Our paper deals with the design of mathematical backgrounds to study 2nd-order dynamical systems' phase portraits in planes with non-orthogonal coordinates. We design our approach by using afine coordinate transformations such as shift, rotation, and scale to define the new system phase portrait. Since coordinate transformations are defined by algebraic expressions one can find it is not convenient to use both diferential equations to deifne system motions and algebraic ones to perform the transformations. That is why we ofer to diferentiate transformation expressions and use the obtained derivatives to write down system diferential equations in the transformed coordinate system. Such diferentiation in the most common case gives us the possibility to take into account coordinate systems motions. Moreover, one can define a coordinate system in which motions depend on the motions of the initial dynamical system and time. So, in the most general case, our transformation allows us to define the variable-structure time-dependent systems. We show the use of our approach by considering the dynamic of the Dufing pendulum in the non-orthogonal coordinates. Such transformation makes it possible to design and implement a chaotic generator that forms oscillations diferent from known ones.
Today various nonlinear systems are widely used in diferent scientific and engineering applications because their unique properties [1, 2, 3]. These systems are used to study various biological, meteorological, communication and process control, and technical systems modeling [4, 5, 6, 7, 8, 9, 10].
One of the applications for such systems is control plant prototyping to design novel control algorithms which are used in cyber-physical systems [11, 12, 13, 14, 15] and Industrial Internet of Things’ systems [16, 17, 18, 19, 20, 21].
A lot of various systems with nonlinear dynamic are known [22, 23, 24, 25, 26, 27, 28] and used [29, 30, 22, 23] in diferent communication applications as well [ 31, 32, 33]. However, many authors design and study these systems without any explanation of how and why they use one or another block in their system. Thus, the main drawback of known chaotic systems is their subjective design.
We ofer to avoid this drawback by using a well-known coordinate transformation to give the considered system the desired features and put it attractor in the desired domain of coordinate system. Our method is demonstrating by considering a simple 2nd order chaotic system in the 2D plane, but due to the matrix methods usage it can easily be extended to any N-dimensional dynamical system.
The paper is organized as follows: firstly, we afine transformation from one orthogonal coordinate system into another one and define system matrix motion equation as derivative of transformation equation. Then, we generalize this equation for the case of non-orthogonal coordinates. We illustrate the use of our approach by studying a well-known Dufing equation’s transformation and implementing it by using modern single-chip MCU. Finally, we make a conclusion.
Let us show the principle of the proposed dynamical system transformation by considering the 2nd order harmonic-driven nonlinear dynamical system
¨ 1 = (1, ̇1) + sin( + 0), where 1 is a system state variable, (.) is some nonlinear function, and are parameters of an external harmonic signal.
But our studies can be easy generalized for the case of N-dimensional dynamical system as well.
If one takes into the consideration derivative of system state output as some new state variable , he
can rewrite (
̇1 = 2; ̇2 = (1, 2) + sin( + 0) = (︀ 1 2 )︀ ; ( ) = (︀ 2 (1, 2) )︀ ; (, , 0) = ︂(
0 sin( + 0) In the most general case, here and further one should use n-th sized vectors and matrices to define system dynamic in N-dimensional state space and perform its transformations. Solution of this system allows us to define some phase portrait 1(2) (Figure. 1). in another coordinate system 1′(2′). We think that old 1(2) and new 1′(2′) coordinate systems are interrelated each other by using gain, shift, and rotate transformations, which allows us to define system’s state variables in the new coordinates by using following matrix expression ′ = + 0, ′ =
; =
︂( 1′ )︂
′
2
︂( 1
0
0 )︂
2
; 0 =
here are some gain factors, 0 are shift factors, and is angle between new and old coordinate
systems (Figure 1), is a gain matrix, is a rotation matrix, and 0 is a shift matrix. Here and later
we assume that all terms in matrices , 0, and are time-depended ones.
︂)
︂)
(
From the control theory viewpoint, one can consider (
Solution of (
We ofer to avoid this drawback by transforming (
˙ ′ = ˙0 + ˙ + ˙ + ˙ .
