Study of A Nonlinear Finance System

Study of a nonlinear finance system
Manuel Luci
Università di Pisa
1 Introduction
2 Mathematical Model
3 Model analysis
4 Results of Calculation
5 Results
6 Bibliography
Manuel Luci (Università di Pisa) Study of a nonlinear finance system 2 / 28

Introduction
Introduction
Chaotic phenomenons occur in the economic system, macro economic

operation has in itself the inherent indefiniteness (definiteness is caused by
the system internals).

Introduction
Introduction

Government can adopt such macro control measures (financial policies,

monetary policies), effectiveness really limited.

Introduction
Introduction


In the field of finance complexity is given by the interaction between

nonlinear factors and due to the evolution process from low dimensions to
high one.

Introduction
Introduction


In the field of finance complexity is given by the interaction between

nonlinear factors and due to the evolution process from low dimensions to
high one.
It is important studying internal structural characteristics to reveal the

reasons why such complicated phenomena occur and to provide theory
basis and practical ways for the analysis, prediction and control of the
complicated continuous economic esystem.

Mathematical Model
Mathematical Model
Financial model composed by:

parts of product;
money;
bound;
labour force
model is extremely sensitivity to the initial value and changing of
parameters.

Mathematical Model
Mathematical Model
Financial model composed by:

parts of product;
money;
bound;
labour force
model is extremely sensitivity to the initial value and changing of
parameters.
We consider:
x : interest rate;
y : investment demand;
z : price exponent.
So that we are focus in the sensitivity of parameter, we consider changing
rates about time : ẋ = dx /dt, ẏ = dy /dt, ż = dz/dt.
Mathematical Model
x: Interest rate
ẋ is:
the contradiction from the investment market, i.e. the surplus
between investment and savings;
it is the structure adjustment from good prices.

Mathematical Model
x: Interest rate
ẋ is:
the contradiction from the investment market, i.e. the surplus
between investment and savings;
it is the structure adjustment from good prices.
ẋ = f1 (y − SV )x + f2 z (1)
SV is the amount of saving, f1 , f2 constants.

Mathematical Model
y: Investment demand
ẏ is:
proportional with the rate of investment;
in proportion by inversion with the cost of investment and interest
rate.

Mathematical Model
y: Investment demand
ẏ is:
proportional with the rate of investment;
in proportion by inversion with the cost of investment and interest
rate.
ẏ = f3 (BEN − αy − βx 2 ) (2)
BEN is benefit rate of investment, f3 ,α and β are all constants
(considering benefit rate of investment costant in a certain period of time)

Mathematical Model
z: Price exponent
ż is:
controlled by the contradiction between supply and demand of the
market
influenced by inflation rate
We suppose the amount of the supplies and domands is constant in a
certain period of time and proportion by inversion with the price.

Mathematical Model
z: Price exponent
ż is:
market
Change of the inflation rate can be represented by the changes of the real
interest rate and the inflation rate equals the norminal interest rate
subtracts the real interest rate.

Mathematical Model
z: Price exponent
ż is:
market
Change of the inflation rate can be represented by the changes of the real
interest rate and the inflation rate equals the norminal interest rate
subtracts the real interest rate.
ż = −f4 z − f5 x (3)

Mathematical Model
Dynamic System
Changing coordinate system and simplifying:


ẋ = z + (y − a)x


ẏ = 1 − by − x 2 (4)

ż = −x − cz

with:
a ≥ 0 saving amount;
b ≥ 0 is the per-investiment cost;
c ≥ 0 is the elasticity of demands.

Model analysis
Model analysis
First of all we have to find the stationary point:


z + (y − a)x = 0
 
−ax + xy + z 

 2
F (x , y , z) = −x − by + 1 = 0 ⇒ 1 − by − x 2 = 0 (5)

−x − cz

−x − cz = 0


Model analysis
Model analysis
First of all we have to find the stationary point:


z + (y − a)x = 0
 
−ax + xy + z 

 2
F (x , y , z) = −x − by + 1 = 0 ⇒ 1 − by − x 2 = 0 (5)

−x − cz

−x − cz = 0

Se:
se (c − b − abc ≤ 0), we have a single stationary point
Vs = (0, 1/b, 0)
se c − b − abc ≥ 0, we have 3 stationary point:
Vs1 = (0,
p1/b, 0) p
Vs2 = ( p (c − b − abc)/c, (1 + ac)/c, −1/c p(c − b − abc)/c)
Vs3 = (− (c − b − abc)/c, (1 + ac)/c, 1/c (c − b − abc)/c)

