The Dynamics of the Fixed Points to Modified Jerk Map
Samah Abdulhadi Al-Hashemi, Zainab Abdulmunim Sharba and Najlaa Adnan
Department of Computer Science, College of Science for Women, University of Babylon, Hillah, Iraq
Abstract: Recently, Jerk equation is the third-order explicit autonomous differential equation, is noticed to be a motivating sub-class of dynamical systems. Where many of regular and chaotic motion features can be reveal from these systems. In this paper, a simplified version Jerk map is presented. Different properties of dynamical behavior is acquired by replacing three dimensional systems to two dimensional one. Where a new parameter is added with the same properties. Moreover, we study the fixed point of modified Jerk map with the form Mj a,b = (y ? ax + by2) = (xy) and the general properties of them, so, we find the contracting and expanding area of this map. Also, we determine fixed point attracting repelling or saddle.
Key words: Modified Jerk Map, Fixed Point, Attracting-Expanding Area, dimensional, properties, saddle
INTRODUCTION
Non-linear dynamical happens and covered a large area of engineering, physics, biology, and many other scientific disciplines. There is a large amount of interest in the chaos literature in finding out of chaos in natural and physical system. Poincare way the first to notice the possibility of chaos according to which a deter monistic system to show periodic behavior that depends on the initial condition. There by rendering long term could be expected impossible.
Three dimensional system which is simple and chaotic have been tried to find by Gottlieb (1996), Hoover (1995), Linz (1997), Patidar and Sud (2005) and Posch et al. (1986). By sprott (1994) found 19 clear chotic models (one of them is conservative and the lasting are dissipative, mentioned as models A –S) with three- dimensional vector fields that involve of five terms including two non-linearities or of six terms with non- linearity one quadratic.
Hoover (1995), stated that the special case of nose that has been created by Sportt (1994) is simply conservative system (Model A) which is a known Hoover thermostat dynamical system. This system shows a time reversible Hamiltonian chaose (Linz, 1997). Apart from this the others models from B-S are unknown. Moreover,
Gottliebl indicated that Jerk function is a noticeable third- order form y=J(Y,.Y,..Y) produces from reorder the sprott’s Model A. This function contains the third derivative of x. According to Gottlieb (1996) study, an exciting question has been an annoyance “what is the simplest jerk function that gives chaos?”.
Linz (1997) shows that there models can be reduce to a a jerk form named original R..Ossler Model, Lorenz Model and Sprott Model R. Moreover the difficulty of
Ossler and Lorenz Models are higher and imappropirate
R
..
candidate for the Gottlieb s simplest jerk function.
In Patidar and Sud (2005), the global dynamic of some member of a special family of a dynamic system has been explored by Patidar. The following form represents this dynamic system under consideration:
?? + ???? +? ??? = ??(??)
Where the system parameter represented by A and the nonlinear function that have three arguments which are one nonlinearity, one system parameter and a constant term represented by G(X).
In this research, we simplified the Jerk map by replacing three dimensional systems to two dimensional systems and include the new parameter so we get the difference (new various) properties of dynamical behavior from Jerk
Corresponding Author: Samah Abdulhadi Al-Hashemi, Department of Computer Science, College of Science for Women, University
of Babylon, Hillah, Iraq
2296J. Eng. Applied Sci., 13 (Special Issue 1): 2296-2300, 2018
map also there exist the same properties of it. Also, we show that the modified Jerk map is a diffeomorphism map and it has two fixed points, we determinate the type of fixed points.
MATERIALS AND METHODS
Preliminaries: "For any map G define from R2to R2, we say G is C?, if it’s continuous for all k ? Z+and mixed K-th partial derivatives exist, and we say that G is diffeomorphism map if it is onto, one to one, C? and its inverse is C?. Let U be a subset of R2 and v0 be any element in R2. Consider G: V ? R2 be a map. Furthermore, assume that the first partials of the coordinate maps f and gof G exist at s0. The differential of ?? at ??0 is the linear map ????(??0) defined on ??2 by:
Proof: By definition of fixed point:
MJa,b = ( y
?ax + by
2) = (
x y)
Then x = y and ?Ay + By2 ? y = 0, by2 ? (a + 1) y =
0 , y [by ? (a + 1)] = 0 then y = 0 or y = a + 1/b and x = a + 1/b. Therefor:
a+1
2=(
b)
0
0 ) and P
b
P1=(
a+1
Proposition (3.2): Let MJ: R2 ? R2 be modified Jerk map
then the Jacobin of MJa,b is a:
Proof:
0 1 ) so DMJa,b (yx) = det ( 0 1 ) = a
DMJa,b (yx) = (?a 2by
?a 2by
?f
?f
Proposition (3.3): The eigen value of:
DMJa,b (yx) are ?1 ‚2 = by ? ?(by)2 ? a at p2 ‚ ? (
?x ?g ?x
?y ?g ?y
DG(s0) =
x
y)?R
)
(
2
Proof: If ? is eigen value of DMJa,b then must be satisfied the following equation:
for all s0 in R2. The determinate of DG(s0) is called the Jerk of G at s0 and it is denoted by JG(s0) = det DG(s0), so G is said to be area-contracting at v0 if |detDG(v0)| < 1, G is said to be area-expanding at v0 if |detDG(v0)| > 1.