Since the last term in (
It is clear that right-hand expression in (
= ()( ′ − 0),
where (.) means determination of inverse matrix,
and substituting (7) into (
We call (8) as dynamical system’s motion equation in the generalized orthogonal coordinate system. The main feature of this equation is possibility to take into account motion of coordinate system’s origin, axes rotation and scaling as some known time functions.
The particular case of this equation is a system motion’s equation in the stationary rotated, shifted, and scaled coordinate system, which allows us to neglect the first summand in (8)
˙ ′ = ( (()( ′ − 0)) + (, , 0)).
One can use (9) and (8) to define novel motions for known dynamical systems. It is clear that for the nonstationary coordinate systems these motions can significantly difer from known ones.
One of the features of (9) and (8) is a rotation matrix . This matrix has the simplest form in the
case of orthogonal coordinate system. At the same time one can define transformation (
Let us consider the transformation of such a system in details we think that the angle between axes is equal to some angle . Performing of simple trigonometric transformations allows us to rewrite transformation matrix as follows = ︂( only one axis . The position of dynamical system in this axis depend on in this case. When is considered as /2 the previously defined matrix can be obtained.
We show the use of our approach by studying a well-known Dufing’s pendulum equation ¨ + 1̇ + 2 + 33 = 1 cos(),
︂) (︂ . = ︃( ︂( ( ) = 231 =
cos() 2 sin()
sin() 12 = − 2 sin()
11(1′ − 10) + 12(2′ − 20) 22(2′ − 20) + 21(1′ − 10) + 3(232(2′ − 20) + 231(1′ − 10))3 ; 11 = ; 21 = sin( + ) 1 sin() ; 22 = 1 sin() 2 sin( + ) − 1 cos( + ) 1 cos() −
2 sin() 2 sin() ; 232 = − ︂)
Replacing with ′ in (8) gives us the possibility to define the generalized system motion equation for the case of non-orthogonal coordinate system ˙ ′ = ˙0 + (˙′ + ˙′)()( ′ − 0)+ + ′( ((′)( ′ − 0)) + (, , 0)).
Thus, in the most common case, (11) can be used to define motion equations of the generalized dynamical system in some arbitrary coordinate system which can be non-orthogonal one.
Analysis of (15) shows that contrary to (13) the terms of vector-function (15) depends on both pendulum parameters and transformation factors. Moreover, detailed study of the first row in matrix (15) shows that after transformation the pendulum position depends on both its speed and position. This fact proves the above-given sentence about the generalized diferential operator because control theory shows that phase shift in dynamical system with inner feedback is not equal /2.
If one substitutes (15) and (13) into (9) he can write down the Dufing equation in the non-orthogonal coordinate system ⎛ ⎝
11010 + 111′ + 12020 + 122′+ ̇ ′ = ⎜⎜ +13(232(20 − 2′) + 231(10 − 1′))3 + 1 cos() ⎟⎟ ,
21010 + 211′ + 22020 + 222′+
+23(232(20 − 2′) + 231(10 − 1′))3 + 2 cos()
⎞
⎠
are driving force factors.
with matrices
here¨ , ̇, and are pendulum acceleration, speed, and position, are pendulum parameters, 1 and
Since our approach is based on the use of matrix equations let us rewrite (12) into matrix form (
0 1 cos() ︂) 1 sin() (1′ − 10) − 2 sin() (2′ − 20) )︃ sin(+) sin() cos(+) − 1 sin() (1′ − 10) + 2 sin() (2′ − 20) Then, we substitute (13) into matrix-function ( ) 1′ )︂ ′ 2 −
111 cos(); 11 = 111 cos() − 120 = − 112 cos() + 122 sin() 12 = 112 cos() − 121 sin(); 122 sin(); 1 = 11 sin(); 210 = 221( + ) − 22 = − 222 sin( + ) +
212 cos( + ); 2 = 21 sin( + ); 23 = 23 sin( + ).
211 cos( + );
Analysis of (16) shows that the equation has the similar components but with diferent factors. We believe that this fact can be used while control problems are being solved to reduce the order of controlled system but checking of this hypothesis is leaved for future studying.