Model analysis Study of the local topological structure
Study of the local topological structure

Let’s study Vs = (0, 1/b, 0) for (c − b − abc ≤ 0), translating the
coordinate:
X = x, Y = y − 1/b Z =z =⇒ Vs∗ (0, 0, 0)


coordinate:
X = x, Y = y − 1/b Z =z =⇒ Vs∗ (0, 0, 0)

Ẋ = (1/b − a)X + Z + XY


Ẏ = −bY − X 2 (6)

Ż = −X − cZ



coordinate:
X = x, Y = y − 1/b Z =z =⇒ Vs∗ (0, 0, 0)

Ẋ = (1/b − a)X + Z + XY


Ẏ = −bY − X 2 (6)

Ż = −X − cZ

Evaluate the Jacobian of the system in Vs ∗:

 
1
−a + b 0 1
∗
J(Vs ) =  0 −b 0  (7)
 
−1 0 −c

Characteristic polynomial:
P(λ) = (λ + b)(λ2 + λ(c + a − 1/b) + ac − c/b + 1) (8)
A first eigenvalue is λ = −b, the other 2 comes from:
λ2 + λ (c + a − 1/b) + ac − c/b + 1 = 0 (9)

| {z } | {z }
T D
Solutions can be divided into 3 cases

Case 1) c − b − abc < 0, T > 0
λ2 + λ (c + a − 1/b) + ac − c/b + 1 = 0
| {z } | {z }
T D
If c − b − abc < 0 ⇒ D > 0 and T > 0, we find λ1 , λ2 , λ3 < 0 so

Vs (0, 1/b, 0) is a sink, the point is stable.

Case 2) c − b − abc < 0, T < 0
λ2 + λ (c + a − 1/b) + ac − c/b + 1 = 0
| {z } | {z }
T D
If c − b − abc < 0 ⇒ D > 0 and T < 0 we find λ1 < 0, λ2 , λ3 > 0 so

Vs (0, 1/b, 0) is a saddle

Case 3.1) c − b − abc = 0, (1 − c 2 )/c > 0
λ2 + λ (c + a − 1/b) + ac − c/b + 1 = 0
| {z } | {z }
T D
If c − b − abc = 0 ⇒ λ2 = 0, λ3 = −(c + a − 1/b) and λ1 = −b < 0, we

need to study 2 different case:

Case 3.1) c − b − abc = 0, (1 − c 2 )/c > 0
λ2 + λ (c + a − 1/b) + ac − c/b + 1 = 0
| {z } | {z }
T D
If c − b − abc = 0 ⇒ λ2 = 0, λ3 = −(c + a − 1/b) and λ1 = −b < 0, we

need to study 2 different case:
If (1 − c 2 )/c > 0 → 0 < c < 1 than λ1 < 0, λ2 = 0, λ3 > 0 and for the
theorem of existence of stable, central and unstable manifold, we can
conclude Vs (0, 1/b, 0) is unstable.

Case 3.2) c − b − abc = 0, (1 − c 2 )/c < 0
λ2 + λ (c + a − 1/b) + ac − c/b + 1 = 0
| {z } | {z }
T D
If (1 − c 2 )/c < 0 → c > 1 than we can use central manifold theorem:
 
0
λ1 = −b < 0 → v¯1 = 1
 
0
 
1
λ2 = 0 → v¯2 =  0 
 
−1/c
 
−1/c
2
λ3 = (1 − c )/c < 0 → v¯3 =  0 
 
1
Case 3.2) c − b − abc = 0, (1 − c 2 )/c < 0
v¯1 , v¯3 are extended into the stable subspace E s and v¯2 is extended into the
central subspace E c
   
1 −1/c 0 c 2 /(c 2 − 1) 0 c/(c 2 − 1)
T = 0 0 1 T −1 =  c/(1 − c 2 ) 0 c 2 /(1 − c 2 )
   