For any nxn matrix B, the scalar ? is called aneigenvalue of n × n matrix B if there exists a non zero vector Z ? Rn such that BZ = ?Z, such an Z is called eigenvector of B corresponding to eigen value ?. Any pair for which f(pq) = p. g(pq) = q is called a fixed point of the two dimensional dynamical system it is repelling fixed point if ?1and ?2 are >1 in absolute value , and it is an attracting fixed point if ?1 and ?2 is<1 in absolute value B ? GL(2, Z) with det (B) = ±1 is called hyperbolic matrix if |?i| wher ?i are the eigenvalues of B" (Denny, 1992).
RESULTS AND DISCUSSION
General properties of modified jerk map
Proposition (3.1) Let MJ: R2 ? R2 be modified Jerk map
?? 1 | ?a 2by ? ?
|=0
Then:
(??)(2by ? ?) + a = 0 ? 2 ? 2by? + a = 0
And the solution of characteristic equations are:
2by ? ?(2by)2 ? 4a
2
?1‚2 =
?1‚2 = by ? ?(by)2 ? a
Remark:
? It is clear that if |by| > ?a then the eigen value of DMJa,b is real.
0
1 are ?1,2
? The eigen value of DMJa,b at fixed point P
=
???a and ?1‚2 = a + 1 ? ?(a + 1)2 ? a for the fixed point P2.
Proposition (3.4): Let MJ: R2 ? R2 be modified Jerk map if a ? 0 then MJa,b is diffeomorphism.
Proof: To prove MJa,b is one to one. Let
then MJ has two fixed points:
0). ( (0
a+1
b)
(x y1
1 ) , (x2
) ?R
b
y2
a+1
2297J. Eng. Applied Sci., 13 (Special Issue 1): 2296-2300, 2018
Such that:
x
Ker (MJa,b) = (y) : x. y ? R
Then MJa.b is not one to one hence MJa,b is not diffeomorphism.
Proposition (3.5):
x1 y1
(x y2
MJa,b (
) = MJa,b
2)
Then:
y1
y2
(
2) = (
2)
1 + by1
2 + by2
?ax
?ax
x y)
DMJa,b (
?a
Then:
2
y1 = y2 and ?ax1 + by1
2 2 + by2
= ?ax
is non hyperbolic if ? b also it is non hyperbolic matrix if a = ?1
Proof: Let y = ? ?a
b then b2y2 = a and b2y2 ? 2by?a + a =
(by)2 ? ?a. Hence:
x = |by ? ?(by)2 ? a| = 1
Hence, – ax1 = ?ax2 then x1 = x2,MJa,b is C?:
xy
?ax + by2)
y)=(
MJa,b (
then all first partial derivative exist and continuous, note that:
?nf(x. y)
?xn ? nf(x. y)
?yn ? ng(x. y)
?xn
=0?n?N
And
0 = ???a the eigenvalue at p1.
?1 = ?2 = 1, ?1,2
So
=0?n?2 =0?n?3
If a = ?1 then ?0
1,2
= ?1.
