DOI: 10.5817/FAI2016-3-1 No. 3/2016
5
Application of the Nonlinear Oscillations Theory
to the Study of Non-equilibrium Financial Market
Nikolay Markov, Viktor Dmitriev, Svetlana Maltseva, Andrey Dmitriev
National Research University Higher School of Economics
Faculty of Business and Management, School of Business Informatics
33 Kirpichnaya Str., 105187 Moscow, Russia
E-mail: a.dmitriev@hse.ru
Abstract: The research deals with the construction, implementation and analysis of
the model of the non-equilibrium financial market using econophysical approach and
the theory of nonlinear oscillations. We used the scaled variation of supply and
demand prices and elasticity of these two variables as dynamic variables in the
simulation of the non-equilibrium financial market. View of the dynamic variables
data was determined based on the strength of econophysical prerequisites using the
model of hydrodynamic type. As a result, we found that the non-equilibrium market
can be described with a good degree of accuracy with oscillator models with nonlinear
rigidity and a self-oscillating system with inertial self-excitation. The most important
states of model of oscillation non-equilibrium model of the market were found,
including the appearance of chaos and its mechanisms. We have made the
calculations of the correlation dimension for the financial time series. The results
show that all observed time series have a clearly defined chaotic dynamic nature.
Keywords: ask price, bid price, correlation dimension, financial market, financial
time series, Lorenz system, low-dimensional chaos, non-equilibrium system,
nonlinear oscillations
JEL codes: C610, C620, C690, G100
Introduction
Since the second half of the XX century, the general trend of science development
has been the penetration of the ideas and methods of physics in natural and
traditional humanities. Methods of physical modeling are often used in sciences such
as demography, sociology, linguistics and economics.
In the mid-1990s there was an interdisciplinary research field, known as
Econophysics, applying theories and methods originally developed by physicists in
order to solve problems in economics, usually those including uncertainty or
stochastic processes and nonlinear dynamics. The term “econophysics” was coined by
H. Eugene Stanley to describe the large number of papers written by physicists on
the issues of (stock and other) markets (for econophysics reviews see Chakraborti et
al., 2011; Richmond et al., 2013; Savoiu, 2013).
In 1988 Robert Savit presented the first studies of chaos in the financial markets
(Savit, 1988). “Deterministic chaos” (or low-dimensional chaos) is a term used to
denote the irregular behavior of dynamic systems arising from a strictly deterministic
6
time evolution without any source of noise or external stochasticity (Korsch and Jodl,
1999). Most surprisingly, it turned out that such chaotic behavior can already be
found for systems with a very low degree of freedom and it is, moreover, typical of
most systems.
Since the end of the XX century there have been two fields of research devoted to the
low-dimensional chaos in the financial markets. Some researchers have focused on
the detection of the chaos in the financial time series using fractal dimensions (for
examples, box counting dimension, information dimension and correlation dimension)
as various measures of chaos (Lai and Ye, 2003). So, Frank and Stengos examined
gold and silver returns and found that the correlation dimension is between 6 and 7
while the Kolmogorov entropy is about 0.2 for both assets (Murray and Stengos,
1989). In 1991 Steven Blank analyzed the futures prices for the S&P500 index and
soybeans (Blank, 1991). All of their results were consistent with those of markets
with underlying generating systems characterized by deterministic chaos (they give
necessary, but not sufficient, conditions to prove the existence of deterministic
chaos). Gregory Decoster, Walter Labys and Douglas Mitchell searched for evidence
of chaos in commodity futures (silver, copper, sugar and coffee) prices and found
evidence of nonlinear structure (Decoster et al., 1992). Scott Mayfield and Bruce
Mizrach estimated the dimension of the S&P500 (sampled at approximately 20second
intervals) and concluded that the data are either of low dimension with high
entropy or nonlinear but of high dimension (Mayfield and Mizrach, 1992). Yang and
Brorsen found evidence of nonlinearity in several futures markets, which was
consistent with deterministic chaos in about half of the cases (Yang and Brorsen,
1993). Abhyankar, Copeland and Wong (1995) tested for the presence of chaos in
the FTSE 100 Index using a six-month sample of about 60,000 minute-by-minute
returns and found little to support the view that the process is chaotic at any
frequency. Andreou, Pavlides and Karytinos (2000) examined four major currencies
against GRD and found evidence of chaos in two out of four. Panas and Ninni (2000)
found strong evidence of chaos in daily oil products for the Rotterdam and
Mediterranean petroleum markets. Antoniou and Vorlow (2005) investigated the
“compass rose” patterns revealed in phase portraits (delay plots) of FTSE 100 stock
returns and found a strong nonlinear and possibly deterministic signature in the data
generating processes. Other relevant works in this area are Hafner and Reznikova,
(2012); Urrutia et al., (2002).