Numerical solution of (16) is performed by using a well-known Arduino Due board. Numerical simulation program, which solves (16), implements the simplest integration method which is based on the use of finite backward diference with sample time 0.001 s. Signals from the board where obtained to PC by using standard serial communication. In Figure.2 and Figure.3 we show the transformed and classical system attractors.
As one can see our approach allows to scale, rotate, and shift the transformed attractor in comparison with the known one. The above-given results are obtained for the following transformation factors here , = /6 , = /4 and pendulum equation factors 1 = 1, 2 = 0.02, 3 = 5, 1 = 8, = 0.5.
It is clear that the attractor changing is caused by changing chaotic oscillations in the transformed dynamical system. We show transformed and classical pendulums oscillations in Figure.4 and Figure.5 respectively.
Analysis of the above-given results of numerical simulations allows us to claim that even in a quite short time range statically-transformed and classical systems produce diferent outputs. This fact allows us to claim the possibility to generate novel chaotic oscillations by transforming known ones. Here we study the simplest afine transformation with the constant factors but we believe that the use of nonlinear one with variable factors which depend on system state variables and time can dramatically change system motion.
Now we turn our attention into the most general case when transformations factors are some time-depended functions. In this case we use (11) instead of (9) to perform the pendulum model transformation. Also, we take into account that the last summand in (11) is the system motion equation in stationary coordinates which is defined by (9). This fact allows us to use the previously defined equation (16) as the part of following one ⎛ 2(1 − 01)̇1 sin() + 1 ċos() cos( + )+ ⎜ +2(2 − 02)̇1 sin() + 1 ċos() sin( + )+ ⎜ ⎜⎜ +12(̇( 2 − 02) cos2() − ()( 1 − 01) cos() − 2 + 02) +̇ ⎜⎜ +1((1 − 01) cos() + sin()( 2 − 02)) cos() ̇1 + ̇01+ ⎜⎜ +11010 + 111′ + 12020 + 122′+ ̇ ′ = ⎜⎜⎜ +13(232(20 − 2′) + 231(10− 1′ ))3 + 1 cos() ⎜ −( ̇2(2 − 02) − 2(1 − 01)( ̇+ )̇) 2 cos2( + )+ ⎜⎜ +(2((1 − 01)(̇2 + 2(2 − 02)( ̇+ )̇) sin( + )+ ⎜⎜⎜⎜⎜ −( ̇+ +1)̇̇2(1(((1−1− 010)1c)ocso(s)(+)+sinsi(n)(()( 2 −2− 020)2))c)os2(s+in() −+ )+ ⎜⎝ +2(2 − 02)̇2 + ̇02 + 21010 + 211′ + 22020 + 222′+ +23(232(20− 2′ ) + 231(10− 1′))3 + 2 cos() ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (17)
Equation (17) use the most general terms to define derivatives for components of matrices , 0, and ′ by assuming all these components are some functions. The main feature of the transformed system is its determination by using both polynomial and trigonometric functions. One can use this fact to define the classes of system’s nonlinearities which use cause chaotic motions.
One can specify (17) by taking into account diferent transformation functions’ formulas. It is clear that there is no any restriction of the functions’ formulas. Also, we think that these functions make great efect in system dynamic that is why we leave study and design of chaotic systems with specified transformation functions for future research and now show some examples of numerical solution (17). We use the same numerical methods to solve this equation as the above-used. Also, we leave unchanged the MCU board to implement the considered chaotic system.
Implementation of all above-considered dynamical systems are performed by using the simplest feedback diference approximation for derivative operator ̇ ≈ 1 − −1 ℎ , here ℎ is a system discretization period and −1 means backward signal shifting on one discretization period.
It is clear that contrary to transformation of pendulum motion for the case of constant values of transformation factors, in the most general case of variable transformation factors one should take into account both value and derivative of such factor. This fact causes the necessity to define trajectories of these factors’ motions.
Since these trajectories can be defined by tons of diferent expressions, we consider here several of them and leave detailed system studies for future research.