−1/c 1 0 0 1 0
   
X u
Y  = T  v 
   
Z w
Put this in (6) we find

Case 3.2) c − b − abc = 0, (1 − c 2 )/c < 0
      
u̇ 0 00 u c 2 w (u − v /c)/(c 2 − 1)
20   v  +  cw (u − v /c)/(1 − c 2 ) 
(c − 1)/c
 v̇  = 0
      
ẇ 0 −b0 w −(u − v /c)2
(10)
c s u
Therefore E = u axis, E = span {(v , w )},E = ∅

Case 3.2) c − b − abc = 0, (1 − c 2 )/c < 0
We find:
c2
u̇ = (u − v /c)w (11)
c2 − 1
! ! ! !
c
v̇ (c 2 − 1)/c 0 v̇ (1−c 2 )
w (u− v /c)
= + (12)
ẇ 0 −b ẇ −(u − v /c)2
Central manifold W c is the curve tangent the central subspace E c at Vs∗
and passing at it.

Case 3.2) c − b − abc = 0, (1 − c 2 )/c < 0
We find:
c2
u̇ = (u − v /c)w (11)
c2 − 1
! ! ! !
c
v̇ (c 2 − 1)/c 0 v̇ (1−c 2 )
w (u− v /c)
= + (12)
ẇ 0 −b ẇ −(u − v /c)2
Central manifold W c is the curve tangent the central subspace E c at Vs∗
and passing at it.
To find the equation of W c we can use the method of form power series:
! !
v a1 u 2 + b1 u 3 + c1 u 4
= h̄(U) =
w a2 u 2 + b2 u 3 + c2 u 4
Coefficient of u and costant term is 0 due to h̄ = 0, D h̄(0) = 0

Case 3.2) c − b − abc = 0, (1 − c 2 )/c < 0
So that ḣi = ∂hi (u)/∂u = 2ai u + 3bi u 2 + 4ci u 3 + · · ·
˙ c 2 −1
! !
h1 (u) c2 c h1 (u)
c
+ 1−c 2 (u − h1 (u)/c)h2 (u)
˙ (u − v /c)h2 (u) =
h2 (u) c2 − 1 −bh2 (u) − (u − h1 (u)/c)2
(13)

Case 3.2) c − b − abc = 0, (1 − c 2 )/c < 0
So that ḣi = ∂hi (u)/∂u = 2ai u + 3bi u 2 + 4ci u 3 + · · ·
˙ c 2 −1
! !
h1 (u) c2 c h1 (u)
c
+ 1−c 2 (u − h1 (u)/c)h2 (u)
˙ (u − v /c)h2 (u) =
h2 (u) c2 − 1 −bh2 (u) − (u − h1 (u)/c)2
(13)
Comparing the coefficients we obtain:
 2
a1 = 0, b1 = − b(c 2c−1)2 , c1 = 0
(14)
 a2 = − b1 , b2 = 0, c2 = − b 3 (c2c 2
2 −1)2 (b + c(c − 1))

Case 3.2) c − b − abc = 0, (1 − c 2 )/c < 0
Putting this in (11) and consider only the first term:
c2
u̇ = − u 3 + o(u 4 ) (15)
b(c 2 − 1)
We can conclude that the stationary point Vs∗ and also Vs is gradually
inclined to be stable.

Case 3.2) c − b − abc = 0, (1 − c 2 )/c < 0
Putting this in (11) and consider only the first term:
c2
u̇ = − u 3 + o(u 4 ) (15)
b(c 2 − 1)
We can conclude that the stationary point Vs∗ and also Vs is gradually
inclined to be stable.
Synthesizing this 2 cases, we can conclude that if c = 1 a bifurcation
occurs at Vs

Case 4) c − b − abc < 0, c + a − 1/b = 0
We get c 2 < 1, λ2 , λ3 are purely imaginary roots, with λ1 = −b < 0.

Case 4) c − b − abc < 0, c + a − 1/b = 0
We get c 2 < 1, λ2 , λ3 are purely imaginary roots, with λ1 = −b < 0.

