Studying the Chaotic of Modified Jerk Map based on Lyapunov
Exponents, Topological Entropy, and Sensitivity
Samah Abdulhadi AL-hashemi
Department of Computer Science, College of Science for Women, University of Babylon, Babylon, Iraq
Email of corresponding author: samah_hadi1@yahoo.com
Abstract:
In the last four decades, Chaos has been studied intensively as an interesting practical phenomenon. Hence, it is considered to be one of the most important branches in mathematics science that deals with the dynamic behavior of systems which are sensitive to the initial conditions. It has therefore been used in many scientific applications in the sciences of chemistry, physics, computers, communications, cryptography, and engineering as well as in bits generators, and psychology. However, there are many issues that need to be considered and highlighted, such as future prediction, computational complexities, and unstable behavior of dynamic system. The dynamic system must contain three characteristics in order to be considered a chaotic system which is first, to be sensitive to the initial conditions; second, to have dense periodic orbits and finally to be topologically mixing. In the previous work, we studied the fixed point of
??
????? + ????
=?
2 ? in order to find the contracting and expanding area of this
a modified Jerk map with the form ???? ??,??
map, as well as to define the area in which the fixed points of attracting, repelling, or saddle are located. In this paper, we continue to address the same problem by Modified Jerk Map. We prove that it has a positive Lypaunov exponent if |a|=1 and has sensitivity dependence to initial condition if |a|>1 and we give an estimate of topological entropy. Finally, to simulate our equations and obtain related results, we have used Matlab program by implementing a Lypaunov exponent and drawing the sensitivity of ???? ??,?? .
Key words: Chaos, Modified Jerk Map, Lypaunov Exponent, Sensitivity Dependence, Entropy
existing state between a specific and randomized state [4].
One of the most widely accepted and popular definitions of anarchy was defined by R. L. Devaney. In this definition, the systems must be based on exhibit topological transitivity sensitive dependency to initial conditions and dense periodic orbits [5] . Later, several works (such as [4] [6]) were implemented to prove that if a system is transitive with dense periodic orbits then it should obviously show sensitivity dependency to an initial condition. Lyapunov defines the Chaos as follows: the continuously differentiable map ???? can be chaotic if and only if the ???? has a positive Lyapunov exponent
1. Introduction
The dynamical system is a theory that has mostly been studied as an abstract concept subject in branches of mathematics, physics and computer science [1]. Mostly, it is considered Chaotic according to either the metric properties or topology of the system [2]. Yorke et al., in 1976 (see [3]) addressed how three-period orbits with a dynamical system emphasize that the dynamical system is chaotic. Hence, Chaos can generally be defined as an In term of physics, it mean that the jerk map is presented as a third derivative of the position with respect to time. In other words, according to Equation (1), x is the third derivative of x that represents the rate of change of the acceleration in a mechanical system. Jerk dynamics can be described by a set of three first-order synchronous differential equations, where the dependent variables are the position x, velocity x?, and acceleration x¨. It is generally as follows [19]:
????
and if it is topologically transitive [7][4]. Chaos has been studied intensively as an interesting practical phenomenon in the last four decades. Hence, it is considered as one of the most important branches in mathematics science and also can be used in many significant applications in the sciences of computers and cryptography [8], bits generators [9], ecology[10], economy [11][12], biology [13][14], and communications [15][16].
The Lyapunov exponents give the average exponential rate of convergence or divergence for near orbits in the phase-space. Thus, Chaotic will be defined for any dynamics system containing at least one positive Lyapunov s exponent. Any small initial differences initially in a system may affect its ability, consequently leading to less predictability. The dynamics system becomes unpredictable with the magnitude of the exponent indicating the time scale [17].
One of the important measures that are used to measure the complexity of the dynamics system is topological entropy. It represents the exponential growth rate of the number of distinguishable orbits iterates. It must therefore be a non-negative real number [18].
In general, sensitivity is employed to nonlinear equations models. The idea of sensitivity is derived from the effect of the butterfly. The reason behind it is that lost patterns and the great effects of inputs are as marginal or negligible as the flap of the butterfly wings. Any change in initial condition, even if it were small, may lead to an undesired result. It is therefore impossible to predict future behavior. However, this does not mean that the system is not deterministic [2].
Mendoza et al., in [19] which is referred to [20] presented a new form in the explicit third order called Jerk map, as in the following equation:
?? … = ??(??, ??? , ??¨) … (1)
= ?? . = ?? … (2)
????
??2??
= ?? .. = ?? … (3)
???? 2
??3??
= ?? … = ????? .. ? ???? . + ??(??) … (4)
???? 3
Sprott called Equation (4), the last of these three equations as the Jerk equation [20] , where the parameters A and B are numerical constants and Q(x) is a nonlinear function.
As is customary, the Jerk map is a 3-D dynamical system. In our previous work [21], we transformed Jerk map 3-D dynamical system into a 2-D dynamical system to reduce the computational time and space based on Equation (5), as follows:
??
????? + ????
=?
2 ? … (5)
??????,??
