1. Beatrice Venturi1 Economic Faculty STABILITY AND DINAMICAL SYSTEMS prof. Beatrice Venturi

2. mathematics for economics Beatrice Venturi 2 1.STABILITY AND DINAMICAL SYSTEMS §We consider a differential equation: with f a function independent of time t, represents a dynamical system.

3. mathematics for economics Beatrice Venturi 3 a = is an equilibrium point of our system x(t) = a is a constant value. such that f(a)=0 The equilibrium points of our system are the solutions of the equation f(x) = 0 1.STABILITY AND DINAMICAL SYSTEMS

4. mathematics for economics Beatrice Venturi 4 Market Price

5. mathematics for economics Beatrice Venturi 5 Dynamics Market Price §The equilibrium Point

6. mathematics for economics Beatrice Venturi 6 Dynamics Market Price The general solution with k>0 (k<0) converges to (diverges from) equilibrium asintotically stable (unstable)

7. mathematics for economics Beatrice Venturi 7 The Time Path of the Market Price

8. mathematics for economics Beatrice Venturi 8 1.STABILITY AND DINAMICAL SYSTEMS Given

9. mathematics for economics Beatrice Venturi 9 1.STABILITY AND DINAMICAL SYSTEMS Let B be an open set and a Є B, §a = is a stable equilibrium point if for any x(t) starting in B result:

10. Mathematics for Economics Beatrice Venturi 10 A Market Model with Time Expectation : Let the demand and supply functions be:

11. A Market Model with Time Expectation mathematics for economics Beatrice Venturi 11 In equilibrium we have

12. Mathematics for Economics Beatrice Venturi 12 A Market Model with Time Expectation We adopt the trial solution: In the first we find the solution of the homogenous equation

13. Mathematics for Economics Beatrice Venturi 13 A Market Model with Time Expectation We get: The characteristic equation

14. Mathematics for Economics Beatrice Venturi 14 A Market Model with Time Expectation We have two different roots the general solution of its reduced homogeneous equation is

15. A Market Model with Time Expectation mathematics for economics Beatrice Venturi 15 The intertemporal equilibrium is given by the particular integral

16. A Market Model with Time Expectation §With the following initial conditions mathematics for economics Beatrice Venturi 16 The solution became

17. mathematics for economics Beatrice Venturi 17 The equilibrium points of the system STABILITY AND DINAMICAL SYSTEMS

18. mathematics for economics Beatrice Venturi 18 STABILITY AND DINAMICAL SYSTEMS §Are the solutions :

19. mathematics for economics Beatrice Venturi 19 The linear case

20. mathematics for economics Beatrice Venturi 20 We remember that x'' = ax' + bcx + bdy §by = x' ax §x'' = (a + d)x' + (bc ad)x x(t) is the solution (we assume z=x) z'' (a + d)z' + (ad bc)z = 0. (*)

21. mathematics for economics Beatrice Venturi 21 The Characteristic Equation If x(t), y(t) are solution of the linear system then x(t) and y(t) are solutions of the equations (*). The characteristic equation of (*) is p(λ) = λ 2 (a + d)λ + (ad bc) = 0

22. mathematics for economics Beatrice Venturi 22 Knot and Focus The stable case

23. mathematics for economics Beatrice Venturi 23 Knot and Focus The unstable case

24. mathematics for economics Beatrice Venturi 24 Some Examples Case a) λ 1 = 1 e λ 2 = 3

25. mathematics for economics Beatrice Venturi 25 Case b) λ 1 = -3 e λ 2 = -1

26. mathematics for economics Beatrice Venturi 26 Case c) Complex roots λ 1 = 2+i and λ 2 = 2-i,

27. mathematics for economics Beatrice Venturi 27 System of LINEAR Ordinary Differential Equations §Where A is the matrix associeted to the coefficients of the system:

28. mathematics for economics Beatrice Venturi 28 STABILITY AND DINAMICAL SYSTEMS §Definition of Matrix §A matrix is a collection of numbers arranged into a fixed number of rows and columns. Usually the numbers are real numbers. Here is an example of a matrix with two rows and two columns:

29. mathematics for economics Beatrice Venturi 29 STABILITY AND DINAMICAL SYSTEMS

30. mathematics for economics Beatrice Venturi 30

31. mathematics for economics Beatrice Venturi 31 STABILITY AND DINAMICAL SYSTEMS §Examples

32. mathematics for economics Beatrice Venturi 32 STABILITY AND DINAMICAL SYSTEMS

33. mathematics for economics Beatrice Venturi 33

34. Eigenvectors and Eigenvalues of a Matrix The eigenvectors of a square matrix are the non-zero vectors that after being multiplied by the matrix, remain parellel to the original vector.

