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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 STABILITY AND DINAMICAL SYSTEMS prof. Beatrice Venturi Beatrice Venturi
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
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
Market Price mathematics for economics Beatrice Venturi
Dynamics Market Price • The equilibrium Point mathematics for economics Beatrice Venturi
Dynamics Market Price The general solution withk>0 (k<0) converges to (diverges from) equilibrium asintotically stable (unstable) mathematics for economics Beatrice Venturi
The TimePath of the Market Price mathematics for economics Beatrice Venturi
1.STABILITY AND DINAMICAL SYSTEMS Given mathematics for economics Beatrice Venturi
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
A Market Model with Time Expectation : Let the demand and supply functions be: Mathematics for Economics Beatrice Venturi
A Market Model with Time Expectation In equilibrium we have mathematics for economics Beatrice Venturi
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
A Market Model with Time Expectation We get: The characteristic equation Mathematics for Economics Beatrice Venturi
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
A Market Model with Time Expectation The intertemporal equilibrium is given by the particular integral mathematics for economics Beatrice Venturi
A Market Model with Time Expectation • With the following initial conditions The solution became mathematics for economics Beatrice Venturi
STABILITY AND DINAMICAL SYSTEMS The equilibrium points of the system mathematics for economics Beatrice Venturi
STABILITY AND DINAMICAL SYSTEMS • Are the solutions : mathematics for economics Beatrice Venturi
The linear case mathematics for economics Beatrice Venturi
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
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
Knot and Focus The stable case mathematics for economics Beatrice Venturi
Knot and Focus The unstable case’ mathematics for economics Beatrice Venturi
Some ExamplesCase a)λ1=1 e λ2= 3 mathematics for economics Beatrice Venturi
Case b) λ1= -3 e λ2= -1 mathematics for economics Beatrice Venturi
Case c) Complex roots λ1 =2+i and λ2 = 2-i, mathematics for economics Beatrice Venturi
System of LINEAR Ordinary Differential Equations • Where A is the matrix associeted to the coefficients of the system: mathematics for economics Beatrice Venturi
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
STABILITY AND DINAMICAL SYSTEMS mathematics for economics Beatrice Venturi
STABILITY AND DINAMICAL SYSTEMS mathematics for economics Beatrice Venturi
STABILITY AND DINAMICAL SYSTEMS • Examples mathematics for economics Beatrice Venturi
STABILITY AND DINAMICAL SYSTEMS mathematics for economics Beatrice Venturi
STABILITY AND DINAMICAL SYSTEMS mathematics for economics Beatrice Venturi
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.
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
Eigenvectors and Eigenvalues of a Matrix • This equation is called the eigenvalues equation. mathematics for economist Beatrice Venturi
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
Eigenvectors and Eigenvalues of a Matrix • Example mathematics for economist Beatrice Venturi
STABILITY AND DINAMICAL SYSTEMS • We consider mathematics for economics Beatrice Venturi
STABILITY AND DINAMICAL SYSTEMS • We get the system: mathematics for economics Beatrice Venturi
STABILITY AND DINAMICAL SYSTEMS mathematics for economics Beatrice Venturi
The CharacteristicEquation mathematics for economics Beatrice Venturi
STABILITY AND DINAMICAL SYSTEMS The CharacteristicEquation of the matrix A is the same of the equation (1) mathematics for economics Beatrice Venturi
STABILITY AND DINAMICAL SYSTEMS EXAMPLE it’s equivalent to : mathematics for economics Beatrice Venturi
STABILITY AND DINAMICAL SYSTEMS mathematics for economics Beatrice Venturi
Eigenvalues • p( λ) = λ2 − (a + d) λ + (ad − bc) = 0 The solutions are the eigenvalues of the matrix A. mathematics for economics Beatrice Venturi
STABILITY AND DINAMICAL SYSTEMS mathematics for economics Beatrice Venturi
STABILITY AND DINAMICAL SYSTEMS Solving this system we find the equilibrium point of the non-linear system (3): : mathematics for economics Beatrice Venturi
STABILITY AND DINAMICAL SYSTEMS mathematics for economics Beatrice Venturi
STABILITY AND DINAMICAL SYSTEMS mathematics for economics Beatrice Venturi