Asymptotic Synchronization of Modified Logistic
Hyper-Chaotic Systems and its Applications
Shu-Ming Chang
∗Ming-Chia Li
†Wen-Wei Lin
‡Abstract
In this paper, we propose a map, Modified Logistic Map (MLM), and give a theoretical proof to show that the MLM is a chaotic map of Devaney’s defini-tion. The MLM is not only no chaotic window but also uniformly distribution in [0, 1] for γ≥ 4. Furthermore, basing on MLMs, we establish a Modified Logistic Hyper-Chaotic System (MLHCS) and apply to develop a symmetric cryptogra-phy algorithm, Asymptotic Synchronization of Modified Logistic Hyper-Chaotic System (ASMLHCS). In numerical simulation, we analyze spectrum of wave-forms of the sequence generated from MLM to express that the orbit wave-forms uniform distribution in [0, 1]; on the other hand, we compute Poincar`e recur-rences to indicate that the MLM possesses the positive topological entropy.
∗Department of Mathematics, National Tsing Hua University, Hsinchu, 300, Taiwan.
Email: [email protected]
†Department of Applied Mathematics, National Chiao Tung University, Hsinchu, 300, Taiwan.
Email: [email protected]
‡Department of Mathematics, National Tsing Hua University, Hsinchu, 300, Taiwan.
1
Introduction
Logistic map of the form
x = γx(1− x) (1.1)
is an essential quadratic map in discrete dynamics, which has been extensively studied not only theoretically but also numerically by mathematicians, physicists and biolo-gists. It is well-known that the logistic map has chaotic behavior for 3.57 < γ ≤ 4 [6, 7, 9]. However, the set of chaotic windows is open and dense [4], that is, the set of visualized chaos is small and sparse for γ ∈ (3.57, 4). On the other hand, logistic map is also proved to be chaotic on an invariant Cantor set for all γ > 4 which is unstable [15, 11].
Pecora and Carrol [12] have shown that a chaotic system (respond system) can be synchronized with a separated chaotic system (drive system) provided that the conditional Lyapunov exponents of the difference equations between drive and re-sponse systems are all negative. In the secure-communication, the chaotic signals are used as masking streams to carry information which can be recovered by chaotic synchronization behavior between transmitter (drive system) and receiver (respond system).
Sobhy and Shehata [18] attacked the chaotic secure-system by reconstructing the map with the output sequence. Because of the unique map pattern of each single-chaotic system, it is easy to distinguish with the other single-chaotic systems and rebuilding the equations. The MatLab routines are used to approximate the parameters. Once parameters are known, the secure information is recovered.
Therefore, many papers focus in enhancing the complexity of the output sequence. Heidari-Bateni and McGillem [8] use a chaotic map to initialize another chaotic map. Utilizing a multi-system with serval chaotic maps are switched by the specific mech-anism [10] or combined into a chaotic system chain [20]. Peng et al. [14] combine
with above two types.
In this paper we will propose a robust map, Modified Logistic Map (MLM). MLM is a chaotic map proved by the definition of Devaney’s chaos and invariant in [0, 1]. Furthermore, MLM is no window. In numerical computation, utilizing computing Poincar´e recurrences to indicate MLM has chaotic phenomena. Basing on two MLMs we establish Modified Logistic Hyper-Chaotic System (MLHCS) and apply consisting of two MLHCSs to develop a symmetric cryptography algorithm, Asymptotic Syn-chronization of Modified Logistic Hyper-Chaotic System (ASMLHCS). There are two parts in ASMLHCS, asymptotic synchronization phase and Encrtyption/Decryption phase. The details will be introduced later sections.
The rest of the paper is organized as follows. In Section 2, we present Modified Logistic Map and prove that MLM is a chaotic map. In Section 3, to utilize spectrum of waveforms and Poincar´e recurrences to study the properties of the modified logistic mao. In Section 4, Modified Logistic Hyper-Chaotic System and its applications will be proposed. In the Section 5, we apply consisting of two MLHCSs to develop a symmetric cryptography algorithm. Finally concluding remarks are given in Section 6.
2
Modified Logistic Map: Devaney’s chaos
For γ≥ 0, define a Modified Logistic Map (MLM) fγ(x) : [0, 1]→ [0, 1] by
fγ(x) = γx(1− x) (mod 1), if x ∈ [0, 1] \ (η1, η2), γx(1−x) (mod 1) γ 4 (mod 1) , if x∈ (η1, η2), (2.1) where η1 = 12 − q 1 4 − [γ 4] γ and η2 = 1 2 + q 1 4 − [γ 4]
γ , in which [z] is the greatest integer
less than or equal to z.