35. mathematics for economist Beatrice Venturi 35 Eigenvectors and Eigenvalues of a Matrix §Matrix A acts by stretching the vector x, not changing its direction, so x is an eigenvector of A. The vector x is an eigenvector of the matrix A with eigenvalue λ (lambda) if the following equation holds:

36. Eigenvectors and Eigenvalues of a Matrix §This equation is called the eigenvalues equation. mathematics for economist Beatrice Venturi 36

37. Eigenvectors and Eigenvalues of a Matrix §The eigenvalues of A are precisely the solutions λ to the equation: §Here det is the determinant of matrix formed by A - λI ( where I is the 2×2 identity matrix). §This equation is called the characteristic equation (or, less often, the secular equation) of A. For example, if A is the following matrix (a so-called diagonal matrix): mathematics for economist Beatrice Venturi 37

38. mathematics for economist Beatrice Venturi 38 Eigenvectors and Eigenvalues of a Matrix §Example

39. mathematics for economics Beatrice Venturi 39 §We consider STABILITY AND DINAMICAL SYSTEMS

40. mathematics for economics Beatrice Venturi 40 §We get the system: STABILITY AND DINAMICAL SYSTEMS

41. mathematics for economics Beatrice Venturi 41 STABILITY AND DINAMICAL SYSTEMS

42. mathematics for economics Beatrice Venturi 42 The Characteristic Equation

43. mathematics for economics Beatrice Venturi 43 STABILITY AND DINAMICAL SYSTEMS The Characteristic Equation of the matrix A is the same of the equation (1)

44. mathematics for economics Beatrice Venturi 44 STABILITY AND DINAMICAL SYSTEMS its equivalent to : EXAMPLE

45. mathematics for economics Beatrice Venturi 45 STABILITY AND DINAMICAL SYSTEMS

46. mathematics for economics Beatrice Venturi 46 Eigenvalues §p( λ) = λ 2 (a + d) λ + (ad bc) = 0 The solutions are the eigenvalues of the matrix A.

47. mathematics for economics Beatrice Venturi 47 STABILITY AND DINAMICAL SYSTEMS

48. mathematics for economics Beatrice Venturi 48 STABILITY AND DINAMICAL SYSTEMS Solving this system we find the equilibrium point of the non-linear system (3): :

49. mathematics for economics Beatrice Venturi 49 STABILITY AND DINAMICAL SYSTEMS

50. mathematics for economics Beatrice Venturi 50 STABILITY AND DINAMICAL SYSTEMS

51. mathematics for economics Beatrice Venturi 51 Jacobian Matrix

52. mathematics for economics Beatrice Venturi 52 Jacobian Matrix

53. mathematics for economics Beatrice Venturi 53 Jacobian Matrix

54. mathematics for economics Beatrice Venturi 54 Stability and Dynamical Systems.

55. mathematics for economics Beatrice Venturi 55 Stability and Dynamical Systems §Given the non linear system:

56. mathematics for economics Beatrice Venturi 56 Stability and Dynamical Systems

57. mathematics for economics Beatrice Venturi 57 Stability and Dynamical Systems

58. mathematics for economics Beatrice Venturi 58 Stability and Dynamical Systems

59. mathematics for economics Beatrice Venturi 59 Stability and Dynamical Systems

60. mathematics for economics Beatrice Venturi 60 Stability and Dynamical Systems

61. 61 LOTKA-VOLTERRA Prey – Predator Model

62. The Lotka-Volterra Equations,

63. 63 We shall consider an ecologic system PREyPREDATOR

64. mathematics for economics Beatrice Venturi 64 The Model

65. Steady State Solutions a x 1 -bx 1 x 2 =0 c x 1 x 2 – d x 2 =0 a prey growth rate; d mortality rate

66. The Jacobian Matrix J= a 11 a 12 a 21 a 22

67. mathematics for economics Beatrice Venturi 67 Eigenvalues §p( λ) = λ 2 (a + d) λ + (ad bc) = 0 The solutions are the eigenvalues of the matrix A.

68. 68 TrJ = a 11 + a 22 a 11 a 12 a 21 a 22 J = THE TRACE

69. 69 THE DETERMINANT Det J = a 11 a 22 – a 12 a 21

70. 70 The equilibrium solutions x = 0 y = 0 Unstable x = d/g y = a/b Stable center

71. 71

72. 72 Cycles

73. 73