In our opinion, the most comprehensive review of mathematical models of financial
markets is presented in the book by R. J. Elliott and P. E. Kopp (Elliot and Kopp,
2005). Despite the many pithy models presented in the book and various articles (Cai
and Huang, 2007; Chen, 2008; Holyst et al., 2001), we could not find a mathematical
model of the financial market whose solutions are changes in ask and bid prices. The
importance of building such a model has various reasons, especially - the possibility
of obtaining of financial time series and their further analysis and prediction.
We have built a model of the "big" market that is such a market which operates many
economic agents pursuing their own respective interests and goals. We believe that
the market is a system that generates aggregated factors from the combined action
of individual interests and needs. In this case, economic dynamics are based on some
7
basic dynamic structure. Aggregated factors are manifested on a macroscopic scale
and act according to the laws of deterministic connections and relationships.
During the constructing of the model we proceeded from the assumption that the
market in terms of generalized macroscopic flows of capital, goods and services in
the phase space of a corresponding economic dynamic system is homeomorphic to
the so-called dynamic systems of hydrodynamic type. If there are interaction counter
flows in such systems, then they tend to be a phenomenon of generalized turbulence
(Mantegna and Stanley, 1996) generating a crisis mode of the states of dynamic
systems.
Modern stock markets are armed with powerful computer technology that allows for
the management and documentation of real economic processes, which unfold in
such markets in real time. Changing the conditions of such markets is very fast, and
this fact allows you to receive time series, representing them on a small scale of
time, which is a prerequisite for the detection of the deterministic component of
dynamic processes.
Thus, the aim of the study is building a model of the non-equilibrium financial market
that can generate the deterministic component of empirical time series using the
econophysical assumptions and the nonlinear oscillation theory.
This paper is organized as follows. In section 1 we present the nonlinear dynamic
model of the non-equilibrium financial market model, including the definition of the
dynamic variables. In section 2 we present the nonlinear dynamic model of the nonequilibrium
financial market model as a nonlinear oscillator. In section 3 we present
the results and discussions, including the definition of the regular and the chaotic
states of the financial market. In section 4 we give the numerical results of
correlation dimension from the financial time series. In section 5 we conclude this
paper.
1 Construction of the Non-equilibrium Financial Market Model
1.1 Dynamic Variables
We suppose that the financial market can be considered as a non-equilibrium open
system (or dissipative system). In physics, a dissipative system is an open system
which is operating out of, and often far from, equilibrium in an environment with
which it exchanges energy and matter (Chen, 2015). Open systems have input and
output flows, representing exchanges of matter, energy or information with its
surroundings. The input and output flows on the financial market are the information
flows and money flows. We suppose also that the financial market can be in
equilibrium and non-equilibrium states. If the intensity of the external information
flow is small, then the financial market is in equilibrium. If the intensity of the
external information flow is large, then the financial market is in non-equilibrium.
A dynamic system can be described simply as a system of N first-order differential
equations
8
( )1,..., , 1,...,i
i i N
dx
x f x x i N
dt
≡ = =& ,
where the independent variable t can be read as time and the ( )ix t are dynamic
variables whose time dependence is generated by the equations, starting from
specified initial conditions ( )0 , 1,...,ix i N= . It should be noted that the dynamic
system is autonomous because it is not explicitly t -dependent (Ruelle, 1989).