Proposition 3.6: For all:
?a
( yx) ? R2, y ? b |by| > ?a , DMJa,b (yx)
We find all its MJa,b exist Kth partial derivative exist
and continuous for all k. MJa,b is onto:
And:
Let (
w)?R v
2
Such that:
is hyperbolic matrix if and only if |a| = 1. Proof: Since DMJa,b ? GL(2. z) and:
det (DMJa,b (yx)) = |a|
then a = 1 or a = ?1
Conversely: Let a = 1, since
det (DMJa,b (yx)) = a
y = v and ? Ax + By2 = w
w ? bv2 x=
So:
?a
?w)?R2
a v
Then, there exist:
Such that:
( bv2
x y)=(
MJa,b (
w) v
1
we have ?1?2 = a = 1 Then ?1 =
if ?1 > 1 ?2 < 1 or
?2
Then MJa,b is onto MJa,b has an invers. Remark: if a = 0 then:
?1 < 1 then ?2 > 1. So, DMJa,b is hyperbolic matrix.
if |a| < 1 then DMJa,b is area–
Proposition (3.7):
y2) by
MJa,b = (
contracting and its area expanding if |a| > 1.
So:
2298J. Eng. Applied Sci., 13 (Special Issue 1): 2296-2300, 2018
a+1
(
) where a > 1
Proof: if |a| < 1 then by proposition (Jorobian):
|det (DMJa,b (yx))| < 1
that is DMJa,b is area–contracting map and if |a| > 1 then
|det (DMJa,b (yx))| > 1
This implies that MJa,b is an area-expending. Fixed Point Properties of Modified Jerk Map
Proposition (4-1): If |a| > 1 thenThe fixed point:
b a+1
b
MJa,b has attracting fixed point:
a+1
(
b )where a < 1
b
a+1
0|=
1
Proof (1): Since, ?1 < ?? then we have ???? < 1 and |??
0|<1
2
|??
By proposition (3-1) and (3-3). Hence:
(00)
is attracting fixed point.
Now to show that P2 is a saddle fixed point by proposition (2-3), we have two eigenvalue:
?1 = a + 1 + ?(a + 1)2 ? a, ?2 = a + 1 ? ?(a + 1)2 ? a (1) Since:
?a2 + a + 1 < ?(a + 2)2=|a + 2| = ?(a + 2) (2)
Hence:
a + 1 + ?a2 + a + 1 < 1 (3)
So:
|a + 1 + ?a2 + a + 1| = |?1| < 1 (4)
For ?2 since -1 < a < 0 we have:
a+1
(
b)
b
a+1
of modified Jerk map is repelling where a > 1. The fixed point:
(00)
of modified Jerk map is repelling where a < ?1. Proof (1): Since:
+1 > a, ?(a + 1)2 ? a > ?(a ? 2)2
Then:
a + 1 ? ?(a + 1)2 ? a > a ? ?(a ? 2)2
By proposition (3-1and 3-3)Thuse |?1| = |?2| > 1. Then:
a+1
b)
(
is repelling fixed point.
b
a+1
?a2 + a + 1>?a2=|a|=-a By adding (a+1) for both side we get:
a+1-?(a + 1)2 ? a> 1 so |?2| > 1
From Eq. 3-5 and by proposition (3-1) we obtain
(5)
(6)
Proof (2): Since a < ?1 then we have ??a>1 by
0 | = |?0| > 1. Hence:
proposition (2-1and 2-3) |? is repelling fixed point.
2
1
(00)
a+1
(
b)
b
a+1
Proposition (4-2): if |a| < 1then, MJa,b has attracting fixed point:
(00)
and saddle fixed point:
is a saddle fixed point. Proof (2): Since:
?a2 + a + 1 < ?(a + 1)2 =|a + 1| = ?(a + 1).
2299J. Eng. Applied Sci., 13 (Special Issue 1): 2296-2300, 2018
Patidar,V and K.K Sud, 2005. Bifurcation and chaos in simple jerk dynamical systems. Pramana. J. Phys., 64, 75-93.
Posch, H A, W G Hoover and F J Vesely, 1986. Canonical dynamics of the nose oscillator: stability, order, and chaos, Phys. Rev. A., 33:4253-4263253.
Sprott, J C,1994. Some simple chaotic flows. Phys. Rev. E., 50: R647-R650 R647.
hence by adding (a+1) for both side, we get:
a + 1 + ?a2 + a + 1 < 1 (6) ?a2 + a + 1 > ?a2=|a|= a So,??a2 + a + 1 < ?a By adding (a+1) for both side we get:
a+1-?(a + 1)2 ? a < 1 (7) From Eq. 6 and 7 and by proposition (3-1) we obtain
a+1
(
)
b a+1
b
is attracting fixed point.
CONCLUSIONS
In this study, a simplified version of Jerk map is presented. Different properties of dynamical behavior is acquired by replacing three dimensional systems to two dimensions by adding new parameters that have the same properties.