For γ ≤ 4, the MLM of Eq. (2.1) is the same as the classical logistic map. It is well-known that the classical logistic map has chaotic behavior for 3.57 < γ ≤ 4.
That is, the sequence generated by the classical logistic map never settles down to a fixed point or a periodic orbit, instead of the aperiodic long-time behavior. From the bifurcation diagram [5] we see that attractors route from period doubling to chaos (strange attractor). The range of strange attractors becomes larger and larger, whenever γ increases from 3.57 to 4. For γ = 4, the length of strange attractor is one. However, the chaotic window can happen for γ ∈ (3.57, 4). In fact, the attractor of a chaotic window visually forms periodic points which has been proved to be open and dense. For instance, we observe numerically that there are very narrow intervals of γ near 3.63, 3.73 or 3.83 which form chaotic windows.
Since the MLM has a large range of visually chaotic windows for γ < 4 which is unusable as a chaotic mask in secure-communication, in following we shall show that the MLM has chaotic behavior according to Devaney’s definition [6] for γ ≥ 4. In these cases the lengths of strange attractors are always one and the chaotic behavior is topologically equivalent to that of γ = 4. In other words, for γ ≥ 4 the MLM has no chaotic windows which produces a large key space in secure-communication.
Figure 2.1 plots a modified logistic map with γ = 10 which is a piecewise monotonic transformation.
Fig. 2.1 is near here.
Definition 2.1. Let f : I → I be a map, where I is a closed interval. We say that f exhibits Devanvey’s chaos on I if the following conditions are satisfied:
1. the set of periodic points is dense in I;
2. the map f is topologically transitive, i.e., for any given pair of nonempty open sets U and V in I, there is a positive integer n such that fn(U)∩ V 6= ø; and
3. the map f has sensitive dependence on initial conditions, i.e., there exists α > 0 such that for any x ∈ I and any ǫ > 0 , there are y ∈ I and n ∈ N such that
|x − y| < ǫ and |fn(x)
− fn(y)
| > α.
We also need the following definition. For a C3 map g : I → I, where I is an interval, the Schwarzian derivative of g is defined by
Sg(x) = g′′′ (x) g′(x) − 3 2 g′′ (x) g′(x) 2 for x∈ I with g′
(x)6= 0. By using the chain rule, one has that Sg < 0 implies Sg2 < 0.
Hence,
if Sg < 0, then Sgn < 0 for all n≥ 1. (2.2)
Moreover, Sg < 0 implies that g′ cannot have a positive local minimum or a negative
local maximum. Indeed, if c is a critical point of g′
, then gg′′′′(c)(c) = Sg(c) < 0 and hence
g′′′
(c) and g′
(c) have opposite signs. Therefore, by continuity of g′
, we have that if g′
6= 0 and Sg < 0 on [a, b], then for any x∈ (a, b),
either g′ (x) > min{g′ (a), g′ (b)} > 0 or g′ (x) < max{g′ (a), g′ (b)} < 0. (2.3)
Return to our study on the model fγ in (2.1). We show the existence of Devaney’s
chaos.
Theorem 2.1. If γ ≥ 4, then fγ exhibits Devaney’s chaos on [0, 1].
Proof. Without loss of generality, we may assume that 4 < γ < 8. For convenience, we write f = fγ. Then there are four fixed points 0 < A < B < C. There is
a point B− such that A < B− < B and f (B−) = B. Let k = [γ/4] − 1. Let
J = {x ∈ [B−, B] : fn(x) ∈ [B−, B] for some integer n ≥ 1}. For x ∈ J, define
τ (x) = min{n ∈ N : fn(x)
∈ [B−, B]}. Then τ(x) is well defined.
First, we claim
|(fτ(x))′
For n≥ 1, let In = {x ∈ (1/2, B] : τ(x) = n} and ˆIn = {x ∈ [B−, 1/2) : τ (x) = n}.
Then J = ∪∞
n=1(In∪ ˆIn), I1 = {B}, ˆI1 = {B−}, and |f′(x)| > 1 for x ∈ I1 ∪ ˆI1.
Consider n≥ 2. The definition of f implies that In consists of finite disjoint intervals,
say Jn,i’s, and f maps each interval homeomorphically onto [B−, B]. By the mean
value theorem, there exists a point in each components of [1/2, B]\In at which the
absolute value of (fn)′
is greater than one. The discontinuity points are sent to the fixed point 0 by f2. Since S
f < 0, by (2.2) we have Sfn < 0 and hence by (2.3)
applied to fn, we obtain that the absolute values of (fn)′
at the end points of each interval Jn,j are greater than 1. By (2.3) again, we get that |(fn)′(x)| > 1 for all
x ∈ In. By using the same argument as above, we have that |(fn)′(x)| > 1 for all
x∈ ˆIn. The desired claim follows.