Let’s define the dynamic variables of the mathematical model:
• ( )1Y t states the localized variable of the demand function,
• ( )2Y t states the localized variable of the supply function,
• ( )1X t states locally varying demand price,
• ( )2X t states locally varying supply price,
•
( ) ( )0 0
1 2 0Y Y Q= = states the equilibrium values of the functions of supply and
demand in a market equilibrium condition,
•
( ) ( )0 0
1 2 0X X P= = states the equilibrium values of the price of supply and
demand in the equilibrium state of the market ( )0 0,R P Q= ,
• ( ) ( ) ( )0 , 1,2i i iY t Q Y t y iδ− ≡ ≡ = states variations of the volume of demand
and supply near ( )0 0,R P Q= ,
• ( ) ( ) ( )0 , 1,2j j jX t P X t x jδ− ≡ ≡ = states variations of prices of supply and
demand near ( )0 0,R P Q= ,
• ( )1 1 2,F x x states function of aggregated demand in cumulative product of
the market as a function of various kinds of price,
• ( )2 1 2,F x x states function of aggregated supply in cumulative product of the
market as a function of various kinds of price.
Let’s consider that ( )1 1 2,F x x and ( )2 1 2,F x x are continuous and differentiable in the
variables 1x and 2x in the vicinity of the equilibrium point ( )0 0,R P Q= .
Let’s construct a matrix of first partial derivatives of functions ( )1 1 2,F x x and
( )2 1 2,F x x near the point ( )0 0,R P Q= :
1 1
1 2
2 2
1 2
F F
x x
L
F F
x x
∂ ∂ 
 ∂ ∂
 =
 ∂ ∂
 ∂ ∂ 
(1)
9
Given that the price variations 1x and 2x are small, use the L . Onsager’s relations
(Onsager, 1931) of reciprocity in matrix form:
1 11 12 1
2 21 22 2
y L L x
y L L x
    
=    
    
(2)
where { }ijL states phenomenological matrix coefficients (1). The matrix equation (2)
connects the extensive ( )1 2,y y and intensive ( )1 2,x x variables of the model; in
terms of the thermodynamics of irreversible processes – flows and driving forces
associated with data flows.
Economic dynamics of non-equilibrium market due to deviations of demand ( )1Y t
and supply ( )2Y t from their equilibrium state
( ) ( )0 0
1 2 0Y Y Q= = should induce a
temporary change in the variation of prices ( )1 2,x x . In the first approximation, these
dynamics can be represented as a system of ordinary differential equations:
1 11 12 1
2 21 22 2
x K K y
x K K y
    
=    
    
&
&
(3)
where ix& states first order derivative with respect to time, { }ijK states the entries
of dynamic market conditions.
The joint solving of equations (2) and (3) gives the equation describing the dynamics
of economic and non-equilibrium dynamic system:
1 11 12 1
2 21 22 2
x A A x
x A A x
    
=    
    
&
&
(4)
where
11 12 11 12 11 12
21 22 21 22 21 22
A A K K L L
A A K K L L
    
=    
    
states matrix, which determines the
dynamics of such a non-equilibrium system.
1.2 Bid Price Dynamics
Let’s proceed to the nonlinear dynamic analysis of economic system (4). If 1 0x = ,
then 2 22 2x A x=& . Locally varying supply price ( )2X t comes near 0P , therefore
22 0A < . In this case time of relaxation 2 0X P→ is 2 221 Aτ = . Also there should
appear states 2 0x =& around the state of dynamic equilibrium 1 2x x= . This is
possible if 22 21 21A A τ= = . Consequently, up to the first order values ( 1x and 2x )
the second equation of the system (4) takes the following form:
10
( )2 2 1
2
1
x x x
τ
= − −& (5)
Similar arguments about the coefficients of the first equation of (4) lead to the fact
that 11 11A τ= , where 1τ states characteristic time of relaxation 1 0X P→ . If 1x
and 2x are small enough, then 12A is constant. In this case, the solution of (3) is a
noise relaxation oscillations near the equilibrium state ( )0 0,R P Q= . Relaxation
means the return of a perturbed system into equilibrium.
According to the Le Chatelier-Braun principle (Atkins, 1993), if a system deviates
from the state of the stable equilibrium, then the forces arise and try to return the
system back to the equilibrium state. If ( )1 0 0X t P P−  , then as a first
approximation it can be considered that these forces are proportionate to deviation.
