Example Consider the explosion/extinction equation 在=kxc-M,k>0,M>0 It has also two critical points x=0 and x=M.The solution Mxo x()= xo+(M-xo)ekMr satisfying the initial condition x(0)=xo. 4口卡0y至，无2000
Example Consider the explosion/extinction equation dx dt = kx(x− M), k > 0, M > 0 It has also two critical points x = 0 and x = M. The solution x(t) = Mx0 x0 + (M −x0)e kMt satisfying the initial condition x(0) = x0
01。 'c0 0 I=0 I=M ut业 Louhe Phuse diogyun (g) (h) 4日1①y至，无2000
t x x=M 0 Typicalsolution curves (g) x=0 x=M x '>0 x '<0 x '>0 Stable Unstable Phase diagram (h)
Stability and Phase Plane A wide variety of natural phenomena are modeled by two- dimensional first-order systems of the form Definition dx =F(x,y), dt =G(x,) d (2) in which the independent variable t does not appear explicitly. Such a system is called an autonomous system.The xy-plane is called the phase plane for(2). 4口0y至无3000
Stability and Phase Plane A wide variety of natural phenomena are modeled by twodimensional first-order systems of the form Definition dx dt = F(x, y), dy dt = G(x, y) (2) in which the independent variable t does not appear explicitly. Such a system is called an autonomous system. The xy− plane is called the phase plane for (2)
According to the existence and uniqueness theorems,given to and any point (xo,yo)of R,there is a unique solutionx= x(t),y=y(t)of(2)that is defined on some open interval (a,b) containing to and satisfies the initial conditions x(to)=xo,y(to)=yo x(1),y(t)describe a parametrized solution curve in the phase plane,called a trajectory of the system(2).A critical point of the system (2)is a point (x,y)such that F(x*,y*)=G(x*,y*)=0 The constant solution (x,y)is called an equilibrium solution of the system(2). 口0，克元，2000
According to the existence and uniqueness theorems, given t0 and any point (x0, y0) of R, there is a unique solution x = x(t), y = y(t) of (2) that is defined on some open interval (a,b) containing t0 and satisfies the initial conditions x(t0) = x0, y(t0) = y0 x(t), y(t) describe a parametrized solution curve in the phase plane, called a trajectory of the system (2). A critical point of the system (2) is a point (x∗, y∗) such that F(x∗, y∗) = G(x∗, y∗) = 0 The constant solution (x∗, y∗) is called an equilibrium solution of the system (2)
Example Find the critical points of the system d =14x-22-y, 空-6-3- 4日10y至，无2000
Example Find the critical points of the system dx dt = 14x−2x 2 −xy, dy dt = 16y−2y 2 −xy