The paper is devoted to the development of a multichannel chaotic dynamical system. Our study is based on the design of theoretical backgrounds to transform a phase portrait of the generalized second-order dynamical system from cartesian coordinates into biangular ones. Such a transformation is defined by two nonlinear functions, which depend on the system's position and the positions of some base points. The system angular position is considered as the relative one respectively to base points. The differentiating of these functions and considering the initial system dynamic in the cartesian plane allows us to construct differential equations that define system dynamics in the novel state space. Since in the most general case the considered system can be quite nonlinear the obtained equations become enough complex. To reduce their complexity, we offer to replace the system nonlinearities as well as transformation nonlinearities with piecewise dependencies to approximate nonlinearities. Such an approach gives us the possibility to design a highly formal matrix-based approach to transform the piecewise linear dynamical system from cartesian coordinates into biangular ones. We show the example of this approach usage by designing a chaotic generator which is based on the well-known Duffing equations. The developed novel chaotic system can be easily implemented by using finite difference approximations for derivative operators with modern digital devices. Simulation results prove the significant difference between the designed and known systems.
In today's world, chaotic systems are widely used in various scientific and engineering applications
due to their unique ability to produce unpredictable signals [
At the same time, the primary application of chaotic systems is in data encryption [16, 17, 18] and secure data transmission [19, 20, 21]. The increasing need for highly secure communication systems is driven by the necessity to transmit large amounts of data for critical infrastructure cyber-physical systems [22, 23, 24], while restricting access to unauthorized persons. This interest is further fueled by the emergence of Industry 4.0 and Industry 5.0.
Consequently, numerous chaotic systems have been developed [25, 26, 27], with many authors studying chaotic systems in the real domain and designing channels to provide desired system features. However, these systems are often considered as single-channel dynamical systems, which limits their ability to perform multichannel data transmission.
Considering a dynamical system with multichannel observability equation can automatically solve this issue [28, 29] and facilitate the design of a multichannel chaotic system for parallel data transmission [30, 31, 32]. This approach also eliminates the need for subjective design of new chaotic systems as it involves transforming known systems [33, 34]. We demonstrate our method by considering a well-known Duffing system, but it can easily be extended to any system with chaotic and regular dynamics [35, 36].
The paper is organized as follows: firstly, we consider the generalized 2nd order nonlinear differential equation and transform it from cartesian coordinates into biangular ones. Then, we show the practical usage of our approach by replacing the system nonlinearities with piecewise linear functions. Such an approach allows us to design highly-formalized mathematical background for system designing. Thirdly, we illustrate the use of our approach by considering a well-known chaotic system based on the Duffing pendulum equations. Finally, we make a conclusion.
Let us consider the generalized 2nd order dynamical system which motion is given in normal form as follows
= , ; = , , (1) where x and y are system state variables, (.) and (.) are some nonlinear functions.
Here and further, we consider a free motion of closed-loop dynamical system and in more complex case some external control signal should be taken into account while the considered system is modeled.
It is clear that the system motion can be given by some trajectory in the phase plane (Figure 1).
Y yA yP1 yP2
P1 α1 xP1
A xA , , , , , , , + , , − − − − − − − − − , , , ,
− − , , , , , , , , , , , , − + , , , , , , , −
− − −
+ ⎛ ⎜ − ⎝ − ⎛ ⎜ − −
, , , + + − − − − , ⎞ ⎟ = 0; ⎠ + , ⎞ ⎟ = 0.
Analysis of (8) shows that the fourth first summands in each equation define its motion by using current values of base points position as well as the fifth and sixth summands define influence of the base points motions on the system dynamic. This fact allows to rewrite (8) in more specific way for the case of the stationary base points (9) , , ,
+ ,
− , , , , , , , , , , , , , , , , + − − − − − − , − − − − , , , , , , − , , , , , , −
+ , − , , , , , , + , − , + , ⎝ , − , ⎠
Equations (8) and (9) are given in the implicit form. One can rewrite them into explicit normal form by solving these equations for derivatives of system angular position components. Such an approach allows us to rewrite notion equations in the normal form as follows
= ' , , , , , ; = ' , , , , , , (10) here ' (.) and ' (.) shows nonlinear functions which are solutions (8) and (9). Due to the complexity of the general solution of these equations we do not give them here and think that functions ' () and ' () differ for (8) and (9).
