SYSTEMS OF DIFFERENTIAL EQUATIONS

1. SYSTEMS OF DIFFERENTIAL EQUATIONS

2. Outline • Preliminary Theory • Homogeneous Linear Systems • Distinct Real Eigenvalues • Repeated Eigenvalues • Complex Eigenvalues • Solution by Diagonlization • Matrix Exponential

3. Introduction In many problems with several dependent variables the formulation leads to a system of simultaneous differential equations. Often these equations are nonlinear and exceedingly difficult, if not possible. But, in certain important cases, they are linear and can be solved. If the system is linear, the equations can be written as where pij is a polynomial operator in D = d/dt. Theorem. If the determinant of the operational coefficients of a system with constant coefficients is not identically zero, the total number of independent constants in any complete solution is same as the degree of the determinant of the operational coefficients.

4. (cont) To solve a system like this, the elementary operations that will reduce it to an equivalent system of the form Example 1. Find a complete solution of the system.

5. (cont) The system can be rewritten in terms of the differential operator, D, as Now, the D for x in the first equation can be eliminated by multiplying the first equation by 2 and subtract the second equation. The x in the first equation now is substituted into the second equation. Thesecond equation now is solved first and then x(t) is then obtained from the first equation.

6. (cont) Thus, the solution can be expressed in vector form as Example 2. Find a complete solution for the system. Example 3. Three tanks are connected as shown in the figure. The first tank contains 100 gal of pure water; the second contains 100 gal of brine containing 1 lb of salt per gallon; the third contains 100 gal of brine containing 2 lb of salt per gallon. Liquid circulates through the tanks at a constant rate of 5 gal/min. If the brine in each tank is kept uniform by stirring, find the amount of salt in each tank as a function of time.

7. (cont) Example 4. The system below can be written as the standard form by solving for x’ and y’. Example 5. Reduce each equation to a first-order system and write it as a matrix equation.

8. Preliminary Theory The standard form of first order equations is When the right-hand sides are not functions of t, the system is called autonomous. This system can be written in compact matrix differential equation. Theorem. Any n particular solutions of a homogeneous first-order system in n unknowns, over an interval I over which the system is normal, are linearly independent if and only if their wronskian has a nonzero value at each point of I..

9. (cont) Trajectory and Phase Plane. The solution vector in the following form can provide the trajectory of the solution in the x-y plane for two or three-dimensional problems.

10. (cont) Fundamental Set of Solutions: Any set of X1, X2, …, Xn of n linearly independent solution vectors of the homogeneous system on an interval I is said to be a fundamental set of solutions on the interval. Theorem. General Solution for Homogeneous Systems: Let X1, X2, …, Xn be a fundamental set of solutions of the homogeneous system on an interval I. Then, the general solution of the system in the interval I is a linear combination Theorem. General Solution for Nonhomogeneous Systems: Let Xp be a given solution of the nonhomogeneous system on an interval I. Then, the general solution of the system in the interval I is a combination of homogeneous solution (i.e., complimentary solution) and the particular solution.

11. Homogeneous Linear Systems Systems with Constant Coefficients Eigenvalues and Eigenvectors • Distinct Real Eigenvalues

12. Ex. 1 Example 1. Distinct Eigenvalues The eigenvalues and eigenvectors are Thus, the solution is

13. (cont)

14. Repeated Eigenvalues • Repeated Eigenvalues • When all linearly independent eigenvectors are known, the solution has the same form as before. • When there is only one eigenvector corresponding to the eigenvalue 1 of multiplicity m, the m linearly independent eigenvectors can be obtained as

15. (cont) The eigenvectors for repeated roots can be determined by

16. Ex. 2 Example 2. Repeated Eigenvalues The eigenvalues are {-1, -1, 5}. For 1 = -1, the system for eigenvector becomes for any q2 and q3.

17. (cont) Thus, two independent eigenvectors for the repeated eigenvalue can be determined and the results are Since there are three independent eigenvectors, the solution is Alternative method cannot be used in this case because two independent eigenvectors already exist.

18. Ex. 3 Example 3. Solve the repeated eigenvalue problem. The eigenvalues are {2, 2, 2} and only one eigenvector exists as Since there is only one eigenvector found, the solution can be written as

19. (cont) The K1, K21, and K31 are same and nothing but K as The K22, and K32, K33 are determined by Then, the final solution becomes

20. Complex Eigenvalues Example 4. Solve the complex eigenvalue problem. The eigenvalues are {5 + 2i, 5 – 2i} and the eigenvectors are Thus, the solution is This can be further written in real terms by Euler’s formula as

21. Ex. 5 Example 5. Solve the complex eigenvalue problem. The eigenvalues are {2i, – 2i} and the eigenvectors are Thus, the solution in real terms is given below with the plot.

22. Solution by Diagonalization The coupled system, X’ = A X, can be decoupled by using eigenvector matrix P as below to produce a diagonal matrix D. This is called similarity transformation that transforms the matrix without changing the eigenvalues. Special case of the similarity transformation is orthogonal transformation when the matrix P satisfies P-1 = PT, and the matrix is called orthogonal matrix and usually new notation Q is used. Thus, the orthogonal transformation is given by QTAQ = D. The similarity transformation results from the substitution as The resulting system then becomes

23. Nonhomogeneous Linear System A fundamental matrix is a set of fundamental solutions of the homogeneous system X’ = AX. The homogeneous solution can be written in vector form The fundamental matrix satisfies the original homogeneous equation as

24. (cont) A particular solution can be found by the method of variation of parameters. Assume the solution as As for the case of single differential equation, the particular solution is found and given below

25. Ex. 6 Example 6. Solve the nonhomogeneous problem. The eigenvalues are {–2, –5} and the eigenvectors are Thus, the fundamental matrix is

26. (cont) Thus, the solution is given below with the plot.

