Page 692 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 692
GENERAL TOPICS
Dynamical analysis of rural-urban migration using a chaotic map
Samuel Ogunjo, stogunjo@futa.edu.ng
Federal University of Technology Akure, Nigeria
Coauthors: Emmanuel Dansu, Ibiyinka Fuwape
Global and local migration patterns are generating a lot of attention in political circles. This is
a challenge for mathematical scientists to develop and simulate the various impact of migration
patterns using models. In this research, the dynamics of migration between two communities
was investigated using the discrete logistic map. The specific case of rural-urban migration was
considered. Phase space and Lyapunov exponents were employed to investigate the dynamical
complexity of the model. Results obtained elucidate the implication of low and high rates of mi-
gration from rural areas on urban centres. For instance, a rural area with a population growth rate
of 2.5% will cause chaotic population "explosion" in an urban centre with population growth
of 3.8%, if the rural-urban migration rate is between 0.0 and 0.7. The model proposed will be
useful for policy, infrastructural and social development planning by city administrators.
Geometry in space and orthogonal operators (Optimal linear prognosis)
Andrey Valerianovich Pavlov, a_pavlov@mirea.ru
Moscow Institute of Radiotechnics, Electronics and Automatics-RTU, Russian Federation
We will consider two facts ( the facts are interesting from point of algebra, geometries and the
theory of linear estimation).
We can use the two scalar productions: the first production is the source production (x, y) =
(x, y)1; the second production we can determine with help of the equality (X1, X2)2 = C1R1 +
C2R2+C3R3, X1 = C1E1+C2E2+C3δ3, ||δ3|| = 1, X2 = R1E1+R2E2+R3δ3, , E1, E2, δ3 are
the vectors (E1, E2) = 0, ||E1|| = ||E2|| = 1, E1 = x1/||x1||, where E2 = x2/||x2||, ||X|| =
(X, X)1 = (X, X)2, on the x1, x2 sides of the rhombus ||x1|| = ||x2||. We can write
x3 = x3 + ∆ = e1||x3||1 + c3δ, ||e||1 = 1, x3 = x3 + ∆ = e2||x3||2 + c3δ, ||e||2 = 1, if the x3
vector is the projection of the x3 vector on the plane with x1, x2 basis vector (the optimal linear
estimation). The two equalities are possible, if and only if ||e1|| = ||e2||, ||x3||1 = (x1, x1)1 =
||x3||2 = (x1, x1)2, (from the linear independents of the vectors), but the length are the same
for both metrics on the x1, x2 sides of the rhombus. It is the first fact.
To consider the second fact we can use the equality (x1, x2) = (AQ1, AQ2) = (Q1, Q2) =
0; the x1, x2 vectors are the result of the orthogonal A transformation of the two Q1, Q2 diag-
onals of rhombus to the x1, x2 sides of the rhombus: A = A−1 = A , AQ1 = x1, AQ2 =
x2, A−1x1 = Q1, A−1x2 = Q2,
√ 1 1 , A2 = E,
A = (1/ 2) 1 −1
(with help of the (AQ1, AQ2) = Q1kA AQ2k = (Q1, Q2) = 0, AA = E equalities, where
Q1k and Q2k are the vector-line and vector-column of the E1, E2 coordinates in Q1/||Q1||,
Q2/||Q2|| basis). We obtain x1⊥x2, but (x1, x2) = 0 as a primary assumption.
690
Dynamical analysis of rural-urban migration using a chaotic map
Samuel Ogunjo, stogunjo@futa.edu.ng
Federal University of Technology Akure, Nigeria
Coauthors: Emmanuel Dansu, Ibiyinka Fuwape
Global and local migration patterns are generating a lot of attention in political circles. This is
a challenge for mathematical scientists to develop and simulate the various impact of migration
patterns using models. In this research, the dynamics of migration between two communities
was investigated using the discrete logistic map. The specific case of rural-urban migration was
considered. Phase space and Lyapunov exponents were employed to investigate the dynamical
complexity of the model. Results obtained elucidate the implication of low and high rates of mi-
gration from rural areas on urban centres. For instance, a rural area with a population growth rate
of 2.5% will cause chaotic population "explosion" in an urban centre with population growth
of 3.8%, if the rural-urban migration rate is between 0.0 and 0.7. The model proposed will be
useful for policy, infrastructural and social development planning by city administrators.
Geometry in space and orthogonal operators (Optimal linear prognosis)
Andrey Valerianovich Pavlov, a_pavlov@mirea.ru
Moscow Institute of Radiotechnics, Electronics and Automatics-RTU, Russian Federation
We will consider two facts ( the facts are interesting from point of algebra, geometries and the
theory of linear estimation).
We can use the two scalar productions: the first production is the source production (x, y) =
(x, y)1; the second production we can determine with help of the equality (X1, X2)2 = C1R1 +
C2R2+C3R3, X1 = C1E1+C2E2+C3δ3, ||δ3|| = 1, X2 = R1E1+R2E2+R3δ3, , E1, E2, δ3 are
the vectors (E1, E2) = 0, ||E1|| = ||E2|| = 1, E1 = x1/||x1||, where E2 = x2/||x2||, ||X|| =
(X, X)1 = (X, X)2, on the x1, x2 sides of the rhombus ||x1|| = ||x2||. We can write
x3 = x3 + ∆ = e1||x3||1 + c3δ, ||e||1 = 1, x3 = x3 + ∆ = e2||x3||2 + c3δ, ||e||2 = 1, if the x3
vector is the projection of the x3 vector on the plane with x1, x2 basis vector (the optimal linear
estimation). The two equalities are possible, if and only if ||e1|| = ||e2||, ||x3||1 = (x1, x1)1 =
||x3||2 = (x1, x1)2, (from the linear independents of the vectors), but the length are the same
for both metrics on the x1, x2 sides of the rhombus. It is the first fact.
To consider the second fact we can use the equality (x1, x2) = (AQ1, AQ2) = (Q1, Q2) =
0; the x1, x2 vectors are the result of the orthogonal A transformation of the two Q1, Q2 diag-
onals of rhombus to the x1, x2 sides of the rhombus: A = A−1 = A , AQ1 = x1, AQ2 =
x2, A−1x1 = Q1, A−1x2 = Q2,
√ 1 1 , A2 = E,
A = (1/ 2) 1 −1
(with help of the (AQ1, AQ2) = Q1kA AQ2k = (Q1, Q2) = 0, AA = E equalities, where
Q1k and Q2k are the vector-line and vector-column of the E1, E2 coordinates in Q1/||Q1||,
Q2/||Q2|| basis). We obtain x1⊥x2, but (x1, x2) = 0 as a primary assumption.
690
