The paper deals with developing a mathematical framework to design novel discrete-time chaotic systems based on the known ones. Our development is based on applying coordinate transformation to the domain where the initial system dynamic is defined. We study the shift of 2D system coordinate origin and use it to define novel system state variables, which take into account this shift. The dynamical system obtained in such a way is considered the interval one with piecewise linear interval boundaries. This fact gives us the possibility to consider possible uncertainty caused by changes in system parameters and the presence of nonlinear functions and rewrite the system into a linear-like form. Unlike the initial nonlinear ones, performing all coordinate transformations for such type systems is easy. Our approach is based on transforming the continuous-time system dynamic into a discrete-time domain due to the possibility of its implementation in modern digital devices. Transformation into a discrete-time domain allows us to define system dynamics using its previous states to define the piecewise constant factors in the system equations. The system equation is designed in such a way that it is created on a solid background and uses information about previous system motions as well as its motion in the moved coordinate system and motions of the considered moved coordinate system. To make the system dynamic more complex, we ofer to consider its perturbed motions as the diference between motions in the moved and stationary coordinate system.
Nowadays, data transmission using chaotic systems [1] refers to the practical usage of chaos theory [2] and chaotic signals [3] to secure information transmission. Chaotic systems are susceptible to initial conditions and exhibit complex, unpredictable behavior over time [4]. These facts about chaotic systems make them useful in secure communication [5] because chaotic signals can be dificult to predict, intercept, or reproduce without knowing the exact system parameters [6].
Such unique chaotic systems’ features cause several key concepts in chaotic communication: • Chaotic modulation involves embedding information into a chaotic signal [7]. The chaotic signal acts as a carrier wave, which is then modulated by the data. Only receivers knowledgeable about the chaotic system’s parameters can demodulate and recover the original message. • The noise-like signals concept assumes that chaotic signals appear similar to noise, making them hard to distinguish from random background noise in the communication channel [10]. This property provides inherent security, as an eavesdropper without the system parameters will find it challenging to extract meaningful data.
The above-shown concepts find their practical implementation in designing various chaotic modulation schemes. The main ones are chaos shift keying (CSK) [11] and chaotic phase modulation (CPM) [12]. These modulation schemes find their applications in establishing wireless communications. In this case, chaotic communication can be applied in wireless systems where robustness to interference is crucial. Since chaotic signals are noise-like and spread across a wide bandwidth, they can be used in environments with high electromagnetic interference. Also, various optical fiber communication systems use laser signals to transmit data securely. Optical chaos can be generated using semiconductor lasers, and synchronization between transmitter and receiver can be achieved with optical feedback.
In summary, chaotic systems ofer a promising approach to secure data transmission by leveraging the unpredictable and noise-like nature of chaos, making it dificult for unauthorized parties to intercept or decode the communication.
The main drawback of known chaotic systems, which are used to implement chaotic generators and produce chaotic signals, is some subjectivism in the design of these systems. Since authors do not explain the influence of terms and factors in their equations, modifying and improving them is tough. We ofer to avoid this drawback by designing novel chaotic systems with applying some transformations to known ones. Thus, our paper’s goal is to design a novel chaotic system by combining motion equations of known ones and motions of the origin of the coordinate system where the above-mentioned chaotic system is considered. We believe that the goal achieving makes a systematic basis in chaotic system design.
Our paper is organized as follows: at first, we consider the generalized chaotic system and transform its equations into interval matrix form to represent it in a piecewise linear form. Then, we consider the system in the discrete-time domain to avoid solving any diferential equations. Such a discretetime dynamical system is viewed as a system in some coordinate system in which the origin changes its position relatively to a stationary one. We define the chaotic system position in the stationary coordinates as the sum of the chaotic system and the origin position. At last, we show the use of our approach by considering Dufing pendulum equations.
2.1. Interval discrete-time model of the generalized second order dynamical system
Y˙ = F (Y) + U, Y = where are system state variables and (.) are some nonlinear functions, and are some input signals.
System nonlinearities make analysis of its motions, their transformations, or synthesis a quite hard problem which should be solved for each particular case in the separate way. That is why we ofer to use interval methods and replace the system nonlinear functions (.) with domains Ωi where these functions are defined and do not exceed them on whole system operation range.
The boundaries for these domains can be defined in a diferent way. Thus, one can approximate boundaries for function (.) by nonlinear function (.) which are more simple than system nonlinearities and which use during system study of design does not cause any dificulties. One of such functions is a piecewise linear function which for the case of system with two arguments can be written down as follows (1, 2) = here are factors of piecewise linear approximation which are defined in i-th subdomain Ωij of system state variables’ values.
It is clear that one can use diferent mashes to define subdomains Ωij. We believe that the most accurate one is a triangular mesh which we ofer to use to define the subdomains Ωij.
Such an approach allows us to redefine i-th component of vector F in (
if (y1 ∈ Ωi1) and (y2 ∈ Ωi1) ; . . .
if (y1 ∈ Ωin) and (y2 ∈ Ωij) , ≈ ℎ ︀( 1, − 1)︀ , ≈ 1 − − 1 − 1
.
