DE 4

DE 4

1. Previously, the phase plane has been a one dimensional system. By introducing two dependent variables.

   • In general, A solution of the IVP d/dt(x(t))=f(x,t) : x(t0)=x0 x0∈R^n, on an open interval I that contains t0 is a cts function x:I→R^n with x(t0)=x0 and x'(t)=f(x,t) for all t∈I
     • The jacobian matrix of a function f:R^n→R^n is the matrix of partial derivatives as follows:
       • • Thm: (Existence and Uniqueness) If f(x,t) and Df(x,t) are cts functions ("nice") for x∈U⊆R^n, a<t<b, then for any x0⊆U and t0∈(a,b), there exists a unique solution to the IVP on some open interval containing t0.
           • Important special case: with second order ODEs we assume existence and uniqueness. Consider x''=f(x',x,t) where x∈R. By introducing a new variable y=x' we can rewrite this as: x'=y=f1(x,t), y'=x''=f2(x,t). In this case, we have that the Jacobian matrix is:
             • • so to guarantee unique solutions for second order ODEs we require that the partial derivatives of f wrt x & y are "nice", and we also require an initial condition (x0,y0) i.e. x(t0)=x0, x'(t)-y0

2. Coupled 2x2 Linear Systems with Constant Coefficients

   • A general linear coupled 2x2 system may look like
     • • One approach to solve this is try to combine these systems into one 2nd order ODE, in a similar method to simultaneous equations

3. Homogeneous linear 2x2 Systems with Constant Coeffs.

   • Consider systems of the form
     • • • Then we try a solution of the form x=e^(λt)*v
           • So finding a solution x=e^(λt)*v is equivalent to finding the eigenvalues (λ) and eigenvectors (v) of A, if we have two such solutions, then by linearity, x(t)=e^(λ1t)*v1+e^(λ2t)*v2
             • If the eigenvalues are distinct, be they real or complex, then so are the eigenvector, and so the two solutions will be linearly independent
               • With complex eigenvalues p±iq or λ,λ' (Where λ' is the conjugate of λ). It can be shown that if λ and v are eigenvalue and eigenvector of A, respectively, then so are λ' and v'. Splitting v into real and imaginary parts, we can write v=v1+i*v2, giving that v'=v1-iv2. The solution now looks like: x(t)=e^(λ1t)*(c*v1)+e^(λ2t)*(c*v2)'. Where the choice of c, c' is such that x(t) is real.
                 • Noting that for z∈C, z+z'=2R[z] then we have that x(t)=2R[(c*e^(λt))*v]
                   • Letting λ=p+iq and c=α+iβ we have the solution

4. Phase Portraits and Change of Variables

   • • Note that (0,0) is a "fixed point" i.e. x'=0. Therefore: Any solution with initial value a=0 b=/=0 i.e. x(0)=(k,-4k) will give x(t)→(0,0) as t→∞. In the language of the 1-D case, (0,0) is stable. Any solution with initial value a=/=0, b=0 ie.. x(0)=(k,k) will give x(t)→∞ as t→∞. For such point, (0,0) is "unstable". By linearity, if a=/=0, b=/=0, then the solution will be a combination of these two cases. All this gives us the phase diagram below
       • • Diagonalisation and Uncoupling
           • Consider the system x'=Ax, where A has distinct eigenvalues. Recall for a matrix A with distinct eigenvalues λ1, λ2 and corresponding eigenvectors v1, v2, we can call the matrix P=(v1|v2) and then:
             • • Then changing coordinates we have
                 • • Then phase portraits to the right show (left to right) the cases where λ1<λ2<0. Here (0,0) is a stable fixed point also known as a "sink". The phase diagram when λ1>λ2>0. Here (0,0) is an unstable fixed point, also known as a source.
                     • • When we have distinct, real eigenvalues, we have to take into account whether the eigenvalues are positive or negative. This gives rise to three cases: λ1<0<λ2. Here (0,0) is called a "saddle point". If λ1<λ2<0 all solutions tend to (0,0)-but they will move faster in the v1 direction, and hence we will have a sink. Alternatively, if λ1>λ2<0, we will get a sink at (0,0).
                         • Recall the solution to a general system x'=Ax where A has complex eigenvalues . Then note the e^pt term, if R[λ]<0, then (0,0) is stable, but if R[λ]>0 then (0,0) is unstable. The imaginary part, makes the solution spiral about the origin, where the direction depends on the sign of q. To study this further we change coordinates and convert to polar:
                           • • With solution r(t)=ae^pt θ(t)=-qt(+c)
                               • The phase diagrams when the eigenvalues are complex; the left showsR[λ]>0, while on the right R[λ]<0
                                 • • Repeated real eigenvalue
                                     • Then we have solution, x(t)=Be^(λt)*v+Ce^(λt)*(a+tv)
                                       • Where v is the eigenvector, and f1 and f2 are the solutions of the linear system described by the matrix A in x'=Ax,
                                         • • We can decouple a 2nd order ODE by substituting x'=y and gaining two equations. Then solving for the determinant of the corresponding matrix, we gain the auxillery equation

5. Functions of Two variables

   • If we no longer restrict our functions by linearity. Then we consider the surface f(x,y) in R^3 and define the directional derivative and multivariable chain rule as defined in GM
     • Again level curves are defined in R^2 as all points such that f(x,y)=k
       • Parametric curves are also defined in the same way and we make use of the thms
         • At a point (x0,y0)
           • Closed to fixed points of nonlinear systems, solutions behave like x'=Df(x*)x and hence the behaviour of solutions to nonlinear systems can be approximated by a linear system near fixed points. Hence we can now compute the stability of fixed points.
             • Notes, At every point on a solution x(t) (x',y') points in the direction of the solution. The collection of such arrows is called a vector field.For any (x,y)] we can draw (x',y'). Hence we can draw integral curves without an explicit solution.
         • The maximum value of Dvf(x,y) occurs in the direction of ∇f, with maximum value |∇f(x,y)|, as this is the direction of steepest ascent

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