With the growth of the amplitude values of deviations ( )1X t and ( )2X t from
( )0 0,R P Q= variations in prices of supply and demand start to "cling" to each
other.There is a simple nonlinear interaction between supply and demand in the
system (4). Consequently, there is a relation between variables 1x and 2x ,
describing the elasticity of 1x due to 2x . In this case, elasticity is the measurement
of how responsive the ask price is to a change in the bid price.
Consider,
( ) ( )12 12A t cE t= (6)
where ( )12E t – the elasticity of 1x due to 2x , c – parameter, used for dimensions
matching. With help of choosing of measure units we can prove that 1c = .
Introduce the dimensionless time 1T t τ= . Multiplying the equation (5) to 1τ , we get
the following equation:
( )2
2 1
dx
x x
dT
σ= − − (7)
where
1 22
2 11
A
A
τ
σ
τ
= = . Factor σ shows the relation between the relaxation speeds
1 0X P→ and 2 0X P→ , i.e. it represents the relative sensitivity to changes in market
prices of supply and demand. If 2 1τ τ< , then we can observe an effect of delay of
the reaction of demand price to rapid change in the supply price, which leads to
certain dynamic effects.
11
1.3 Ask Price Dynamics
In the view of (6) and (7), the first equation of system (4) takes the following form:
( )1
1 1 12 2
dx
x cE t x
dT
τ= − + (8)
1.4 Elasticity Dynamics
The value ( )12E t is regarded as an independent dynamic variable, provided in the
following form:
( ) ( )
( )0
12 12 12E t E E tδ= + (9)
where
( )0
12E states elasticity in the equilibrium state.
The essence of the non-equilibrium dynamics of the market leads to the conclusion
that in the first nonlinear approximation, the differential equation for ( )12E t takes
the following form:
12 12 1 2E eE kx x= − +& (10)
where , 0e k > – constants, 1 Ee τ= , Eτ states characteristic time of relaxation
( ) ( )0
12 12E t E→ .
Let’s introduce 1 21C Lτ= , 1K kτ= , 1 Eβ τ τ= and multiply (10) to 1τ . Then we get
the following equation:
12 12 1 2E E Kx xβ= − +& (11)
Let’s also introduce some characteristic elasticity scale λ and new renormalized
quantities:
( )0
12 12
1 2, , , , , ,
E E K K dx dy dz
z x x y x x y z
t C C dT dT dT
ρ ρ
λ
= = − = = = = =& & & (12)
Therefore the equations (7), (8) and (11) respectively take the following forms:
( )x y xσ= − −& (13)
y x y xzρ= − −& (14)
z z xyβ= − +& (15)
The system of equations (13), (14) and (15) is well known as Lorenz system (Lorenz,
1963).
12
2 Nonlinear Oscillations
Now let’s reduce the number of equations describing the non-equilibrium market.
Introduce a new variable:
2
2u z xσ≡ − (16)
Taking into account (16), equation (13) takes the following form:
( ) ( )31 1
1 1
2 2
x x x u xσ σ ρ
 
= − + − + − −  
&& & (17)
Taking into account (16), equation (15) takes the following form:
( ) 2
2u u xβ σ β= − + −& (18)
The system of equations (17) and (18) corresponds to the dynamic model of an
oscillator with inertial nonlinear inflexibility (Hayfeh and Mook, 1995). Equation (17)
represents the dynamics of inharmonic oscillations along the coordinate axis Ox in
two-humped potential pit, damped with the member ( )1 xσ− + & and described by the
function
( ) ( )4 2 4 21 1 1 1 1
, 1
8 2 2 8 2
V u x x u x x xσ ρ ω
 
= − − − ≡ −  
(19)
The value ( ) ( )
1
1
2
u uω σ ρ= − − is not a constant. Consequently, the system
generates complex auto-oscillatory relations between ( )u x and ( )x u , defined by
equations (17) and (18).
The system of equations (17) and (18) generates a complex irregular movement in a
plane of the variables ( ),u x that under certain relations between the parameters
, ,σ β ρ generate a phenomenon called dynamic chaos (Wiggins, 2003).
The system of equations (17) and (18) is equivalent to Lorenz system, but now it can
interpreted in the following way.
First of all, let’s u const= . Then (17) is an equation of nonlinear oscillator
V
x x
x
γ
∂
+ = −
∂
&& & , where 1γ σ= + - dissipation parameter and potential function
depends on ρ , σ , u parameters and has a view (19).