Thus, one can consider (10) as the model of the considered dynamical system in biangular coordinates. This model shows angular system position relatively some base points. It is clear that in the plane two base points allows us to define the exact system angular position.
Analysis of the above-given formulas their enough complex structure which makes usage of the proposed approach quite difficult. That is why we offer to replace a nonlinear functions in the initial dynamical system model (1) as well as transformation equations (3), and motion equations for the base points (7) with some piecewise linear functions. Such an approach gives us the possibility to rewrite the above-mentioned equations as follows
= + + (; = + + (, (11) = ) + ) + ) * + + ) , + + ) - + + ) . + + ) (; (12) 72 = ?)) ((@ ; 52 = = ((> ; 62 = ? // ((@ ; 58 = = > ; 78 = ?// // @ ;
56 = =52 62 > ; 56 2 = =6522 >, here s is a Laplace operator, Y0, X0, and α0 are initial condition's matrices.
If one substitutes the last equation of (14) into the first one, the dynamical system motion in terms of angular coordinates can be written down
789 + 7:4 + 72 − 02 = 3 789 + 37:4 + 372 + 32 . (16) here we consider the third and fourth summands in the left-hand expression as weighted δ-functions to define the generalized system initial conditions which are caused by piecewise linearity of the considered system.
The second summand in the left-hand expression (16) defines the components of system dynamic which are caused by motions of the base points. It is clear that in the case of motionless base points one can equals this summand to zero and consider only the one summand in tight-hand expression as a variable one and others are constants
789 + 72 − 02 = 3 789 + 37:4 + 372 + 32 . (17) Thus, equations (16) and (17) can be considered as piecewise linear analogues for (8) and (9).
It is clear that left-hand expression in both of (16) and (17) have some weight matrix B1 which makes system study more complex. That is why we offer to rewrite (17) in the normal form as follows (18) 9 = 78A8 3 789 + 37:4 + 372 + 32 − 72 + 02 .
Equation (18) define system motion in the biangular coordinates with the motionless base points. Analysis of matrices in (18) shows that dynamic of a transformed system depends on initial system matrices as well as transformation ones. Now we turn our attention into the case of the moved base points and define their position as the result of solution the second equation in (14) 4 = B − 56 A8 56 2 + 42 , (19) where E is 4x4 identity matrix.
Substitution of (19) in (16) makes it possible to redefine the transformed system dynamic as follows 9 = 78A8 3 789 + 37: B − 56 A8 56 2 + 42 + 372 + 32 − (20) − 7: B − 56 A8 56 2 + 42 + 72 − 02 .
It is understood that in the most general case system motion in the biangular coordinates depend on its initial conditions and approximation factors which are defined as piecewise linear functions of system state variables in the cartesian and biangular coordinates.
Since this equation is defined by using inverse characteristic matrix of second equation in (14), one can rewrite it in terms of characteristic polynomial-adjugated matrix
9 ∙ DEF B − 56 = 78A8 3 789 ∙ DEF B − 56 + +32 ∙ GHI B − 56 − 7: ∙ JGK 56 − B 56 2 + 42 + + 72 ∙ GHI B − 56 − 02 ∙ GHI B − 56 . (21)
If one takes into account an order of matrices E and CD, he finds that (21) define the dynamic of system with moved base points as 5th order motion trajectory. This trajectory can be defined in the normal form if one puts only the highest derivatives of system position's components in left-hand expression of (21) the and writes its lower derivatives in the right-hand-expression. Aa a result, after some trivial transformations one can rewrite (21) as follows
,
- 9 = L
M7N9 + 37: − 7: ∙ JGK B − 56
+ 372 + 32 + 72 − 02 ∙ GHI B − 56 .