Second, we claim that for every x ∈ [0, 1] whose orbit does not go through 1/2, there exists a positive integer nx such that
|(fnx)′(x)| > 1. (2.5)
If x ∈ J, claim (2.5) follows (2.4) by taking nx = τ (x). If x ∈ [B−, B]\J, fn(x) /∈
[B−, B] for all n ≥ 1, hence |fn(x)| > 1 for all n sufficiently large since |f′| > 1 on
[0, 1]\[B−, B]. If x6∈ [B−, B], we can let n = 1.
Third, we claim that for any nonempty open set U ⊂ [0, 1], there exists a positive integer n such that
fn(U)
⊃ [0, 1]. (2.6)
Let U be an interval in [0, 1]. Then there are a positive integer n and a subinterval U0 ⊂ U such that fn(U0)⊂ J. For convenience, we denote R(x) = fτ(x)(x) for x∈ J.
The claim (2.4) says that R expands the lengths of intervals in J and hence there is a k > 0 and a subinterval V0 ⊂ fn(U0) such that Rk(V0) contains a discontinuity point
of R. Thus there exists m > 0 such that p ∈ fm(V
fm+ℓ(V
0) = [0, 1] for some ℓ > 0. Let E be the point in the interval (B, C) such that
f (E) = 0. Since f maps [B, E] homeomorphically onto [0, B], there exists a unique d∈ [B, E] such that f(d) = 1/2.
Then fm+2ˆℓ(V
0) ⊃ [1/2, d] for some ˆℓ > 0. Thus, there exists ˆℓ > 0 such
that fm+2ˆℓ(V
0) ⊃ [1/2, d]. Since f2([1/2, d]) = f ([1/2, 1]) = [0, 1], fm+2ˆℓ+2(V0) ⊃
f2([1/2, d]) = [0, 1]. The proof of the desired claim is complete.
Finally, we are in position to obtain the three properties of Devaney’s chaos. Let U be any nonempty open interval in [0, 1]. Then there exist a nonempty open interval V and a closed interval W such that V ⊂ W ⊂ U. By claim (2.6), there exists a positive integer n such that fn(V )
⊃ [0, 1] and hence fn(W )
⊃ W . By the fixed point theorem, fn has a fixed point in W . Therefore, f has a periodic point in
W and so in U. We have proved that the set of periodic points is dense in [0, 1]. The claim (2.6) immediately implies that f is topologically transitive. For sensitive dependence of f , we take η = 1
4. Let x ∈ [0, 1] and ǫ > 0 be arbitrary. Take U to
be the interval (x, x + ǫ
2) or (x− ǫ
2, x) provided it is well defined. By claim (2.6), we
have fn(U)
⊃ [0, 1]. Thus there exists y ∈ U such that |fn(x)
− fn(y)
| > 1
4 = η. The
proof of the theorem is complete.
3
Numerical study of MLM
In this section, we present numerical experiments on MLM by computing spectrum of waveforms to observe that no chaotic window occurs and orbits form uniform distributions in [0, 1]. On the other hand, we compute Poincar´e recurrences to testify that the MLM possesses the positive topological entropy, which shows that the MLM is a chaotic map.
3.1
Spectrum of waveforms
In order to characterize the motion of MLM we compute spectrum of waveforms of the system (2.1) with different γ. The spectrum of waveform is computed using FFT subroutine in MATLAB and the spectrum distribution is displayed by the frequency versus log10(|fft(·)|2). Here FFT subroutine is the discrete Fourier transform,
some-times called the finite Fourier transform, is a Fourier transform widely employed in signal processing and related fields to analyze the frequencies contained in a sampled signal. Therefore, we generate a sequence from MLM to be a signal processing and assume data sampled at 1,000 Hz.
Figures 3.1 and 3.2 present attractors of (2.1) and plot the spectrum of waveforms at γ = 5.9 and 10.8, respectively. However, not only at γ = 5.9 or 10.8 but also at the other γ ≥ 4 we observe that all attractors form uniform distributions (continuous spectrum??) in [0, 1] and the spectrum of waveforms reveals to contain any frequency in the signal processing. Moreover, in the numerical viewpoint, there is no chaotic window of MLM.
Fig. 3.1 is near here. Fig. 3.2 is near here.
3.2
Poincar´
e recurrences
We wish to briefly recall some of the definitions about the fractal dimension for Poincar´e recurrences which are main indicators and characteristics of the repetition of behavior of dynamical systems in time. It is to study statistical properties of the quantity τ (x, U), the first return time of the orbit through x into a set U (see [19] and references therein). Typical motions in dynamical systems repeat their behavior in time. Simplicity or complexity of orbits often can be displayed in terms of Poincar´e
recurrences. Furthermore, Poincar´e recurrences also could be to describe what hap-pens for the map in the regions of the phase space where regular or chaotic motions [16].
