Rural/Urban Migration: The Dynamics of Left and Right Eigenvectors

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1 Overview Rural/Urban Migration: The Dynamics of Left and Right Eigenvectors Left & Right Eigenvectors {v i } Calculation Interpretation and use Left and right eigenvectors Additional discussion as appendix to these slides Population migration example EGR 326 March 31, 2017 Left Eigenvectors (1) Define a scalar function of the state variables a) Migration example: total population (2) Are governed by a first order dynamic equation a) With eigenvalue as governing dynamic parameter (3) Are the expansion coefficients of any state vector expanded w.r.t the right eigenvectors a) x i = z 1 v 1 + z 2 v z n v n Left Eigenvector Definition Right e.v.: Av i = l i v i Define: Left eigenvectors: f it A = l i f i T The notation f it indicates a row vector (the transpose of a column vector) A new variable: z[k] = f T x[k] Note that written this way, z[k] is a scalar function of the state variables 1

2 Analysis of the Dynamic Structure of a System 1. Develop the system model and create the Matlab model 2. Plot and interpret the system behavior and system evolution 3. Calculate the eigenvectors and eigenvalues the eigenvectors and eigenvalues in terms of the state variables and the system behavior Model Physical System Know the state variables, their order, and their meaning in order to make sense out of the eigenanalysis Be familiar with the coefficients of the state variables and their physical significance In order to interpret the significance of the modes with respect to the physical system structure. Migration Example R[k] and u[k]: Population of country divided into rural and urban, r[k] and u[k] a: Annual growth in each sector Typically > 1 b: Migration factor from rural to urban areas Positive and < a g: The optimal rural base The percentage of population in rural areas to support the total population This parameter is a measure of rural productivity Original Migration Model r[k +1] = αr[k] β{r[k] γ(r[k]+ u[k])} u[k +1] = αu[k]+ β{r[k] γ(r[k]+ u[k])} x[k +1] = Ax[k] & α β(1 γ) βγ ) with A = ( + ' β(1 γ) α βγ* & x[k] = r[k] ) ( + ' u[k] * 2

3 Expected Behavior? (handout) What do you anticipate for the system behavior under the following parameters? α = 1, β = 0.5, γ = 0.7 α = 1, β = 0.7, γ = 0.7 α = 1, β = 0.5, γ = 0.95 α = 1.02, β = 0.5, γ = 0.7 α = 1.02, β = 0.5, γ = 0.95 α = 1.02, β = 2, γ = 0.7 3

4 Deduce Eigenvalues & -vectors Rather than writing out and solving the characteristic equation, use some deduction for this very stylized problem A characteristic of the system is that it grows at a rate of Try this as the first eigenvalue This is the growth rate of what element of the population? So the first, associated, left eigenvector is Verify with f T A = λf T Deduce 1 st Right Eigenvector The corresponding right eigenvector defines the r[k] and u[k] values (the population distribution) required for these variables to grow at the rate λ 1 = For both r[k] and u[k] to grow at rate there must be no net migration Zero rural imbalance The ratio of r[k]:u[k] that will be maintained for all time Thus r[k] = γ(r[k] + u[k]), or the ratio of r[k]:u[k] must = γ : (1 γ) Therefore $ e = γ ' & ) % 1 γ( ( Second λ, f, e ) Using the 3 rd property of left eigenvectors, that f it e j = 0 if i j = 1 if i = j We can find f 2 and e 2 And we find the left eigenvectors can be interpreted as: 1. Total population 2. Rural imbalance 4

5 Original Migration Model 5

6 Discussion 1. The total population grows at rate α each year 2. If initially, more than a fraction γ of the population is rural, there will be migration to urban areas, that grows by the factor (α β) 3. If 0 < β < α then the growth of the rural imbalance is less that that of the population, so eventually the imbalance disappears 4. If β > α, then migration oscillates, from rural to urban one year and urban to rural the next. 6

7 Plotting the State Vector Plotting the State Vector Plotting State Vector w/ β = 2 Diagonalized Migration Model 7

8 Plotting State Vector w/ β = 2 Summary Eigenvalues are the characteristic values of the system Population growth factor, for example Right eigenvectors are interpreted as vectors, state vectors, in state space Left eigenvectors are interpreted as scalar valued functions of the state variables Rank of popularity? eigenvalue > right e.v. > left e.v. 8

9 Appendix Slides discussing and defining left eigenvectors. Comparison of left and right eigenvectors Note the mathematical development is in terms of discrete time dynamics. Be comfortable with the similar development in continuous time (2) Left Eigenvectors Given a dynamic system, x[k+1] = Ax[k] Define f it A = l i f i T Combine f it x[k+1] = f it Ax[k] = l i f it x[k] Recalling z[k] = f T x[k] Write z i [k+1] = f it x[k+1] Altogether then f it x[k+1] = f it Ax[k] = l i f it x[k] = l i z i [k] And we have: z i [k+1] = l i z i [k] So the z i [k+1] are dynamic variables (3) Left Eigenvectors Recall that we can expand any state vector w.r.t the {v i } x[k] = z 1 [k]v 1 + z 2 [k]v z n [k]v n Claim: The weighting coefficients of the right eigenvectors, z i [k], are the scalar values defined by the left eigenvectors Thus, Left Eigenvectors (1) Define a scalar function of the state variables (2) Are governed by a first order dynamic equation (3) Are the expansion coefficients of any state vector expanded w.r.t the right eigenvectors 9

10 Interpret: Right Eigenvectors A mode of the system Represent a mode shape the relative activity of the state variables when a specific mode is excited Represent special directions in the state space once pointing in this direction, the state vector remains so Analyzed & used as a vector Interpret: Left Eigenvectors A linear combination of state variables Identify which combination of the state variables will result in system behavior of only the i th mode Analyzed/used as a scalar value Left and Right Eigenvectors x[k] = z 1 [k]v 1 + z 2 [k]v z n [k]v n Any state vector is a linear combination of the right eigenvectors The weighting coefficients of the right eigenvectors, z i [k], are the system variables defined by the left eigenvectors 10