Chaotic Properties of the Modified Hénon Map
1 Samah Abd Alhadi Abbas and 2Hussein Alawi Jasim
1 Department of Computer Science, College of Science for Women,
University of Babylon, Babylon, Samah, Iraq
2 Department ofMathematics, University of Babylon, Babylon, Iraq
Abstract: In this study, a dynamical system of modified Hénon map on two dimension with the form MHa,b (yx) =
( 1+ ax + cos 2?y
by
) is studied. We find some general properties, and we show some chaotic properties of it. The
proposed paper prove that the modified Hénon map has positive Lypaunov exponent and sensitivity dependence to initial condition. Fpr applying the suggested scheme, Mat lab programs are used to draw the sensitivity of modified Hénon map and compute the Lyapunov exponent.
Key words: modified the Hénon map, fixed point, attracting- expanding area, Lyapunov exponent, Sensitive dependence
on initial conditions
INTRODUCTION
There are several definitions for chaos were proposed .When the system is sensitive to initial condition on its domain or has positive Lyapanov exponent at each point in its domain then this system will be chaotic (Denny,1992). Chaotic behavior of lows dimensional map and flows has been generally considered and described (Sprott and Chaos, 2003). Previously, the French space expert -mathematician Michel–Henon was scanned for simple two-dimensional squeezing extraordinary properties of more complication system the result was family of the form:
contraction is Independent of x and y. For b = 0, the Hénon map reduces to the quadratic map which Follows period doubling route to chaos. Bounded solutions exist for the Hénon map Over a range of a and b values. Hénon map had two fixed points. Which can be either attracting, saddle or repelling points depending on the choice of parameters (a, b).
Hénon map had two fixed points. Which can be either attracting, saddle or repelling points depending on the choice of parameters (a, b) (Shameri, 2012) In this research, we introduce a new map in two dimension, we will call it the modified Hénon map as:
x 1 ? ax2 + y
y)=(
)
Ha,b (
x 1 + ax + cos 2?y
y) =(
)
MHa,b (
bx
by
Where ??, ?? are parameter and real number (Denny ,1992).This is a nonlinear two dimensional map, which can also be written as a two-step recurrence relation:
Xn + 1 = 1 ? ax2n + byn ? 1
The parameter b is a measure of the rate of area contraction, and the Hénon map is the most general two- dimensional quadratic map with the property that the
Preliminaries: Let I: Rn ? Rn be a map, we say I is C ?if its P-th partial derivatives exist and continuous for all P ? Z+, and it is called diffeomorphism if it is one – to – one , onto C?and its inverse is C?, let W be subset of R2, and µ be any element in R2 concider G: W ? R2 be a map. Furthermore assume that the first partials on R2 by DG:
Corresponding Author: Samah Abd Alhadi Abbas, Department Computer Science, College of Science for Women, University of
Babylon, Samah, Iraq
1095Res. J. Applied Sci. 11(10): 1095-1101, 2016
Since bs = s and b ? 1 then s = 0. Also since 1 + an +
?f1 ?x ?f2 ?x
?g1 ?y ?g2 ?y
cos 2?s = ?? ?????? ?? ? 0 ??????? n = ?2/a ? 1.But this contradiction So:
(u°) (u°)
(u°) (u°)
(u°) =
)
(
[yx]=[
s] n
For all u° ? R2 the determinate of DG(u°) is called Jacobian of G at u° and denoted byJG(u°) = detDG(u°). So G is said to be area expanding at u°if|Det DG(u°)| > 1, G is said to be area contracting at u° if |Det DG(u°)| < 1 .Let B be n × n matrix the real number ? is called Eigen value
Such that if a?0 then:
[ 2/10? a]
is the unique fixed point.
Proposition (3.2): If a ? 1 , b = 1 then MHa,b has infinite fixed point.
Proof: By definition of fixed point:
of B. The point (pq) is called fixed point if G (
q ) = ( pq) it is p
repelling fixed point if ?1and ?2 > 1 in absolute value , and it is an attracting fixed point if ?1 and ? 2 < 1 in absolute value B ? GL(2, Z)with det (B) = ±1 is called hyperbolic matrix if |?1| ? 1 where ?1 are the eigenvalue (Denny, 1992).
MATERIALS AND METHODS
General properties of modified Hénon map: In this study, we find the fixed point and study the general properties of modified Henon map (one to one,
[ 1+ ax + cos 2?y
by
]=[
x y]
Since b = 1 then y = y, 1 + ax ? cos2?y = x ? x = ?1 ?
cos2?y
?? ? 1. Then MHa,b has infinite fixed point:
?1 ? cos2?y y]
a?1
[yx]=[
onto,c?,and invertible)
make
which
it
Proposition (3.3): If a = 1 , b ? 1 then MHa,b has no fixed point
Proof: By definition of fixed point:
diffeomorphism and find the value of a, b which MHa,b has area contracting or expanding.
Proposition (3.1): If a ? 1 and b ? 1 then modified Hénon map MHa,b has unique fixed point.
Proof: By the definition of fixed point, we get:
[ 1+ x + cos 2?y
by
]=[
x y]
Since b ? 1 then by = y ? y = o, x ? x = 1 + cos2?by then Ha,b has no fixed point.
Proposition (3.4): The Jacobian of the modified Hénon map MHa,b is ab
Proof:
x 1 + ax + cos 2?y
]=[
y)=[
x y]
MHa,b (
by
Then by = y since b ? 1 then y = 0 since 1 + ax + cos 2?(0) = x then ??x(a ? 1) = ?2 ? x = ?2/(a ? 1) then
[ 2/10? a]
?f1 ?x ?f2 ?x
?f1 ?y ?f2 ?y
y ) = [ (x
]= [
a ? ?2? sin 2?y 0b]
DHa,b
Then ??= detDHa,b(v°) = ab
Proposition (3.5): Let ??????,?? be modified Hénon map
? If |??| < 1 and |??| < 1 then MHa,bis area contracting map
and |??| < 1.
? If |??| > 1 and |??| > |??| > 1.
x ns]. Also by the definition of
is the fixed point Let [ fixed point
y]?[
1 + an + cos 2?s
bs
MHa,b (
s )=[ n
]=[
s] n
1096Res. J. Applied Sci. 11(10): 1095-1101, 2016