10Ma2aAnNotespart3 - 60 PART III QUALITATIVE ANALYSIS OF 2D...

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60 PART III: QUALITATIVE ANALYSIS OF 2D AUTONOMOUS SYSTEMS 23. Orbits of autonomous systems 23.1. Autonomous systems. ˙ x = v ( x ) , x ( t ) D R n . The domain D (which can be the whole space R n ) is the phase space of the system. We interpret the given vector-valued function v : D R n as a vector field in the phase space: think of v ( x ) as a vector based at x . Thus autonomous systems vector fields [The concept of a vector field, and therefore the concept of an autonomous system, generalizes to arbitrary manifolds, such as spheres and tori, see Arnold. E.g., consider the velocity field of the wind on the surface of the earth.] A solution is a vector-function x = x ( t ) : ( t 1 , t 2 ) D, which we interpret as a motion in the phase space. Geometrically, x ( t ) is a parametrized curve in D . The set x ( t 1 , t 2 ) = { x ( t ) : t 1 < t < t 2 } , is the orbit of the solution [or the trajectory of the motion]. Orbits are also called phase curves of the system. Note that orbits don’t show dependence on time: we can’t tell from the orbit how fast the particle moves along the trajectory. Theorem. Suppose v C 1 ( D ) . Then every IVP ( t 0 , x 0 ) with x 0 D has a unique maximal solution. 23.2. Properties of (maximal) orbits. We always assume v C 1 ( D ). The following properties are immediate consequences of the existence and uniqueness theorem. (a) [ Translation over time. ] If x ( t ) is a solution, then x ( t + t 0 ) is a solution. These two solutions have the same orbit. In the opposite direction, if two solutions x ( t ) and ˜ x ( t ) have the same orbits, then ˜ x ( t ) = x ( t + t 0 ) for some t 0 . (b) Two orbits either coincide or don’t intersect. Pf: let x ( · ) and ˜ x ( · ) be two solutions such that their orbits intersect. By translation, we can assume ˜ x (0) = x ( t 0 ) := x 0 . Then both ˜ x ( t ) and x ( t + t 0 ) are solutions of the IVP(0 , x 0 ). (c) There is one and only one orbit that passes through a given point.
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61 (d) Three possible types of orbits: an orbit is a single point { x * } [these are orbits of stationary solutions = singular points of the vector field, i.e. v ( x * ) = 0]; an orbit is a simple closed curve [these are orbits of non-stationary periodic solutions = ” cycles ”]; an orbit is a non-closed curve without self-intersections. (e) An orbit can not suddenly stop inside D unless the endpoint is a singular point. [ ”Suddenly stop” means that the solution tends to a finite limit at the endpoint of the time interval.] Property (d) can be restated as follows. Let x ( t ) be a maximal solution. FAE: (i) x ( t 1 ) = x ( t 2 ) for some t 1 = t 2 ; (ii) the orbit of x ( t ) is a closed simple curve; (iii) x ( t ) is a periodic function [defined for all t R ]. The proof of this statement is similar to the proof in (b). 23.3. Phase portraits. The phase portrait of a system is the collection of all orbits. Usually we draw a diagram that shows typical orbits. Examples. (a) 1D autonomous systems, see Section 3. If the phase space is R , then the non- stationary orbits are exactly the complementary intervals of the set of stationary points.
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