We adopt another point of view to instead of looking at the mean return time or at the return time of points [1]. We define the smallest possible return time into U by taking the infimum over all return times of the points of the set. We consider a dynamical system (Rd, f ) with f being continuous and d ∈ N. Let A ⊂ Rd be an
f -invariant subset. We follow the general Carath´eodory construction and consider covers of A by open balls. We denote by Bǫ the class of all finite or countable open
covering of A by balls of diameter less than or equal to ǫ. Let the Poincar´e recurrence for an open ball U ⊂ Rd be
τ (U) = inf{τ(x, U) : x ∈ U}, where τ (x, U) = min{n ∈ N : fn(U)
∩ U 6= ∅} is the first return time of x ∈ U. By convention we set the return time τ (x, U) to be infinity provided that the point x never comes back to U. Given C ∈ Bǫ and α, q ∈ R, we consider the sum
M(α, q, ǫ, C) =X
U ∈C
exp (−qτ(U)) |U|α, (3.1)
where, as usual, |U| stands for the diameter of the set U. Now, define M(α, q, ǫ) = inf{M(α, q, ǫ, C) : C ∈ Bǫ}.
The limit
M(α, q) = lim
ǫ→0M(α, q, ǫ)
has an abrupt change from infinity to zero as, for a fixed q, one varies α from zero to infinity. The transition point defines a function αc(q) as follows,
This function is said to be the spectrum of dimensions for Poincar´e recurrences. Moreover, we let q0 := sup{q : αc(q) > 0}. Then, roughly speaking, q0 is the smallest
solution of the equation α(q) = 0. The number q0is called the dimension for Poincar´e
recurrences (see [3] and references therein).
For computational purpose [2], we will derive an asymptotic relation between τ (U), ln ǫ and q0. For the sake of simplicity, we assume that M(αc(q), q) is a finite
number. Then the partition function (3.1) behaves as follows M(αc(q), q, ǫ,C) = X U ∈C exp(−qτ(U))|U|αc(q) ∼ 1, i.e., 1 N X U ∈C exp(−qτ(U))|U|αc(q) ∼ 1 N, (3.2)
where N is the number of elements in the cover C. But we know that if ǫ is small enough that 1/N behaves like ǫb, where b is the box dimension of the set A (provided
that it exists and is equal to the Hausdorff dimension [13]).
Therefore, we may rewrite the asymptotic equality (3.2) as follows hexp(−qτ(U)|U|αc(q)i ∼ ǫb,
where the brackets h·i denote the mean value. For q = q0 we have
hexp(−qτ(U)i ∼ ǫb. (3.3)
Here Eq. (3.3) can be treated as the definition of the dimension q0 for Poincar´e
recurrences.
If (3.3) is satisfied, we may expect that the average value hτ(U)i for Poincar´e recurrences satisfies the following asymptotic equality
hτ(U)i ∼ b q0
where|U| ≤ ǫ and ǫ ≪ 1. Our numerical simulations later will confirm this conjecture to plot hτ(U)i versus (− ln ǫ) and to evaluate the slope b
q0.
Furthermore, the relation of Eq. (3.4) infers the dynamical system (Rd, f )
pos-sesses positive topological entropy [3]. On the other hand, in the paper [17], authors proved that the Lyapunov exponent of some class of f can be estimated from the behavior of the first return times of a ball as the diameter vanishes. More precisely, if f is a piecewise monotonic mapping with a derivative of bound p-variation for some p > 0 and if µ is an ergodic f -invariant measure with non-zero entropy, then for µ-almost every x we have
λµ ≥ lim ǫ→0 τ (x, U) − ln ǫ −1 , (3.5)
where λµis the Lyapunov exponent of an invariant measure µ. Hence, from Eq.s (3.4)
and (3.5), if the slope qb
0 is positive, then it implies that the map f has a positive
Lyapunov exponent.
Remark: For a function f : [0, 1]→ R and p > 0 we define the p-variation of f by
varp(f ) := sup (N −1 X i=1 |f(xi+1)− f(xi)|p ) ,
where the supremum is taken along all finite ordered sequences of points 0 ≤ x1 <
x2 <· · · < xN ≤ 1 and integer N.
Figure 3.3 plots Poincar´e recurrences of the system (2.1) with γ = 4.7 and 11.9. As a result, the plot of hτ(U)i versus (− ln ǫ) has the positive slopes 0.77 and 0.57, respectively. The dispersion of the calculated values of the slopes is about 3 %.