In the field of parameters, corresponding with complicated chaotic dynamics,
( )1 0σ ρ − > , thus for small u the coefficient for quadratic term is negative and
potentially has two symmetrically posed minimums. For big u the coefficient for
quadratic term becomes positive and at least one (Fig. 1). However u in the system
under consideration is not a parameter, but a dynamic variable as a subject of (18).
13
It controls the change of
2
x value in time, not instantly, but with inertness and
smoothing that are determined by β parameter.
Figure 1 Potential function ( )V x
Source: WolframAlpha®
System dynamics can be presented as fluctuation inside one of potential pits with
occasional transfer to another one caused by u parameter change and the
corresponding potential form change.
3 Results and Discussion
3.1 Asymptotically Stable Equilibrium Price
Let’s proceed to a qualitative analysis of the systems of differential equations
(Wiggins, 2003). Let’s find the stationary points and determine their stability.
The system of differential equations (17) and (18) has three stationary points:
( )0,0O , ( ) ( )( )( )1,2 1 , 2 1E β ρ σ β ρ± − − − (20)
If 1ρ < , then the zero point is asymptotically stable. At the same time there are no
other real stationary points 1,2E . Point 1ρ = is a point of supercritical pitchfork
bifurcation. If 1ρ > , then the zero stationary point is unstable.
Real stationary points 1,2E appear if 1ρ > . At the same time, if 1 cρ ρ< < , then
the points are stable; if cρ ρ> , then 1,2E – unstable. Here
3
1
c
σ β
ρ σ
σ β
+ +
=
− −
,
where 1σ β> − . In these cases, ( )1 0β ρ − ≠ and stable equilibrium market price
does not exist.
The condition for the existence of equilibrium (not necessarily stable) market price is
the equality of supply and demand prices ( ) ( )1 2 0X t X t P= = or achieving a state of
14
equilibrium system ( )0 0,R P Q= . In the notation of the Lorenz system for variations
in the price of supply and demand, this condition is equivalent to 0x y= = .
Consequently, if and only if
( )0
12
1
E
λ
< , the market may contain asymptotically stable
equilibrium price (fig. 2). It is the equilibrium state of the financial market. The real
financial market is not in an equilibrium state.
Figure 2 Asymptotically stable solution ( )tx at 7.0=ρ
Source: Wolfram Mathematica®
10.2
3.2 Dynamic Chaos and Financial Time Series
Let’s fix σ and β , i.e. ratio of times of relaxation 1
2
τ
τ
and 1
E
τ
τ
. Consider 1
2
10
τ
τ
=
and 1 8
3E
τ
τ
= . At the same time we vary the parameter
( )0
12E
ρ
λ
= .
For sufficiently large values ρ ( 28ρ ≥ ), i.e. for big elasticity
( )0
12E
λ
, random
(irregular) paths ( )x t and ( )u t appear in the system of equations (17) and (18)
(fig. 3). For Lorenz system, whenT → ∞ phase path fills an attractive special area
near points 1E and 2E , which is called a strange attractor, in this case – the Lorenz
attractor (Lorenz, 1963). Lorenz attractor is stable, i.e. it is preserved under small
changes in the parameters. An attractor is a set of states (points in the phase space),
invariant under the dynamics, towards which neighboring states in a given basin of
attraction asymptotically approach in the course of dynamic evolution (Baker and
Gollub, 1996). The Lorenz attractor has a correlation dimension of 2.05 0.01±
(Grassberger and Procaccia, 1983). For more details, see Lichtenberg and Lieberman
(1983) and Tabor (1989).
15
Figure 3 Chaotic solution ( )tx at 28=ρ
Source: Wolfram Mathematica®
10.2
Strange attractors are one example of the appearance and manifestation of the
evolutionary phenomenon that is called a catastrophe (Arnold, 1992). Accidents in
dynamic systems give rise to such a phenomenon as dynamic chaos. One of the most
difficult and important tasks – to learn how to predict occurrence of the phenomenon
of dynamic chaos and precede the appearance of this kind of phenomena.
Thus, in complex dynamic systems such as financial markets, there are basic
nonlinear dynamics, which generate the observed sequence of their states in its
various modes of display, which are called time series. Time series that are obtained
from empirical observations and that are "well" organized carry meaningful
information about the deterministic component of the dynamics that form these
series.