Where variables ?? and ?? are the states; and prime indicates as differentiation. The work addresses and studies the fixed points of ????????,?? , as well as its general properties. It also found the contracting and expanding area of this map and is thus made to determine the fixed points of attracting, repelling or saddle. Continuing with the previous work, the proposing study aims to prove the properties of chaotic in dynamical system, including sensitivity, Lyapunov exponents, and topological entropy. The remainder of this paper is organized as follows. In Section 2, the Lyapunov Exponents are defined and simulated. In Section 3, the Topological Entropy A
B
(x,y)
L1
L2
0.99
0.05
(0.1,0.2)
-0.0050152848
-0.005035051
0.79
0.05
(0.1,0.2)
-0.117860720
-0.1178616130
0.59
0.05
(0.1,0.2)
-0.2638161836
-0.2638165585
0.39
0.05
(0.1,0.2)
-0.6189371307
-0.6189372253
-0.99
0.05
(0.1,0.2)
-0.0049733468-
-0.0050769890
-0.79
0.05
(0.1,0.2)
-0.1178610782
-0.1178612553
-0.59
0.05
(0.1,0.2)
-0.2638163575
-0.2638163846
-0.39
0.05
(0.1,0.2)
-0.4708042671
-0.4708042727
Continuous, if |??1| < 1, then:
is defined. Sensitive Dependence on Initial is defined and simulated in Section 4, and in Section 5 this paper concludes with a summary.
2. Lyapunov Exponents
In the first instance, Lyapunov Exponents is developed as follows:
Let ?? be a continuous differential map at ??, ? ?? ? ?? in direction ??.
Also Lyapunov exponent of a map F is defined by:
1
? = lim
???? ??
??1 ??
?? ???,??
1 ln?(????????,?? ? ???? ? ??1)?? ? >
1
???? ????? ? ?(????)2 ? ??? … (10)
by hypothesis ??1 > 0. So:
if |??1| > 1, then:
? = lim
???? ??
??2 ??
?? ???,??
1 ln?(????????,?? ? ???? ? ??2)?? ?
n u|| … (6)
x
L±(x, u) = lim
n?? n
ln ||DF
2
whenever the limit exists.
In higher dimensions, ???? map ?? has ?? Lyapunov exponents, based on:
< ???? ????? ? ?(????)2 ? ??? … (11)
This
Exponent
Lyapunov
?? ±(??, ??) = max{??±(??, ??1) , ??±(??, ??2) hence
?? ±1 (??, ??1), ??
± (??, ??
2
± (??, ??3), … … ??±??(??, ????) … (7)
1 ), ??3
2
1
Lyapunov Exponent of ??Ja,b is positive.
In order to implement the Lyapunov Exponents dependence on the initial condition of a map, the points (???? , ???? ) are changed or fixed, where ?? = 1,2 control parameters (a, b). Matlab program is used to simulate and obtain results as illustrated in Table 1 and Table 2.
for a minimum Lyapunov exponent that is:
?? ±(??, ??) = ?????? ? ??
1 ), ??±2 (??, ??2 ), … ??±?? (??, ???? )? … (8)
± (??, ??
1
where u=(u1,u2,…….,un)[22].
So, we proposed a new important theorem of
Lyapunov Exponents as follows:
Proposition (1):
Table 1: |??| < 1
??
??????
???
?? ?????? ???? > ???
For ? ?
2 , if |??| = 1, ?? ?
then the ??????,?? has positive Lyapunov Exponents. Proof:-
??
??????
Let ?
2 the Lyapunov Exponents of ??Ja,b
given by the Equation (9) as follows:
? = lim
???? ??
??1 ??
?? ???,??
1 ln????????? ??,?? ? ???? ? ??1? . . (9)
1
1
In Table (1), all the values of parameter a are less than 1, and the parameter b is set as 0.05 since it may not may affect the results. So, when |??| < 1, all Lyapunov Exponents results are negative.
From proposition (3-6) in [23] |??1| =
??? ??
, where: ?? ?
|??2|A
B
(x,y)
L1
L2
1.001
0.005
(0.1,0.2)
0.0039907258
0.0039774438
1.004
0.005
(0.1,0.2)
0.0019976051
0.0019944162
1.008
0.005
(0.1,0.2)
0.0039907258
0.0039774438
1
0.005
(0.1,0.2)
0.0000004999
-0.0000004999
-1.001
0.005
(0.1,0.2)
0.0009925325
0.0000069679
-1.004
0.05
(0.1,0.2)
0.0030486413
0.0009433799
-1.008
0.05
(0.1,0.2)
0.0071595456
0.0008086240
Let the ??????,??: ??2 ? ??2 be a continuous map ????????????????,??? ? ??????|??|,where ?? is the largest eigen value of DMJ(v) where ?? ? ??2.
Proposition (2):
Table 2: |??| > 1 ???? |??| = 1
1 log(??2 + ?? + 1)
2
If ?? > 0, then: ??????? ???????,?? ? > Proof:-
Since:
?? + 1 + ???2 + ?? + 1 > ???2 + ?? + 1 … (13) ???? + ???2??2 ? 2?????? + ??2 > ???2 + ?? + 1 … (14)
so,
In Table (2), all the values of parameter a are set to be 1 or more than 1, and the parameter ?? is set as 0.005 since it may not affect the results. So, wherever |??| > 1, all Lyapunov Exponent results are positive, but when |??| = 1 , there are two Lyapunov Exponents results; one is positive and the other is negative.
3. Topological Entropy
In this section, the topological entropy is defined as follows:
Let ??: ?? ? ?? be a continuous map of a compact metric space ?? for ? > 0 and ?????+ , we say ??? ?? is an (??, ?) – separated set if for every ??, ?? ??? , then exists i.e., 0? ?? < ?? , such that ???? ??(??), ???? ?? (??)) > ? then the topological entropy of f, denoted by ??????? (????), is defined to be:
therefore,
?????? | ??1| > ?????????2 + ?? + 1, then:
1
2 log(??
??????? ???Ja,b? >
2 + ?? + 1) … (15)
Accord