We offer to consider (22) as a matrix piecewise linear motion equation in a biangular coordinates. Due to the use of piecewise constant elements in the matrices one should use this equation to perform cyclic calculations of system angular position, then define its position in cartesian coordinates, and at last use both of them to specify the elements of system matrices. (22) (24)
Let us show the use of the proposed approach by modeling and simulating the well-known Duffing pendulum which can be considered as 2nd order nonlinear dynamical system
= ; = − − − * * + , - , (23) here x1 and x2 are system state variables, ai are system factors, and t is a system operating time.
The simplest way to perform a piecewise linear approximation for pendulum nonlinearity which can be highly formalized and implemented with various MCU/FPGA/CPU is using secant lines for the nonlinearity. Such an approximation allows us to replace system nonlinearity with piecewise linear function as follows * ≈ Q
+ (, ∈ M, MS , = 1. . U , Q = VVW⬚WXAAVVWXW⬚YYZZ = M + M MA + MA , ( = M* − Q M = − M MA ⬚ ⬚
M + MA here N is a number of fracture points, xi is a coordinate of i-th fracture point.
It is clear that such an approach allows us replace nonlinear function with piecewise linear one, which is designed with some lines with the same parameters between neighbor fracture points.
In a similar way we approximate nonlinear surfaces (3) by some planes in sixth dimensional state space. Equation of such a plane can be given in matrix form as follows
− M] − M] − M] − M] − M] − M] \ MA ] − M] MA ] − M] MA ] − M] MA ] − M] MA ] − M] MA ] − M]\ = 0, (25)
M ]A − M] M ]A − M] M ]A − M] M ]A − M] M ]A − M] M ]A − M] here indices i and j means ij-fracture point in plane.
Intersections of two neighbor planes allows us to define some line which bound the plane and define some triangulars. Equation for such boundaries can be written down similar to (25) ^^ MM]AA−−] −− MM]] MM]] MMA]A−−] −− MM]] MM]] MM]AA−−] −− MM]] MM]] MM]AA−−] −− MM]] MM]] MM]AA−−] −− MM]] MM]] MM]AA−−] −− MM]] MM]]^^ == 00; (26)
Thus, for the considered case the problem of approximation multidimensional transformation expressions (3) can be considered as the triangulation problem in the 6D space.
Replacing the system and transformation nonlinearities with the (24) and (25) gives us the possibility to write down following differential-algebraic equations
= ; = − + *Q − − * ( + , - , = ) + ) + ) * + + ) , + + ) - + + ) . + + ) (; (27) = ) + ) + ) * + + ) , + + ) - + + ) . + + ) (.
One can consider these equations as some state space equations, where the first and second equations are motion equations and the third and fourth equations are observability equations. ) − _ = - a *k+a − _ − *k+a P1 − *k+)a11 )2)21-5b)1222)-2b112 )25 P2 − − *k+a P1 − *k+)a11)2)21-6b)1222)-2b112)26 P2 + (30) + cos - -b10k-y( *-a )10 )22+b12)20 *k+a
)11)22-b12)21 )11)22-b12)21
As one can see, the motion equations in a canonical form (30) is simpler than in a normal one. The main feature of these equations is dependence of system output only from one angular position. So, the second angle should not be defined. Nevertheless, (30) clear depend from coordinates of both base points. It can cause some misunderstanding in system structure, which we offer to avoid by considering the summands with coordinates of the second base as components of the first base point's coordinates ) * −
= _ ; _ = -a*k+a − _ − m *k+a (31) − m *k+a )14)22-b12)24 + *k+a )16)22-b12)26 P2n P1 +
)11)22-b12)21 )11)22-b12)21 P1 + cos - , )22+sin - , - )12 + -b10k-y( *-a )10 )22+b12)20 *k+a
)11)22-b12)21 )11)22-b12)21 We call (31) as Duffing pendulum equations in the canonical form.
Thus, one can define pendulum dynamic in biangular coordinates in canonical form and obtain equations which are similar to the initial pendulum piecewise linear equations. One can consider these equations as some generalization for the known ones because of adding some summands with piecewise constant factors only.