4
Synchronization in Modified Logistic Hyper-Chaotic
System
In Section 2 and 3, from theoretical and numerical point of view, we have shown that MLM is a chaotic map which has no window and is uniformly distributed in [0, 1]. These fine properties are essential in the application to secure-communication. In order to conform a high standard of the secure-communication [18], based on MLMs in (2.1), we construct a multi-system F, called Modified Logistic Hyper-Chaotic System (MLHCS), defined by F(r, x, C) := C fγ1(x1) fγ2(x2) , where x = [x1, x2]⊤, r = [γ1, γ2]⊤, and C = 1− c1 c1 c2 1− c2 is a coupling matrix with coupling strengths 0≤ c1, c2 ≤ 1. Note that a hyper-chaotic system means that
it has at least two positive Lyapunov exponents [?]. When γ1 and γ2 are arbitrary
chosen larger than 4, as well as, c1 and c2 are arbitrary chosen between 0 and 1, it is
no doubt that MLHCS could almost be a hyper-chaotic system. Let G be another MLHCS defined by
G(r, y, C) := C fγ1(y1) fγ2(y2) ,
where y = [y1, y2]⊤, parameters r and C are the same as the systemF.
Now we want to build up a communication system between F and G, called Transmitter and Receiver, respectively. We utilize simplex partial coupling to reach synchronization between Transmitter and Receiver. More precisely, for given initial datum x(0)1 , x(0)2 , y1(0), y2(0)∈ (0, 1), we define the communication system (4.1)-(4.2):
y(i) = G r, y(i−1), C ,
y(i) = [x(i)1 , ¯y2(i)]⊤,
(4.2)
where x(i) = [x(i) 1 , x
(i) 2 ]
⊤
and y(i) = [¯y1(i), ¯y2(i)]⊤
for i = 1, 2, . . .. The vectors x(i) and
y(i) of Transmitter and Receiver can be synchronized by the partial portion x(i)1 with a suitable coupling strength C, as i is sufficiently large. Under the usual metric on R/Z, we obtain a sufficient condition for synchronization below.
Let | · |1 be the usual metric on R/Z defined by
|x − y|1 = min{|x − y|, 1 − |x − y|} for x, y ∈ [0, 1).
For convenience, we define a function δ(γ),
δ(γ) := max x∈[0,1]|f ′ γ(x)| = γ, if γ = 4k for some k ∈ N, √ γ2−4γ[γ 4] γ 4 (mod 1) , if γ ∈/ N. Theorem 4.1. If 1− 1 δ(γ2) < c2 < 1, then |x (i) 2 − y (i) 2 |1 → 0 as i → ∞.
Proof. From the system (4.1)-(4.2), it holds that|x(i)2 − y(i)2 |1 =|(1 − c2)[fγ2(x
(i−1) 2 )−
fγ2(y
(i−1)
2 )]|1. By the mean value theorem applied to fγ2, we have |x
(i) 2 − y (i) 2 |1 ≤ α|x(i−1)2 − y(i−1)2 |1 = αi|x(0)2 − y (0)
2 |1, where α = (1− c2)|fγ′2(ξ)| and ξ ∈ (0, 1). Since
1−δ(γ1
2) < c2 < 1, it implies that 0 < α < 1 and then |x
(i) 2 − y
(i)
2 |1 → 0 as i → ∞.
After proving Theorem 4.1, we understand that both sides of the communication system (4.1)-(4.2) can approach to the same state under the chord norm. However, by using Euclidian norm, x(i)2 and y2(i) can only be shown to be sufficiently close for some i.
Theorem 4.2. Given any small ǫ > 0, if |x(0)2 − y2(0)|1 < ǫ and 1− δ(γ12) < c2 < 1,
Proof. Clearly by Theorem 4.1.
Since there exits jumps (discontinuous points) in MLM, we obtain an equivocal phenomenon in the proof of Theorem 4.2. By using the chord norm, |x(i)2 − y2(i)|1 can
always be reduced when i increases under some convergence condition of c2. However,
by using Euclidian norm, x(i)2 and y2(i)may be separated for some i and then|x(i)2 −y2(i)| is far away from zero which cannot be applied to secure-communication.
From Theorem 4.1, we obtain that the communication system (4.1)-(4.2) can achieve synchronization of x(i)2 and y(i)2 in the chord norm. As discussed above, x(i)2 and y2(i)can be separated far away by using Euclidian norm, provided that x(0)2 and y(0)2 are very close, but are on the left and right hand sides of a jump point, respectively. Namely, non-synchronization can happen after some iterations. Hence, in order to guarantee realizing synchronization under Euclidian norm, we have to set up x(i)2 and y(i)2 to be in the same side with respect to the jump point when they are near the jump point. After this setting and from Theorem 4.2, it is clear that lim
i→∞|x (i) 2 − y
(i) 2 | = 0,
that is, x(i)2 and y2(i) can reach synchronization.
