Chaotifying 2-D piecewise linear maps via a piecewise linear controller function
A simple method for chaotifying piecewise linear maps of the plane using a piecewise linear controller function is given. A domain of chaosin the resulting controlled map was determined exactly and rigorously.
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irk-123456789-1749492021-01-29T01:27:38Z Chaotifying 2-D piecewise linear maps via a piecewise linear controller function Elhadj, Z. Sprott, J.C. A simple method for chaotifying piecewise linear maps of the plane using a piecewise linear controller function is given. A domain of chaosin the resulting controlled map was determined exactly and rigorously. Наведено простий метод хаотизацiї кусково-лiнiйних вiдображень площини за допомогою кусково-лiнiйної функцiї керування. Множину хаосу для отриманого керованого вiдображення визначено точно i строго. 2010 Article Chaotifying 2-D piecewise linear maps via a piecewise linear controller function / Z. Elhadj, J.C. Sprott // Нелінійні коливання. — 2010. — Т. 13, № 3. — С. 328-335. — Бібліогр.: 22 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/174949 517.9 en Нелінійні коливання Інститут математики НАН України |
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A simple method for chaotifying piecewise linear maps of the plane using a piecewise linear controller
function is given. A domain of chaosin the resulting controlled map was determined exactly and rigorously. |
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Article |
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Elhadj, Z. Sprott, J.C. |
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Elhadj, Z. Sprott, J.C. Chaotifying 2-D piecewise linear maps via a piecewise linear controller function Нелінійні коливання |
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Elhadj, Z. Sprott, J.C. |
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Elhadj, Z. |
| title |
Chaotifying 2-D piecewise linear maps via a piecewise linear controller function |
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Chaotifying 2-D piecewise linear maps via a piecewise linear controller function |
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Chaotifying 2-D piecewise linear maps via a piecewise linear controller function |
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Chaotifying 2-D piecewise linear maps via a piecewise linear controller function |
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Chaotifying 2-D piecewise linear maps via a piecewise linear controller function |
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chaotifying 2-d piecewise linear maps via a piecewise linear controller function |
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Інститут математики НАН України |
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2010 |
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Chaotifying 2-D piecewise linear maps via a piecewise linear controller function / Z. Elhadj, J.C. Sprott // Нелінійні коливання. — 2010. — Т. 13, № 3. — С. 328-335. — Бібліогр.: 22 назв. — англ. |
| series |
Нелінійні коливання |
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AT elhadjz chaotifying2dpiecewiselinearmapsviaapiecewiselinearcontrollerfunction AT sprottjc chaotifying2dpiecewiselinearmapsviaapiecewiselinearcontrollerfunction |
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2025-07-15T12:05:10Z |
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2025-07-15T12:05:10Z |
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1837714491231436800 |
| fulltext |
UDC 517 . 9
CHAOTIFYING 2-D PIECEWISE LINEAR MAPS
VIA A PIECEWISE LINEAR CONTROLLER FUNCTION
ХАОТИЗАЦIЯ ДВОВИМIРНИХ КУСКОВО-ЛIНIЙНИХ ВIДОБРАЖЕНЬ
ЗА ДОПОМОГОЮ КУСКОВО-ЛIНIЙНОЇ ФУНКЦIЇ КЕРУВАННЯ
Z. Elhadj
Univ. Tébéssa
(12000), Algeria
e-mail: zeraoulia@mail.univ-tebessa.dz
zelhadj12@yahoo.fr.
J. C. Sprott
Univ. Wisconsin
Madison, WI 53706, USA
e-mail: sprott@physics.wisc.edu.
A simple method for chaotifying piecewise linear maps of the plane using a piecewise linear controller
function is given. A domain of chaos in the resulting controlled map was determined exactly and rigorously.
Наведено простий метод хаотизацiї кусково-лiнiйних вiдображень площини за допомогою кус-
ково-лiнiйної функцiї керування. Множину хаосу для отриманого керованого вiдображення ви-
значено точно i строго.