Here we consider the case when numbers of intervals in upper and lower boundaries equal each
other and equal to . In the most general case when upper and lower boundaries are defined with
diferent numbers of intervals and/or these boundaries are defined for various subdomains Ω , one
should split intervals in (
A = aij = Y˙ = AY + A0 + U, ︂( 11 21 12 )︂ 22 ; A0 = ︂( 01 )︂ 02 ; Y = here z− 1 means backward signal shift in one sample time period , ℎ(.) is a some approximation function.
If one substitute (
Y = − 1 (A1Y) + − 1A10 + − 1 U, A1 = ︂( 11 + 1 21
12 22 + 1 ︂) , A10 = ︂( 01 )︂ 02 .
Contrary to (
− ,
here operator [.] means taking integer part from the number. It is clear that solution of (
Nevertheless, sometimes system motions should be considered in some coordinate system which origin moves relatively some stationary base. To design the model which describe such motions let us determine the system position in stationary coordinates Y0 as linear combination of its position in moved coordinates Y and origin position of moved coordinate system Y1.
Y0 = Y1 + Y, here we think that vectors Y0, Y1, and Y have the same size and contain components which define system position in some phase plane and vector Y1 are defined similar to Y
Y1 = − 1 (B1Y1) + − 1B10 + − 1U1, here B1 and B10 are some matrices which components are defined similar to components of A1 and
Y0 = − 1 (B1Y1) + − 1 (A1Y) + − 1A10 + − 1B10 + − 1U + − 1U1.
Expression (
Y = (︀ E − − 1A1)︀ − 1 − 1 (A10 + U) ; Y1 = (︀ E − − 1B1)︀ − 1 − 1 (B10 + U1) ,
(
det (︀ E − − 1A1)︀ det (︀ E − − 1B1)︀ Y0 = = adj (︀ E − − 1A1)︀ − 1 (A10 + U) + adj (︀ E − − 1B1)︀ − 1 (B10 + U1) , where adj(.) means adjugate matrix and det(.) means matrix determinant.
If one expand multiplication of determinants in left-hand expression of (14), he can rewrite this formula as follows
Y0 = − q1− 1Y0 − q2− 2Y0 − q3− 3Y0 − q4− 4Y0+ + adj (︀ E − − 1A1)︀ − 1 (A10 + U) + adj (︀ E − − 1B1)︀ − 1 (B10 + U1) , where q are interval piecewise constant coeficients of the system characteristic polynomial.
Since the system (15) has piecewise constant factors which depend on motion systems (
We ofer to find these expressions by considering (
− 1Y0 = − 1Y1 + − 1Y,
− 1Y = (A1 − B1)− 1 (︀ Y0 − − 1B1Y0 + − 1U1 + − 1U)︀ ; − 1Y1 = (A1 − B1)− 1 (︀ − 1A1Y0 −
Y0 + − 1U1 + − 1U)︀ .
We call (15) as an interval model of the dynamical system in stationary coordinate system. This model consists of three parts: the first one use system’s previous coordinates in the stationary coordinate system to define its current position. The second one uses the system position in the moved coordinate system and the third one uses information about moving of coordinate system origin.
3.1. Dufing pendulum modeling and simulating in the stationary coordinate system Let us consider the use of the proposed approach to design the system with chaotic dynamic. We use the well-known Dufing pendulum [13]
̇1 = 2; ̇2 = − 11 − 22 − 313 + 1 cos 2 as the basis for our system.
Under some pendulum parameters system (18) has a chaotic dynamic. We think that the base system parameters are 2=0.02, 1=1, 3=5, 1=8, 2=0.5. Also, the parametric uncertainty is assumed in the relative interval =[0,9,1.1], which means possibility to 10% parameters drift. This interval allows us to define intervals of possible pendulum parameters in such a way
The pendulum nonlinearity is approximated by piecewise linear domain which is shown in figure 1.
ai = ; ci = .
y1 = ⋃︁ y1i
=1
(
f (1) = k(1)1 + y0(1),
k(1) = ⋃︁ ki, y0(1) = ⋃︁ y0i.
=1 =1
Intervals in (23) for the considered cubic nonlinearity are shown in table 1. These intervals as well as parameters of piecewise linear function, which replace pendulum cubic nonlinearity, obtained by using Nelder-Mead method from routine minimize() that is included in SciPy 1.11.0 library. We also use routine scipy.integrate() which solves diferential equations from the above-mentioned Python library. All calculated data is stored in csv-files and used to visualize calculation results in package PGFplot which is a part of TEXLive-2023.
The use of intervals (19) and (22) gives us the possibility to rewrite (18) in the interval piecewise linear form y˙ 1 = y2; y˙ 2 = − (a1 + a3k) y1 − a2y2 − a3y0 + 1 cos 2 (24)
We call (24) as interval piecewise linear model of Dufing pendulum. One can use this model to define the boundary motions which shows the maximal and minimal possible amplitude of pendulum oscillations. y2 = − 1y2 (1 − a2 ) − − 1 (a1 + a3k) y1 − − 1 a3y0 + − 1 1 cos 2.