5
Application in secure-communication system
In this section we propose a secure-communication system, called Asymptotic Syn-chronization of Modified Logistic Hyper-Chaotic System (ASMLHCS), which is based on the communication system (4.1)-(4.2). ASMLHCS utilizes an important property of the communication system (4.1)-(4.2), that is, Transmitter and Receiver can realize synchronization. In the ASMLHCS there are two phases, one is asymptotical synchro-nization phase and the other one is Encryption/Decryption phase. First, we need to make both sides (Transmitter and Receiver) carry out asymptotic synchronous. And then to utilize asymptotic sync to accomplish the secure-communication.
The communication scheme is as Figure 5.1. Information is transmitted by Trans-mitter through the channel after Encryption. Receiver recovers the information by Decryption.
Fig. 5.1 is near here.
5.1
Crypto-communication system
In this subsection we present a crypto-communication system, Asymptotic Synchro-nization of Modified Logistic Hyper-Chaotic System (ASMLHCS), which is consisted of Transmitter (5.1)-(5.2) and Receiver (5.3)–(5.5). For i > 0 and given x(0)1 , x(0)2 , y(0)1 , y(0)2 ∈ (0, 1),
x(i) = F(r, x(i−1), C), (5.1)
k(i) = jx(i)1 k
n, (5.2)
where k(i) is a encrypting sequence and ⌊x⌋
n := ˆx denotes the chopped finite n digits
approximation with respect to x, i.e.,|x − ˆx| < 10−n, n∈ N.
y(i) = G r, y(i−1), C , (5.3) ˜
k(i) = jy¯(i)1 k
n, (5.4)
y(i) = [k(i), ¯y2(i)] ⊤
, (5.5)
where ˜k(i) is a decrypting sequence of receiver.
Remark: In Eq. (5.2), k(i) has to carefully pick out one, either jx(i) 1 k n or j x(i)1 k n+
10−n, when the interval (jx(i) 1 k n, j x(i)1 k n+ 10
−n) contains one of discontinuous points
of MLM. Here k(i) needs to be chosen in the same side as x(i)
1 with respect to the
discontinuous point. This artificial tactic also need to apply to x(i)2 and y (i)
2 ofF and
5.2
Asymptotic synchronization phase
In order to finish secure-communication, firstly, ASMLHCS has to connect and sync between Transmitter and Receiver. We use simplex direction from Transmitter (to Receiver) to employ a partial system of F, x(i)1 , to drive the system G in Receiver. It
is called simplex partial driving and makes both sides carry out asymptotic synchro-nization. In this subsection we will estimate to need how many iterations to reach asymptotic sync. {x(0)1 , x (0) 2 } and {y (0) 1 , y (0)
2 } are the initial values of F and G, respectively.
ASML-HCS began transmitting with simplex and partial system of x(i), and should be
as-ymptotic synchronization before information transmission, that is, to achieve that the sequence k(i) and ˜k(i) are asymptotic sync.
Theorem 5.1. Under the suitable artificial tactic, in the system (5.1)–(5.5) if 1−
1
δ(γ2) < c2 < 1, then there exists isyn ∈ N such that
y (i) 2 − x (i) 2 < 1 + c2δ(γ1) 1− (1 − c2)δ(γ2) 10−n as i > isyn.
Proof. From Thm 4.1 we know that|x(i)2 −y2(i)|1can be smaller that any given tolerance
condition after i large enough. From the proof of Thm 4.2 we observe that|x(i)2 − y2(i)| could be always smaller than any given tolerance condition under a suitable artificial tactic to avoid x(i)2 and y
(i)
2 located in different sides with respect to the jump η when
the system (5.1)–(5.5) we have that x (i) 2 − y (i) 2 = (1 − c2) h fγ2(x (i−1) 2 )− fγ2(y (i−1) 2 ) i + c2 h fγ1(x (i−1) 1 )− fγ1(y (i−1) 1 ) i < (1− c2)δ(γ2) x (i−1) 2 − y (i−1) 2 + c2δ(γ1)· 10 −n [let α = (1− c2)δ(γ2) and β = c2δ(γ1)] ≤ αhα|x(i−2)2 − y (i−2) 2 | + β · 10 −ni + β· 10−n ... ≤ αi|x(0)2 − y (0) 2 | + [αi−1+· · · + α + 1]β · 10 −n . Since 1− 1 δ(γ2) < c2 < 1, 0 < α < 1. And then x (i) 2 − y (i) 2 < (1 + β 1−α)10 −n for i > i syn as αisyn < 10−n.
