Definition f网 dx (1) is an autonomous first-order differential equation-one in which the independent variable t does not appear explicitly.The solu- tions of the equation f(x)=0 are called critical points of(1). Ifx=c is a critical point of(1),the constant solution x=c of a differential equation is called an equilibrium solution. 4口0y至无2000
Definition dx dt = f(x) (1) is an autonomous first-order differential equation-one in which the independent variable t does not appear explicitly. The solutions of the equation f(x) = 0 are called critical points of (1). If x = c is a critical point of (1), the constant solution x = c of a differential equation is called an equilibrium solution
Stability of Critical Points Definition A critical point x=c of an autonomous first-order equation is said to be stable provided that,if the initial value xo is suffi- ciently close to c,thenx(t)remains close to c for all t>0.More precisely,the critical point c is stable if,for each e>0,there existsδ>0 such that lxo-cl<8 implies that x(t)-cl<g for all t>0.The critical point x=c is unstable if it is not stable. 4口0y至无2000
Stability of Critical Points Definition A critical point x = c of an autonomous first-order equation is said to be stable provided that, if the initial value x0 is suffi- ciently close to c, then x(t) remains close to c for all t > 0. More precisely, the critical point c is stable if, for each ε > 0, there exists δ > 0 such that |x0 −c| < δ implies that |x(t)−c| < ε for all t > 0. The critical point x = c is unstable if it is not stable
Example Logistic differential equation dx =kx(M-x),k>O,M>0 dr It has two critical pointsx=0 and x=M.The solution Mxo x()= x和+(M-xo)e-M satisfying the initial condition x(0)=xo. 4日10y至，无2000
Example Logistic differential equation dx dt = kx(M −x), k > 0, M > 0 It has 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. Remark The importance of stable critical point
Example Logistic differential equation dx =kx(M-x),k>0,M>0 dt It has two critical pointsx=0 and x=M.The solution Mxo x()= x和+(M-xo)e-M satisfying the initial condition x(0)=xo. Remark The importance of stable critical point. 4口0y至，无2000
Example Logistic differential equation dx dt = kx(M −x), k > 0, M > 0 It has 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. Remark The importance of stable critical point
'c01 r'>0 I=0 I-M tsat业 S业 Phuse diogan (e) () 4日1①y至，无2000
t x x=M 0 Typicalsolution curves (e) x=0 x=M x '<0 x '>0 x '<0 Unstable Stable Phase diagram (f)