In the extended form (25) can be given as follows 1 min =− 11 min + − 1 2 min; 2 min =− 12 min (1 − 2 max) − − 1 3 max0 max− 1 max =− 11 max + − 1 2 max; 2 max =− 12 max (1 − 2 min) − − 1 3 min0 min− − − 1 (1 max + 3 maxmax) 1 min + − 1 1 min cos 2 max.
− − 1 (1 min + 3 minmin) 1 max + − 1 1 max cos 2 min.
Y = − 1A1Y + − 1A10 + − 1U, Y = ︂( [1 min, 1 max] )︂ [2 min, 2 max] ; A10 =
︂( U = ︂(
⎛ A1 = ⎝ ︂[ − (1 max + 3 maxmax) , ]︂ [︂ − (1 min + 3 minmin) 1 0 [1 min cos 2 max, 1 max cos 2 min] [− 3 max0 max, − 3 min0 min] ︂) 1 1 .
0 − 2 max, ]︂ − 2 min ⎞ ⎠ ; ︂) ;
Piecewise constant elements of matrix A1 can be found by known interval system output y1 which
can be defined by (
− 1 a3y0 + − 1 1(2)
system is proved by the fact that the motion of pendulum with exactly-known above-given parameters starts as motion which is bounded by motions in the upper and lower boundaries but after a quite short time which is near 10s it leaves the interval of pendulum boundary motions. If one analyzes these boundary motions, he finds that quite small variation of pendulum parameters dramatically changes its dynamic. 3.2. Dufing pendulum modeling and simulating in the moved coordinate system Let us assume that origin of coordinate system, where the pendulum dynamic is defined, moves and this motion can be defined by using (18) with parameters 2=0.01, 1=-1, 3=3, 1=3, 2=1 and the same relative interval. This assumption allows us to write down equation similar to (27)
Y1 = − 1B1Y1 + − 1B10 + − 1U1.
Matrices B1, B10 and U1 are similar to A1, A10 and U but their elements are defined with replacing ai to bi and to .
Equations (29) and (27) allows us to define system dynamic in the stationary coordinates by rewriting (15) as follows y1 = − q1− 1y1 − q2− 2y1 −
q3− 3y1 − q4− 4y1 + (︀ w4− 4 + w3− 3 + w2− 2)︀ y0+ + (︀ v14− 4 + v13− 3 + v12− 2)︀ 1 cos 2 + (︀ v24− 4 + v23− 3 + v22− 2)︀ 1 cos 2, q1 = (a2 + b2) − 4; q2 = 2 (a2b2 + (a3 + b3) k + a1 + b1) − 3 (a2 − b2) + 6; q3 = (a2b3k + a3b2k + a1b2 + a2b1) 3 − 2(a2b2 + (a3 + b3) k + a1 + b1) 2+ + 3(a2 + b2) − 4; q4 = 1 + (a3b3k2 + (a1b3 + a3b1) k + a1b1) 4− − (k (a2b3 − a3b2) − a1b2 − a2b1) 3 − (a2 + b2) + (a2b2 + (a3 + b3) k + a1 + b1) 2; w4 = − (2a3b3k − a1b3 − a3b1) 3 + (a2b3 + a3b2) 2 − (a3 + b3); w3 = − (a2b3 + a3b2) 2 + 2(a3 + b3) ; w2 = − (a3 + b3); v12 = 1; v22 = 1; v23 = 2a21− 2 1; v14 = 3 (b31k − b11) + 2b21 − 1; v13 = 2b21− 2 1; v24 = 3 (a31k − a11) + 2a21 − 1; (30)
Comparison of (30) and (27) shows that taking into account motion of coordinate system increases order of the studied dynamical system and it is necessary to have information about four previous system positions instead of two ones. Iteration of (30) gives results which are shown in figures 4, 5. (29)
Analysis of curves in figures 4 and 5 and comparison them with figures 2 and 3 allows us to claim
that system motion in the moved coordinate system is more complex than stationary one. One can see
three stationary points (-2.0), (0,0), and (
The considering of chaotic system as dynamical system in moving coordinates gives us the possibility to produce novel chaotic oscillations by using well-known chaotic systems. This fact allows us to claim that novel chaotic system can be designed by changing one or both core system and system, which define motion of coordinate system’s origin. In both cases system dynamic difers the core dynamic very much. The order of designed in such a way system equals to core system order and order of dynamical system which describe motion of coordinate system. The increasing system order requires to use more information about previous system states in case of discrete-time system implementation. Analysis of the obtained discrete-time models shows that chaotic system can be defined in class of discrete-time dynamical systems with piecewise constant parameters. It becomes possible due to the use of interval methods to describe system motions. Defining these parameters in some arrays, lists, tables and so on gives us the possibility to implement the considered systems by using wide range MCU and FPGA. Declaration on Generative AI: We declare the no use of any Generative AI tools while preparing data and the paper’s writing.