After isyn steps transmitting, both sides of Transmitter and Receiver reach
hyper-chaotic asymptotic synchronization. The next subsection, we utilize the synchronized system to encrypt a plaintext p to a ciphertext c in Transmitter and to decrypt c to ˜
pin Receiver. Obviously, ˜p is equal to p.
5.3
Encryption & Decryption phase
After reaching asymptotic sync, ASMLHCS starts utilizing k(i) to mask the plaintext
into the ciphertext.
Theorem 5.2. For j ≥ 1 and i = isyn+ j, then
y (i) 1 − x (i) 1 < (1− c1)δ(γ1) + c1δ(γ2) + c1c2δ(γ1)δ(γ2) 1− (1 − c2)δ(γ2) 10−n. Proof. y (i) 1 − x (i) 1 = (1 − c1) h fγ1(y (i−1) 1 )− fγ1(x (i−1) 1 ) i + c1 h fγ2(y (i−1) 2 )− fγ2(x (i−1) 2 ) i < (1− c1)δ(γ1)· 10−n+ c1δ(γ2)|y2(i−1)− x (i−1) 2 | < (1− c1)δ(γ1) + c1δ(γ2) + c1c2δ(γ1)δ(γ2) 1− (1 − c2)δ(γ2) 10−n. (by Thm 5.1)
In Transmitter:
Given m∈ N and m < n − 1 such that (1− c1)δ(γ1) + c1δ(γ2) + c1c2δ(γ1)δ(γ2) 1− (1 − c2)δ(γ2) 10−n< 1 2× 10 −m.
For j ≥ 1 and i = isyn+ j. The plaintext p is decomposed into the integer sequence
{p(i)}, and each integer is small than 10m. The encryption E(p, k) is:
c(i) = k(i)+ p(j)× 10−m
, where c(i) is the ciphertext and c ={c(i)}.
In Transmitter to transmit the ciphertext c(i) to corresponding connection point
and here we also can do additional encoding before sending c(i). On the other hand,
after obtaining c(i), Receiver has to remove the additional encoding firstly; and then
to handle decrypting process. In Receiver:
There are two important procedures, one is decryption and the other one is implicit driving. For j ≥ 1 and i = isyn+ j, The decryption D(c, ˜k) is
˜
p(j)=⌈c(i)− ˜k(i)⌉m× 10m.
In order to keep on asymptotical synchronization of both sides as the succeeding step in ASMLHCS. Here we propose implicit driving to make k(i) and ˜k(i) always preserve
asymptotic sync in Encryption/Decryption phase, that is, Eq. (5.5) is replaced by y(i) = [k(i)− ˜p(j)× 10−m, ¯y
2(i)] ⊤
,
where ⌈x⌉m := ˆx denotes the rounded m-digit approximation to x, i.e., |x − ˆx| < 1
2 × 10 −m.
Remark: Given x(0)1 , x(0)2 , y1(0), y2(0), c1 & c2 ∈ (0, 1) and γ1 & γ2 ∈ (4, ∞). For i ≥ 1, we have x(i)1 , x (i) 2 , ¯y (i) 1 , ¯y (i) 2 , y (i) 1 , y (i)
2 , k(i), ˜k(i) ∈ (0, 1) and c(i) ∈ (0, 2).
In this section, we present ASMLHCS and prove that ASMLHCS is synchronized in asymptotic synchronization phase, and the system is always keeping synchroniza-tion by implicit driving in Encrypsynchroniza-tion/Decrypsynchroniza-tion phase. Although we exhibit this work not only may well be achieved in the decimal system, it is just for easy to understand, but also can be realized in the hexadecimal system.
6
Conclusion
In conclusion, we show a robust chaotic map, Modified Logistic Map, which is not only no window but also uniformly distribution in [0, 1]. Basing on the modified logistic map we carry out a multi-system hyper-chaotic synchronization system to apply in secure-communications, Asymptotic Synchronization of Modified Logistic Hyper-Chaotic System. The system can gain asymptotical synchronization between Transmitter and Receiver after finite times of simplex partial coupling transmission in mathematical theoretics. Further, the implicit driving technic guarantee always asymptotical synchronization between drive system and respond system when during the plaintext transmitting.
Acknowledgments
This research is supported in part by National Science Council and National Center for Theoretical Sciences in Taiwan. We would like to thank Mr. Shih-Liang Chen for giving us a lot of references and many helpful discussions.
References
[1] V. Afraimovich, J. R. Chazottes and B. Saussol, Pointwise dimensions for Poincar´e recurrences associated with maps and special flows, Discrete Contin. Dyn. Syst. 9, no. 2, pp. 263–280, 2003.
[2] V. Afraimovich, W. W. Lin and N. F. Rulkov, Fractal dimension for Poincar´e recurrences as an indicator of synchronized chaotic regimes. Internat, J. Bifur. Chaos Appl. Sci. Engrg. 10, no. 10, pp. 2323–2337, 2000.