27. Ex. 7 (more examples) Example 7. Find a complete solution of the system The homogeneous solution can be found as From this the fundamental matrix X(t) and its inverse are Thus, the particular integral is

28. Systems of Nonlinear Differential Equations

29. Types of Solutions A solution to the plane autonomous system that satisfies X(0) = X0 (i.e., x(t)=x0 and y(t)=y0) is unique and one of three basic types. (1) A constant solution is called a critical or stationary point. It is also called equilibrium solution. (2) A solution x=x(t), y=y(t) that defines an arc that does not cross itself. (3) A periodic solution x=x(t), y=y(t).

30. Ex. 1 Example 1. Find all critical points of the system. The steady-state solution of the system is the critical points. Thus, the critical points are

31. Ex. 2 (shown again from Ex. 5 in previous section) Example 2. Solve the complex eigenvalue problem. The eigenvalues are {2i, – 2i} and the eigenvectors are Thus, the solution in real terms is given below with the plot.

32. Ex. 3 Example 3. Solve the problem. The eigenvalues are {1+i, 1– i} and the eigenvectors are Thus, the solution in real terms is given below with the plot.

33. Stability of Linear Systems • It has shown that the plane autonomous system gives rise to a vector field V(x,y) = {P(x,y), Q(x,y)}, and the solution may be interpreted as the resulting path of a particle that is initially placed at position X(0) = X0. If X0 is placed near a critical point X1, then the following questions arise: • Will the particle return to the critical point? (b) If the particle does not return to the critical point, will it remain close to the critical point or (c) move away from the critical point? • When the case (a) and (b) are true, the critical point is said locally stable, otherwise in (c), it is said unstable.

34. Linear System The eigenvalues can be expressed in terms of trace of the matrix and determinant as where  and  are trace and determinant of the matrix. As seen from the eigenvalues, the behavior of the solution depends on the signs of the determinant, positive, zero, and negative.

35. Ex. 4 Example 4. Solve the problem. The eigenvalues are Thus, the solution plots are shown for various values of c.

36. Three Cases • Case I. Real Distinct Eigenvalues 2 - 4 > 0 • Both eigenvalues are negative  the critical point is stable node • Both eigenvalues are positive  the critical point is unstable node • Eigenvalues have opposite signs  the critical point is saddle point

37. Ex. 5 Example 5. Solve the problem. The eigenvalues are {4, -1} and the eigenvectors are

38. (cont) • Case II. Repeated Real Eigenvalues 2 - 4 = 0 • Two linearly independent eigenvectors and the eigenvalue is negative  the critical point is degenerate stable node, otherwise, degenerate unstable node. • A single linearly independent eigenvector and the eigenvalue is negative  the critical point is degenerate stable node, otherwise, degenerate unstable node.

39. (cont) • Case III. Complex Eigenvalues 2 - 4 < 0 • the eigenvalues are pure imaginary  the critical point is called center. • the eigenvalue have nonzero real part that is negative  the critical point is spiral stable point, otherwise, spiral unstable point.

40. Ex. 6 Example 6. Solve the repeated and complex eigenvalue problems. (a) The eigenvalues are {-3, -3} and an eigenvector is The solution that satisfy X(0) = {1, 0} approaches (0, 0) from the direction specified by the line y = x/3.

41. (cont) (b) The eigenvalues are {-i, +i} and so (0, 0) is a center. The solution is an ellipse.

42. Stability of Linear System

43. Linearization and Local Stability Let’s consider the following nonlinear system

44. (cont)

45. Linearization Nonlinear equations can be rarely solved analytically. Thus, an approximate solution can be found by linearization for the system The solution of this approximate linearized equation is

46. (cont) For the plane autonomous system (i.e., 2D), the Jacobian matrix can be used for linearization as before that results in the following theorem

47. Ex. 7 Example 7. Solve the eigenvalue problems. The critical points are and the Jacobian matrix is

48. (cont) Finding eigenvalues of the Jacobian matrix at the critical points shows that the critical point is unstable point and the second is stable one. The following phase portrait of three curves with various initial conditions show how the trajectories move away from the unstable node and toward the stable node.

49. Ex. 8. Lotka-Voltera Model Example 7. Find all critical points of the system and plot the phase portrait. The critical points can be found from the steady-state solutions. They are (0, 0), (20,40), (50,0), and (0,100). The following shows the phase portrait.

50. Periodic Solutions, Limit Cycles, and Global Stability Investigate the existence of periodic solutions of nonlinear plane autonomous systems and introduce special periodic solutions called limit cycles. First goal is to determine conditions under which we can either exclude the possibility of periodic solutions or assert their existence. The second goal is to determine conditions under which an asymptotically stable critical point is globally stable. Theorem 11.4. Cycles and Critical Points: If a plane autonomous system has a periodic solution X=X(t) in a simply connected region R, then the system has at least one critical point inside the corresponding simple closed curve C. If there is a single critical point inside C, then that critical point cannot be a saddle point. Corollary If a simply connected region R either contains no critical points of a plane autonomous system or contains a single saddle point, there are no periodic solutions in R.