1. Introduction. A large number of physical and engineering systems have been found to be
governed by a class of continuous or discontinuous maps [1, 4, 5, 7 – 10, 13, 14] where the
discrete-time state space is divided into two or more compartments with different functional
forms of the map separated by borderlines [14 – 17]. The theory of discontinuous maps is in
its infancy, with some progress reported for 1-D and n-D discontinuous maps in [1, 8, 9, 11,
12, 17], but these results are restrictive and cannot be obtained in the general n-dimensional
context [12]. On the other hand, there are many works that focus on the chaotic behavior of
discrete mappings. For example, they have been studied as control and anti-control (chaoti-
fication) schemes using Lyapunov exponents [2, 3, 18, 20] to prove the existence of chaos in
n-dimensional discrete dynamical systems with the goal of making in some way an originally
non-chaotic dynamical system chaotic or enhancing the chaos already existing in such a system.
The goal of this work is to present a simple method for chaotifying an arbitrary 2-D piecewi-
se linear map (continuous or not) using a simple piecewise linear controller function, which
allows one to determine exactly and rigorously which portion of the bifurcation parameter
space is characterized by the occurrence of chaos in the resulting controlled map using the
standard definition of the Lyapunov exponents as the test for chaos.
Consider an arbitrary piecewise-linear map f : D → D, where D = D1∪D2 ⊂ R
2, defined
by
c© Z. Elhadj, J. C. Sprott, 2010
328 ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 3
CHAOTIFYING 2-D PIECEWISE LINEAR MAPS VIA A PIECEWISE LINEAR CONTROLLER FUNCTION 329
Xk+1 = f (Xk) =
{
AXk + b if Xk ∈ D1,
BXk + c if Xk ∈ D2,
(1)
where A =
(
a11 a12
a21 a22
)
and B =
(
b11 b12
b21 b22
)
are 2× 2 real matrices, and b =
(
b1
b2
)
and
c =
(
c1
c2
)
are 2× 1 real vectors, and Xk =
(
xk
yk
)
∈ R
2 is the state variable.
The Lyapunov exponents of a 2-D dynamical system are defined as follows.
Let us consider the system
Xk+1 = f(Xk), Xk ∈ R
2, k = 0, 1, 2, . . . , (2)
where the function f : R2 −→ R
2 is the vector field associated with system (2). Let J (Xk) be
its Jacobian evaluated at Xk ∈ R
2, k = 0, 1, 2, . . . , and define the matrix
Tn (X0) = J (Xn−1) J (Xn−2) . . . J (X1) J (X0) . (3)
Moreover, let Ji(X0, n) be the modulus of the ith eigenvalue of the nth matrix Tn (X0) , where
i = 1, 2 and n = 0, 1, 2, . . . .
Now, the Lyapunov exponents for a two-dimensional discrete-time system are defined by
li (X0) = lim
n−→+∞
ln
(
Ji(X0, n)
1
n
)
, i = 1, 2. (4)
Roughly speaking, chaotic behavior implies sensitive dependence on initial conditions, with at
least one positive Lyapunov exponent. Based on this definition, we give in the next section a
rigorous proof of chaos in the resulting controlled map (7) obtained below via a simple piecewi-
se linear controller function applied to the map (1). While many algorithms for calculating the
Lyapunov exponents would give spurious results for piecewise-linear discontinuous maps, the
algorithm used here and given in [21] works for such cases. It essentially takes a numerical
derivative and gives the correct result provided that is taken to ensure that the perturbed and
unperturbed orbits lie on the same side of the discontinuity. This may require an occasional
small perturbation into a region that is not strictly accessible to the orbit.
2. The chaotification method using a piecewise linear controller function. The main idea of
the chaotification method presented in this work is to introduce a piecewise linear controller
function in such a way, such that the two (in both partitions) system matrices of the resulting
controlled piecewise linear system have the same trace and determinant (invariants) and hence
the same eigenvalues. Indeed, the controlled map is given by
g (Xk) =
{
AXk + b if Xk ∈ D1
BXk + c if Xk ∈ D2
+ U (Xk) , (5)
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 3
330 Z. ELHADJ, J. C. SPROTT
where the controller function U (Xk) is defined by
U (Xk) =
(b11 + b22 − a11 − a22)xk
(−a12a21 + b11a22 − b11b22 + b12b21 + a22b22 − a222
a12
)
xk
if Xk ∈ D1,
(
0
0
)
if Xk ∈ D2.