It is clear that one can implement above-given canonical and normal pendulum equations in the discrete time domain by using known approximations for the derivative operator. We use the Tustin approximation in our studies
≈ o2 ∙ +− ppAA , (32) here pA is a shit operator and T is a discretization time.
The usage (32) allows to consider the generated by (29)-(31) continuous-time chaotic signals as some chaotic signals sequences which can be easy implemented in various FPGA/MCU devices and boards. Results of numerical solution of (29) and (31) with Arduino Due board are shown in Figure 2 and Figure 4 for normal and canonical systems. In Fig.3 we show numerical solution of classical Duffing equations. We use following pendulum parameters a1=1; a2=0.02, a3=5, a4=8, a5=0.5. Base point coordinates are xP1=1, yP1=1, xP2=-2, yP2=-3. As one can see transformation of Duffing pendulum equations in the non-cartesian domain changes its dynamic significantly.
a) System outputs b) System attractor 3.2. Duffing pendulum model in biangular coordinates with moved base points The above-given differential equations are obtained for the case of constant coordinates of base points. Since in the most general case these points can move, we generalize our equations by taking into account of speeds of the base points. We consider the case when both of base points have cyclic trajectories which can be defined by following equations
= ; = −_ ; = ; = −_ , (33) which are used when observability equations in (27) are substituting in the system equations. The solution of obtained equations for the derivatives of system angular position, allows us to rewrite Duffing pendulum equations with moved base points as follows
= ; = −_ ; = ; = *) ) Q + ) )) ) −+) )) ) + ) ) + ) + + *) ) *Q + ) ) * + + ) ) - +
= −_ ; + *) Q + ) + ) )
) ) − ) ) ) ) * + ) ) * − ) ) , _ + ) , ) _ ) ) − ) )
) ) - + ) ) - − ) ) . _ + ) . ) _ ) ) − ) ) ) ) , + ) ) , + ) ) * − ) *) + ) ) ,
) ) − ) ) ) ) . + ) ) . + ) ) - − ) - ) + ) ) .
) ) − ) ) − ) ))( + ) * ( + ) () − , + ) + + + + *) ) - Q + + *) ) , Q + + *) ) . Q + + ) *) (Q + ) )) ( )+ ) )) ) − ) -) ; (34) = − *) Q + ) + ) ) − *) ) Q + ) ) + ) ) + ) )
) ) − ) ) ) ) − ) ) − − *) ) *Q + ) ) * + ) ) * + ) ) * − ) ) , _ + ) , ) _
) ) − ) ) + − *) ) - Q + ) ) - + ) ) - + ) ) - − ) ) . _ + ) . ) _
) ) − ) ) + − *) ) , Q + ) ) , + ) ) , + ) ) * − ) *) + ) ) ,
) ) − ) ) − − *) ) . Q + ) ) . + ) ) . + ) ) - − ) - ) + ) ) .
) ) − ) ) − − *) (Q) + ) ()) )+ −))()) + * () + ) () + )) ) , − ) -) .
We call (34) as full Duffing pendulum normal equations in the biangular coordinates. Contrary to the reduced ones the order of these equations equals to six. Moreover, analysis of terms near state variables in the last equations shows general pattern in terms determination. According to this pattern the terms in system with moved base points are defined as sum of terms in motionless system and some terms which are cause by base points motions. Simulation results for the considered system are shown in Figure.5. We assume that ω1=0.1 s-1 and ω1=0.2 s-1.
a) System coordinates
Analysis of given curves shows that motion of base points changes system motion.
Applying the nonlinear transformations to a chaotic system allows us to change its dynamic significantly. Such a transformation gives us the possibility to change system state variables and state space domain where the motions of the considered system are defined. Nevertheless, the initial system's nonlinearities and the proposed formulas can be defined in an analytical way without using any numerical methods. At the same time, the use of piecewise linear functions gives us the possibility to simplify its motion equations and design novel chaotic systems.
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