Considering α = −(c + a − 1/b), than ∂α /∂a|a=a0 = −1 6= 0, the
conditions of Hopf-bifurcation are satisfied, so that at the point Vs
Hopf-bifurcation occurs and exists periodic solution group

Results of Calculation Case 1
Case 1
a=4.5, b=0.2, c=0.6
y z
5.000 0.0010
4.998
0.0005
4.996
x
-0.0010 -0.0005 0.0005 0.0010
4.994
-0.0005
4.992
x -0.0010
-0.0010 0.0010
5.000
y
z 4.995
0.0010
0.0005
y
-0.0005 4.992 4.994 4.996 4.998 5.000
-0.0010
4.990
0.0010
z 0.0005
0.0000
-0.0005
-0.0010
-0.0010
-0.0005
0.0000
0.0005
0.0010
x

Case 2
a=4.5, b=0.2, c=0.4
y 0.5
5.0
4.8
x
-0.8 -0.6 -0.4 -0.2 0.2 0.4
4.6
4.4
-0.5
4.2
x
-0.8 -0.6 -0.4 -0.2 0.2 0.4
-1.0
z y 5.0 x
4.8 -0.5
4.6 0.0
4.4
0.5 4.2
0.5
y
4.2 4.4 4.6 4.8 5.0 0.0
z
-0.5
-0.5
-1.0
-1.0

Results of Calculation Case 3.1
Case 3.1
a=3, b=0.2, c=0.5,x[0]=2,y[0]=1,z[0]=2 a=4.5, b=0.2, c=0.5,x[0]=0.5,y[0]=3.6,z[0]=0.23
y z z
2.0
4
1.5 0.5
y
1.0 3.7
3
3.6
3.5
0.5 x
3.4 -0.5 0.5
2 x
x -0.5 0.5
-1.0 -0.5 0.5 1.0 1.5 2.0
-0.5 -0.5
1
-1.0
x
-1.0 -0.5 0.5 1.0 1.5 2.0
z y
3.7
3.6
4 3.5 -0.5
3.4
3.3
y 3
z x 0.0
2 0.5
2.0
1 0.5
1.5
2
0.5
1.0
0.5 1
y
z 3.4 3.5 3.6 3.7 z 0.0
y 0
1 2 3 4
-0.5 -1 -0.5
-1.0 -1 -0.5
0
x 1
2

Results of Calculation Case 3.2
Case 3.2
a=4.5, b=0.2, c=2,
0.10
y
5.00
0.05
4.98
4.96
x
4.94 0.05 0.10 0.15
4.92
x
0.05 0.10 0.15 -0.05
-0.10
z 0.00
y 5.00
4.98 0.05 x
4.96 0.10
0.10 4.94
4.92 0.15
0.10
0.05
0.05
z
y 0.00
4.94 4.96 4.98 5.00
-0.05
-0.05 -0.10
-0.10

Case 4
a=4.5, b=0.2, c=2,x[0]=0.5,y[0]=3.6,z[0]=0.23 a=4.5, b=0.2, c=2,x[0]=0.1,y[0]=4.95,z[0]=-0.1
y z y z
5 2.0 5.00
0.010
4.99
4 1.5 0.005
4.98
3
1.0 x
-0.010 -0.005 0.005 0.010
4.97
2
-0.005
0.5
4.96
1
-0.010
x x x
0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 -0.010
-0.005 0.0050.010
5 5.00
4
z y 3 4.98
y
z
2.0
2
0.010
4.96
1.5 1 0.005
1.0 2.0
y 0.010
4.96 4.97 4.98 4.99 5.00
1.5 0.005
0.5 -0.005 z
z
1.0 0.000
-0.010
y 0.5 -0.005
1 2 3 4 5
0.0 -0.010
0.0 -0.010
0.5 -0.005
1.0 0.000
1.5 0.005
x 2.0 x 0.010

Results
Results
From our analysis and simulation we can conclude:

The inappropriate combination of the parameters in the system is the
source that causes chaotic, it is likely to make system inclined to
chaotic and lose control, or make it into the stagnant state;

Results
Results

The elasticity deficiency of variables will cause the lagging down of the
information feedback, therfore strengthening the elasticity of variables
will help stabilize economy and help operation of the financial system;

Results
Results

The elasticity deficiency of variables will cause the lagging down of the
information feedback, therfore strengthening the elasticity of variables
will help stabilize economy and help operation of the financial system;
Saving amount variable must be kept in an appropriate level, smaller
is, greater the fluctuation of the system is; if is too small, it will cause
chaotic situation, if too large will cause economy to lack vigor

