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Economic Faculty

Economic Faculty. STABILITY AND DINAMICAL SYSTEMS. prof. Beatrice Venturi. 1.STABILITY AND DINAMICAL SYSTEMS. We consider a differential equation:. with f a function independent of time t , represents a dynamical system. 1.STABILITY AND DINAMICAL SYSTEMS.

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Economic Faculty

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  1. Economic Faculty STABILITY AND DINAMICAL SYSTEMS prof. Beatrice Venturi 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 . mathematics for economics Beatrice Venturi

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

  4. Market Price mathematics for economics Beatrice Venturi

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

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

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

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

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

  11. A Market Model with Time Expectation In equilibrium we have 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 Mathematics for Economics Beatrice Venturi

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

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

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

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

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

  19. The linear case 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. (*) mathematics for economics Beatrice Venturi

  21. The Characteristic Equation If x(t), y(t) are solution of the linear system thenx(t) and y(t) are solutions of the equations (*). The characteristicequation of (*) is p(λ) = λ2 − (a + d)λ + (ad − bc) = 0 mathematics for economics Beatrice Venturi

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

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

  24. Some ExamplesCase a)λ1=1 e λ2= 3 mathematics for economics Beatrice Venturi

  25. Case b) λ1= -3 e λ2= -1 mathematics for economics Beatrice Venturi

  26. Case c) Complex roots λ1 =2+i and λ2 = 2-i, mathematics for economics Beatrice Venturi

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

  29. STABILITY AND DINAMICAL SYSTEMS mathematics for economics Beatrice Venturi

  30. STABILITY AND DINAMICAL SYSTEMS mathematics for economics Beatrice Venturi

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

  32. STABILITY AND DINAMICAL SYSTEMS mathematics for economics Beatrice Venturi

  33. STABILITY AND DINAMICAL SYSTEMS mathematics for economics Beatrice Venturi

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

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

  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

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

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

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

  41. STABILITY AND DINAMICAL SYSTEMS mathematics for economics Beatrice Venturi

  42. The CharacteristicEquation mathematics for economics Beatrice Venturi

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

  44. STABILITY AND DINAMICAL SYSTEMS EXAMPLE it’s equivalent to : mathematics for economics Beatrice Venturi

  45. STABILITY AND DINAMICAL SYSTEMS mathematics for economics Beatrice Venturi

  46. Eigenvalues • p( λ) = λ2 − (a + d) λ + (ad − bc) = 0 The solutions are the eigenvalues of the matrix A. mathematics for economics Beatrice Venturi

  47. STABILITY AND DINAMICAL SYSTEMS mathematics for economics Beatrice Venturi

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

  49. STABILITY AND DINAMICAL SYSTEMS mathematics for economics Beatrice Venturi

  50. STABILITY AND DINAMICAL SYSTEMS mathematics for economics Beatrice Venturi

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