We start our studies from the case of linear time-depended changing of transformation factors = 1 + 0, where is a generalized transformation factor, are parameters of linear dependency and is a system time.
Simulation results for the system (17) with transformation factors (19) are shown if Figure.6-Figure.8 (18) (19)
Analysis of given in Figure.6-Figure.7 simulation results shows that system trajectories after transformation into moved non-orthogonal coordinates dramatically changed in comparison of the initial system. This fact makes strong backgrounds to the design of novel systems. The motion equations for these systems can be defined in terms of system output variable and its derivatives as well as the system time. Thus, the usage of transformation factors (20) allows us to define the new class of systems with chaotic dynamic which motions depend on both state variables and time.
It is clear that the linear dependency of transformation factors from system time define linear rising of system state variables because of the continuous linear origin shifting and gains increasing.
For the case of polynomial dependency of transformation factors from time
= 12 + 0, one can obtain following simulation results (Figure.9-Figure.11),
Shown in Figure.9-Figure.11 simulation results are obtained for following polynomial time-depended transformation factors 1 = 0.0025( + 1)2; 2 = −0.0025( + 2) 2; 01 = 0.00025( + 3)2; 02 = 0.00025(4 − 2) 2; (20) (21) (22)
Analysis of given inFigure.9-Figure.11 simulation results shows that the use of nonlinear time-depended functions to define transformation factors makes system dynamic more complex and allows to increase the oscillation frequency. Also, it allows us to claim that the amplitude of chaotic oscillations in the transformed coordinates depends from the functions which define the transformation factors. This sentence is proven by quadratic rising of oscillation amplitudes which depend on system time.
The above-given simulation results allow us to claim that use of more complex functions to define the transformation factors allows us to form highly-nonlinear system motions. To prove this sentence, we consider following harmonic transformation functions 1 = sin( + 1); 2 = cos( + 2); 01 = sin(0.1 + 0.3); 02 = cos(0.4 − 0.2); = sin( + 0.5); = cos(1.5 + 0.5). (23) (24) Simulation results for dynamical system(17) which use (23) are shown in Figure.12-Figure.14.
Analysis of the given in Figure.7 curves shows that one can use highly-nonlinear time-depended functions to perform system transformation and design novel chaotic system which are defined by nonlinear equations.
All above-considered transformed systems are designed by using information about system time to define the transformation functions. It is clear that such approach is convenient from practical implementation viewpoint only in case when bounded or harmonic function are used. Because in case of unbounded functions after achieving some time value system coordinates can reach very high values which cannot be physically implemented. So, one should use time-depended transformation factors carefully. At the same time the using information about system state variables allows us to solve this problem and design novel chaotic system without using system time in explicit form.
Thus, in Figure.8 and Figure.9 we show the simulation results for the case when time in (20) and (23) are replaced with pendulum state variables. This replacement is defined by using cross-numbering strategy, according to which the factor 1 depends on 2 and vice versa.
The given simulation results prove above-given hypothesis about possibility to use the pendulum state variables to define transformation factors and allow us to claim that highly-nonlinear chaotic system can be designed in case of using nonlinear transformation functions which depend on system state variables.
Applying of known afine transformations to the phase portrait of the generalized dynamical system allows us to transform this phase portrait into non-orthogonal coordinate system. From the control theory viewpoint, in this case, the motion of transformed system is defined with system of matrix diferential motion equation and algebraic observability equation. To fit the use of these equations to known methods of dynamical system analysis and synthesis we represent them with only diferential equations by diferentiating the observability equation. Such an approach allows us to take into account possible motions of the new coordinate system, where the motion of considered dynamical system are transformed, and define matrix expression for this transformation. The use of this expression allows us to write down diferential equations for any transformed dynamical system without performing any coordinate transformation and using observability equation.
Applying of the proposed approach to the existing chaotic systems equations allows us to define novel chaotic systems which attractors are defined in non-orthogonal coordinate systems. Our studies show that using of the above-given approach while Dufing pendulum is being modeled, allows to change the form of its oscillations and shift, scale and rotate its attractor.
We see the future development of our work in applying nonlinear transformations to the considered nonlinear system.
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