Bibliography
Bibliography
Study for the bifurcation topological structure and the global

complicated character of a kind of nonlinear finance system (Ma
Jun-hai, Chen Yu-shu)

Study of A Nonlinear Finance System

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Study of a nonlinear finance system

Manuel Luci (Università di Pisa) Study of a nonlinear finance system 2 / 28

Chaotic phenomenons occur in the economic system, macro economic

Manuel Luci (Università di Pisa) Study of a nonlinear finance system 3 / 28

Chaotic phenomenons occur in the economic system, macro economic

Government can adopt such macro control measures (financial policies,

Manuel Luci (Università di Pisa) Study of a nonlinear finance system 3 / 28

Chaotic phenomenons occur in the economic system, macro economic

Government can adopt such macro control measures (financial policies,

In the field of finance complexity is given by the interaction between

Manuel Luci (Università di Pisa) Study of a nonlinear finance system 3 / 28

Chaotic phenomenons occur in the economic system, macro economic

Government can adopt such macro control measures (financial policies,

In the field of finance complexity is given by the interaction between

It is important studying internal structural characteristics to reveal the

Manuel Luci (Università di Pisa) Study of a nonlinear finance system 3 / 28

Financial model composed by:

Manuel Luci (Università di Pisa) Study of a nonlinear finance system 4 / 28

Financial model composed by:

Manuel Luci (Università di Pisa) Study of a nonlinear finance system 5 / 28

Manuel Luci (Università di Pisa) Study of a nonlinear finance system 5 / 28

Manuel Luci (Università di Pisa) Study of a nonlinear finance system 6 / 28

Manuel Luci (Università di Pisa) Study of a nonlinear finance system 6 / 28

Manuel Luci (Università di Pisa) Study of a nonlinear finance system 7 / 28

Manuel Luci (Università di Pisa) Study of a nonlinear finance system 7 / 28

Manuel Luci (Università di Pisa) Study of a nonlinear finance system 7 / 28

Changing coordinate system and simplifying:

Manuel Luci (Università di Pisa) Study of a nonlinear finance system 8 / 28

First of all we have to find the stationary point:

Manuel Luci (Università di Pisa) Study of a nonlinear finance system 9 / 28

First of all we have to find the stationary point:

Manuel Luci (Università di Pisa) Study of a nonlinear finance system 9 / 28

Study of the local topological structure

X = x, Y = y − 1/b Z =z =⇒ Vs∗ (0, 0, 0)

Manuel Luci (Università di Pisa) Study of a nonlinear finance system 10 / 28

Study of the local topological structure

X = x, Y = y − 1/b Z =z =⇒ Vs∗ (0, 0, 0)

Manuel Luci (Università di Pisa) Study of a nonlinear finance system 10 / 28

Study of the local topological structure

X = x, Y = y − 1/b Z =z =⇒ Vs∗ (0, 0, 0)

Evaluate the Jacobian of the system in Vs ∗:

Manuel Luci (Università di Pisa) Study of a nonlinear finance system 10 / 28

Study of the local topological structure

P(λ) = (λ + b)(λ2 + λ(c + a − 1/b) + ac − c/b + 1) (8)

A first eigenvalue is λ = −b, the other 2 comes from:

λ2 + λ (c + a − 1/b) + ac − c/b + 1 = 0 (9)

Solutions can be divided into 3 cases

Manuel Luci (Università di Pisa) Study of a nonlinear finance system 11 / 28

Case 1) c − b − abc < 0, T > 0

If c − b − abc < 0 ⇒ D > 0 and T > 0, we find λ1 , λ2 , λ3 < 0 so

Manuel Luci (Università di Pisa) Study of a nonlinear finance system 12 / 28

Case 2) c − b − abc < 0, T < 0

If c − b − abc < 0 ⇒ D > 0 and T < 0 we find λ1 < 0, λ2 , λ3 > 0 so

Manuel Luci (Università di Pisa) Study of a nonlinear finance system 13 / 28

Case 3.1) c − b − abc = 0, (1 − c 2 )/c > 0

If c − b − abc = 0 ⇒ λ2 = 0, λ3 = −(c + a − 1/b) and λ1 = −b < 0, we

Manuel Luci (Università di Pisa) Study of a nonlinear finance system 14 / 28

Case 3.1) c − b − abc = 0, (1 − c 2 )/c > 0