4 Correlation Dimension of Financial Time Series
The determination of the correlation dimension (Grassberger and Procaccia, 1983) for
a supposed chaotic process directly from experimental time series is an often used
means of gaining information about the nature of the underlying dynamics (see, for
example, contributions in Ding et al., 1993; for reviews on dimension measurements
see Grassberger et al., 1991). In particular, such analyses have been used to support
the hypothesis that the time series results from an inherently low-dimensional chaotic
process (Ding et al., 1993).
The geometry of chaotic attractors can be complex and difficult to describe. It is
therefore useful to have quantitative characterizations of such geometrical objects.
One of these characterizations is the correlation dimension 2D . The correlation
dimension has several advantages comparing to other dimensional measures:
• 2D is easy to compute from financial experimental data.
16
• Using 2D one can distinguish a chaotic dynamic system ( 2D is finite) from a
stochastic system ( 2D → ∞ ).
• 2D is the lower bound estimate of attractor’s dimension d ( 2d D≥ ).
The correlation dimension of the attractor of dynamic system can be estimated using
the Grassberger-Procaccia algorithm (GP algorithm) (Grassberger and Procaccia,
1983). The implementation of GP Algorithm in this work is written in Mathematica
Language for Wolfram Mathematica®
10.2.
Table 1 provides summary of calculated correlation dimension for financial time
series. These time series include the following:
• EUR/USD exchange rates (daily, from 04.01.1999 to 06.05.2016),
• GBP/USD exchange rates (daily, from 04.01.1971 to 06.05.2016),
• (HPQ) HP Inc. stock prices (daily, from 01.01.1995 to 06.05.2016).
Table 1 Calculated correlation dimension for financial time series
Signal
Data
Points
Correlation
Dimension
EUR/USD 4361 4.56
GBP/USD 11372 4.36
HPQ 5285 4.88
Source: Authors' calculations
Based on the results, it is possible to state that currency exchange rates have chaotic
dynamic nature, rather than “pure” randomness. The results show that all the
observed time series have a clearly defined chaotic dynamic nature. It is also
noticeable that the financial time series have the correlation dimension close to each
other and equal to 4.5.
It is not strictly proven that any of the financial time series has a chaotic dynamic
nature. However, we can consider the dynamic system (17) and (18) as an
approximate model of the financial market. So, we also suggest that the estimation
of the correlation dimension shall be a preliminary step of financial time series
analysis. Moreover, the implementation of GP algorithm, including all optimizations, is
computationally effective.
Conclusions
The main contributions of this paper are as follows.
• We present the nonlinear dynamic model of the financial market as a nonequilibrium
system, including the definition of regular and chaotic state of the
market. It is pointed out that the system is in a condition of an asymptotically
stable equilibrium when the scaled elasticity is small (the ask price and bid
price eventually tend to their equilibrium values).
• We report new numerical results of the correlation dimension from financial
time series. The results show that all the observed time series have a clearly
17
defined chaotic dynamic nature. It is also noticeable that the time series have
the correlation dimension close to each other and equal to 4.5.
We found that the model of a nonlinear oscillator and the Lorenz model describe a
non-equilibrium financial market. A model in the form of equations (17) and (18)
gives the relationship between the demand price time, i.e. the dynamic component of
financial time series. The analysis of the model suggests that market has an
asymptotically stable equilibrium price at small scaled elasticity (
( )0
12
1
E
λ
< ). If the
scaled elasticity
( )0
12
28
E
λ
≥ , one of the crisis mode arises in the system - dynamic
chaos. Given the dependence (16), these phenomena are characteristic for elasticity
( )12E t . In addition, following from the properties of the Lorenz system that models
financial markets, there is asymptotically stable and chaotic dependence of the
supply price from time to time.
The practical significance of the model of the financial market is associated with the
solution of the problem of reconstruction of a dynamic system (Sugihara and May,
1990). In some sense, this is the inverse problem – recovering of dynamic system,
using empirically observed values of financial time series, generating these financial
time series. Some results of the reconstruction of a dynamic system are presented in
our papers (Dmitriev et al., 2014; Maltseva et al., 2014).
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