping (in the course of time they leave any compact set $ V \subset U $) and the positive semi-trajectories, without leaving $ U $, tend to $ x _ {0} $, winding round it like a logarithmic spiral, or conversely. The point $ x _ {0} $ itself is also called a focus. The nature of the approach of the trajectories of the system to $ x _ {0} $ can be described more precisely if one introduces polar coordinates $ r, \phi $ on the $ ( x _ {1} , x _ {2} ) $- plane with pole at $ x _ {0} $. Then for any semi-trajectory $ r = r ( t) $, $ \phi = \phi ( t) $, $ t \geq 0 $( $ t \leq 0 $), that tends to $ x _ {0} $, the polar angle of the variable point $ \phi ( t) \rightarrow + \infty $( a left focus) or $ - \infty $( a right focus) as $ t \rightarrow \infty $.

A focus is either asymptotically Lyapunov stable or completely unstable (asymptotically stable as $ t \rightarrow - \infty $). Its Poincaré index is 1. The figure depicts a right unstable focus at $ x _ {0} = ( 0, 0) $.

Figure: f040700a

For a system (*) of class $ C ^ {1} $( $ f \in C ^ {1} ( G) $) a singular point $ x _ {0} $ is a focus in case the matrix $ A = f ^ { \prime } ( x _ {0} ) $ has complex conjugate eigen values with non-zero real part, but it may also be a focus in case this matrix has purely imaginary or multiple real eigen values (see also Centre; Centre and focus problem).

For references see Singular point of a differential equation.

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