(6)
The controlled system (5) is now given by
g (Xk) =
{
QXk + b if Xk ∈ D1,
BXk + c if Xk ∈ D2,
(7)
where the matrix Q is given by
Q =
b11 + b22 − a22 a12
b11a22 − b11b22 + b12b21 + a22b22 − a222
a12
a22
. (8)
The Jacobian matrix of the controlled map (7) is given by
J (Xk) =
{
Q if Xk ∈ D1,
B if Xk ∈ D2.
(9)
It is shown in [22] that if we consider a system xk+1 = f (xk) , xk ∈ Ω ⊂ R
n, such that
∥
∥f ′(x)
∥
∥ =
√
λmax (f(x)T f(x)) ≤ N < +∞, (10)
with a smallest eigenvalue of f(x)T f(x) that satisfies
λmin
(
f ′(x)T f ′(x)
)
≥ θ > 0, (11)
where N2 ≥ θ, then, for any x0 ∈ Ω, all the Lyapunov exponents at x0 are located inside
[
ln θ
2
, lnN
]
, that is,
ln θ
2
≤ li (x0) ≤ lnN, i = 1, 2, . . . , n, (12)
where li (x0) are the Lyapunov exponents for the map f and ‖·‖ is the Euclidian norm in R
n.
We remark that J (Xk) given in (9) is not well-defined due to the discontinuity, but, since B and
Q have the same eigenvalues, one has that ‖Q‖ = ‖B‖ =
√
λmax (BTB). Because
BTB =
b211 + b221 b11b12 + b21b22
b11b12 + b21b22 b212 + b222
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 3
CHAOTIFYING 2-D PIECEWISE LINEAR MAPS VIA A PIECEWISE LINEAR CONTROLLER FUNCTION 331
is at least a positive semi-definite matrix, all its eigenvalues are real and positive, i.e.,
λmax
(
BTB
)
≥ λmin
(
BTB
)
≥ 0.
Hence the eigenvalues of BTB are given by
λmax
(
BTB
)
=
1
2
b211 +
1
2
b212 +
1
2
b221 +
1
2
b222 +
1
2
√
d,
(13)
λmin
(
BTB
)
=
1
2
b211 +
1
2
b212 +
1
2
b221 +
1
2
b222 −
1
2
√
d,
where
d =
(
(b11 + b22)
2 + (b12 − b21)
2
)(
(b12 + b21)
2 + (b11 − b22)
2
)
> 0 (14)
for all b11, b12, b21, and b22. Condition (10) gives ‖f ′(x)‖ = ‖B‖ = ‖Q‖ =
√
λmax (BTB) =
= N < +∞, because B and Q have the same eigenvalues. Condition (11) gives the inequality
θ2 −
(
b211 + b212 + b221 + b222
)
θ + (b11b22 − b12b21)
2 ≥ 0 (15)
with the condition
θ <
b211 + b212 + b221 + b222
2
. (16)
Since the discriminant of (15) is equal to d > 0, (11) holds if
θ ≥ λmax
(
BTB
)
, or θ ≤ λmin
(
BTB
)
. (17)
The condition θ ≥ λmax
(
BTB
)
is impossible because of condition (16), so that θ must satisfy
the condition
θ < λmin
(
BTB
)
=
b211 + b212 + b221 + b222 −
√
d
2
. (18)
Now, if
2 < b211 + b212 + b221 + b222 < (b11b22 − b12b21)
2 + 1,
(19)
|b11b22 − b12b21| > 1,
then λmin
(
BTB
)
> 1, i.e., θ = 1, and one has
0 < li (x0) ≤ lnN, i = 1, 2, (20)
i.e., the controlled map (7) converges to a hyperchaotic attractor for all parameters b11, b12, b21,
and b22 satisfying (19).