[3] V. Afraimovich, J. Schmeling, E. Ugalde and J. Ur´ıas, Spectra of dimensions for Poincar´e recurrences, Discrete Contin. Dyn. Syst. 6, no. 4, pp. 901–914, 2000. [4] This is true not only for low-dimensional systems, but also for high-dimensional
chaotic systems. See, for example, E. Barreto, B. R. Hunt, C. Grebogi, and J. A. Yorke, From High Dimensional Chaos to Stable Periodic Orbits: The Struc-ture of Parameter Space, Phys. Rev. Lett., no. 78, pp. 4561–4564, 1997.
[5] D. Campbell, An introduction to nonlinear dynamics, in Lectures in the Sciences of Complexity, ed. D. L. Stein, Addison-Wesley, Reading, MA, 1989.
[6] R. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd ed. Redwood City, CA: Addison-Wesley, 1989.
[7] D. Gulick, Encounters with Chaos, New-York: McGraw-Hill, 1992.
[8] G. Heidari-Bateni and C. D. McGillem, A chaotic direct-sequence spread-spectrum communication system, IEEE Trans. Comm., vol. 42, pp. 1524–1527, 1994.
[9] R. Holmgren, A First Course in Discrete Dynamical Systems, 2nd ed., New-York: Springer-Verlag, 1996.
[10] K. Klomkarn, A. Jansri and P. Sooraksa, A design of stream cipher based on multi-chaotic functions, IEEE Int. Symp. Commmunications and Information Technology, vol. 2, pp. 26–29, 2004.
[11] R. L. Kraft, Chaos, Cantor Sets, and Hyperbolicity for the Logistic Maps, Amer. Math. Monthly, 106, (1999) 400–408.
[12] L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett., vol. 64, pp. 821–824, 1990.
[13] Y. B. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Application, The University of Chicago Press (Chicago and London), p. 304, 1997.
[14] F. Peng, S. S. Qiu and L. Min, An image encryption algorithm based on mixed chaotic dynamic systems and external keys, International Conference on Com-munications, Circuits and Systems, vol. 2, pp. 27–30, 2005.
[15] C. Robinson, Stability, Symbolic Dynamics, and Chaos, Boca Raton, FL: CRC Press, 1995.
[16] L. Rossi, G. Turchetti and S. Vaienti, Poincar´e recurrences as a tool to investigate the statical properties of dynamical systems with integrable and mixing componts, Journal of Physics: Conference Series 7 (Chaotic Transport and Complexity in Fluids and Plasmas), pp. 94–100, 2005.
[17] B. Saussol, S. Troubetzkoy and S. Vaienti, Recurrence, dimensions and Lyapunov exponents, J. Statist. Phys., vol. 106, no. 314, pp. 623–634, 2002.
[18] M. I. Sobhy and A.-E. R. Shehata, Methods of attacking chaotic encryption and countermeasures, IEEE International Conf. on Acoustics, Speech, and Signal Processing, vol. 2, pp. 1001–1004, 2001.
[19] L. S. Young, Recurrence times and rates of mixing, Israel J. of Math., Vol. 110, pp. 153–188, 1999.
[20] H. Zhou and X. T. Ling, Problems with the chaotic inverse system encryption approach, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., Vol. 44, pp. 268– 271, 1997.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 γ = 10 x fγ (x )
0 50 100 150 200 250 300 350 400 0 0.2 0.4 0.6 0.8 1 0 50 100 150 200 250 300 350 400 450 500 −1 0 1 2 3 γ = 5.9 { fγ (x )} lo g10 (| ff t( ·) |2 ) frequency (Hz)
0 50 100 150 200 250 300 350 400 0 0.2 0.4 0.6 0.8 1 0 50 100 150 200 250 300 350 400 450 500 −1 0 1 2 3 γ = 10.8 { fγ (x )} lo g10 (| ff t( ·) |2 ) frequency (Hz)
Figure 3.2: The attractor {fγ(x)} and the spectrum of waveforms of MLM for γ =
4 4.5 5 5.5 6 6.5 7 2.6 3 3.5 4 4.5 5 5.4 γ = 4.7 hτ (U )i − ln ǫ 4 4.5 5 5.5 6 6.5 7 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 γ = 11.9 hτ (U )i − ln ǫ
Figure 3.3: Poincar´e recurrences of MLM for γ = 4.7 and 11.9 with respect to the slopes 0.77 and 0.57, respectively. The dispersion of the calculated values of the slopes is about 3 %.
Information Source p (j) Encryption E(p, k) c(i) Decryption D(c, ˜k) ˜ p(j) Information Acquirement
