Critical Point Behavior Definition In general,the critical point (x,y)is called a node provided that Either every trajectory approaches (x,y+)as t+or every trajectory recedes from(x*,y*)ast→+o∞，and o Every trajectory is tangent at(x,y)to some straight line through the critical point. A node is said to be proper provided that no two different pairs of "opposite"trajectories are tangent to the same straight line through the critical point,otherwise improper.A node is also called a sink if all trajectories approach the critical point,a source if all trajectories recede (or emanate)from it. 0a0
Critical Point Behavior Definition In general, the critical point (x∗, y∗) is called a node provided that Either every trajectory approaches (x∗, y∗) as t → +∞ or every trajectory recedes from (x∗, y∗) as t → +∞, and Every trajectory is tangent at (x∗, y∗) to some straight line through the critical point. A node is said to be proper provided that no two different pairs of “opposite” trajectories are tangent to the same straight line through the critical point, otherwise improper. A node is also called a sink if all trajectories approach the critical point, a source if all trajectories recede (or emanate) from it
Example 米 () G) 4口0y至无2000
Example x y proper nodal sink (i) x y improper nodal sink (j) Definition (Phase portrait) A phase plane picture of its critical points and typical nondegenerate trajectories
Example (k) ( Definition (Phase portrait) A phase plane picture of its critical points and typical nonde- generate trajectories
Example x y proper nodal sink (k) x y improper nodal sink (l) Definition (Phase portrait) A phase plane picture of its critical points and typical nondegenerate trajectories
Definition (Saddle Point) (m) (n) 口，0y至，元2000
Definition (Saddle Point) x y (m) x y (n)
Stability Definition A critical point (x,y+)is said to be stable provided that if the initial point (xo,yo)is sufficiently close to (x,y),then (x(t),y(t))remains close to (x,y)for all t>0.In vector no- tation,with x(t)=(x(r),y(t)),the distance between the initial point xo=(xo,yo)and the critical point x.=(x,y)is x0-x=V(0-x)2+o-y*)2. Thus,the critical point x+is stable provided that,for each g>0, there exists >0 such that xo-x|<6 implies that x(t)-x|<g for all t>0.Otherwise,it is called unstable. 280
Stability Definition A critical point (x∗, y∗) is said to be stable provided that if the initial point (x0, y0) is sufficiently close to (x∗, y∗), then (x(t), y(t)) remains close to (x∗, y∗) for all t > 0. In vector notation, with x(t) = (x(t), y(t)), the distance between the initial point x0 = (x0, y0) and the critical point x∗ = (x∗, y∗) is |x0 −x∗| = q (x0 −x∗) 2 + (y0 −y∗) 2 . Thus, the critical point x∗ is stable provided that, for each ε > 0, there exists δ > 0 such that |x0 −x∗| < δ implies that |x(t)−x∗| < ε for all t > 0. Otherwise, it is called unstable