Finally, the following theorem is proved.
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 3
332 Z. ELHADJ, J. C. SPROTT
Theorem 1. The piecewise linear controller function (6) makes the map (1) chaotic in the
following case:
2 < b211 + b212 + b221 + b222 < (b11b22 − b12b21)
2 + 1,
(21)
|b11b22 − b12b21| > 1.
Finally, we note that Theorem 1 does not guaranty the boundness of the resulting controlled
map (7). This problem is still open for general piecewise-linear maps and flows.
3. Example. In this section, we make chaotic the following piecewise linear map using the
above method:
f (xk, yk) =
(
xk − αyk + 1,
γxk
)
if yk ≥ 0,
(
xk + αyk + 1
−βxk
)
if yk < 0,
(22)
where α, β, and γ are bifurcation parameters. The map (22) is a special case of the map (1)
since one can rewrite it as
f (Xk) =
{
AXk + b, if yk ≥ 0,
BXk + c, if yk < 0,
where
A =
(
1 −α
γ 0
)
, B =
(
1 α
−β 0
)
, and b = c =
(
1
0
)
,
and the two sub regions are D1 =
{
(xk, yk) ∈ R
2/ yk ≥ 0
}
and D2 =
{
(xk, yk) ∈ R
2/ yk < 0
}
.
Thus, the resulting controlled map is given by
g (xk, yk) =
(
1 −α
β 0
)(
xk
yk
)
+
(
1
0
)
if yk ≥ 0
(
1 α
−β 0
)(
xk
yk
)
+
(
1
0
)
if yk < 0
=
1− α |yk|+ xk
βxk sgn (yk)
, (23)
which is the so-called discrete butterfly presented in [19], where sgn (·) is the standard signum
function that gives ±1 depending on the sign of its argument. For the controlled map (23),
condition (21) becomes
|α| > max
(
|β|
√
β2 − 1
,
1
|β|
)
, |β| > 1. (24)
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 3
CHAOTIFYING 2-D PIECEWISE LINEAR MAPS VIA A PIECEWISE LINEAR CONTROLLER FUNCTION 333
For illustration, assume that β < −1. Then we remark that
|β|
√
β2 − 1
=
−β
√
β2 − 1
<
1
|β| =
−1
β
,
and thus conditions (24) become
|α| > −1
β
, β < −1.
Using the obtained analytical results, Fig. 1 shows that for −2 ≤ β < −1, and −0.1 −
− 1
β
< α <
−1
β
+ 0.4, the controlled map (23) converges to bounded hyperchaotic attractors
or unbounded orbits for α > − 1
β
. In this figure, unbounded solutions, periodic solutions, and
chaotic solutions are shown in the αβ-plane for the controlled map (23), where we use 5000
different initial conditions and 106 iterations for each point. A chaotic attractor for the case
with α = 0.6 and β = −2 is shown in Fig 2.
On the other hand, it is necessary to verify the hyperchaoticity of the attractors by calcula-
ting both Lyapunov exponents using the formula: l1 (x0) + l2 (x0) = ln | det(J)| = ln |αβ|
averaged along the orbit where det(J) is the determinant of the Jacobian matrix. The result is
shown in Fig. 3 for 0.4 ≤ α ≤ 0.9, with β = −2.
Fig. 1. Regions of dynamical behaviors in the αβ-plane for the
controlled map (23) with −2 ≤ β < −1 and
−0.1− 1/β < α < −1/β + 0.4.
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 3
CHAOTIFYING 2-D PIECEWISE LINEAR MAPS VIA A PIECEWISE LINEAR CONTROLLER FUNCTION 335
4. Conclusion. A new simple chaotification method for piecewise linear maps of the plane
via a piecewise linear controller function was presented. A rigorous proof of chaos in the resulti-
ng controlled map using the standard definition of the largest Lyapunov exponent was also
given.
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Received 09.12.08,
after revision — 15.04.09
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 3
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