Multi-wing chaotic attractors are highly complex nonlinear dynamical systems. A new four-scroll chaotic attractor was found by energy feedback controlling method in this paper. Spectral analysis shows that the system in the four-wing chaotic mode has very broad frequency bandwidth, verifying its random nature, and indicating the prospect for engineering applications such as secure communications, biology, etc. Creating a chaotic system with a more complicated topological structure such as a multi-scroll or multi-wing attractor, therefore, becomes a desirable task and sometimes a key issue for many engineering applications

Hrušák, Josef

Mayer, Daniel

Štork, Milan

DSpace at University of West Bohemia

ISBN 978-80-261-0386-8, © University of West Bohemia, 2015  A New Chaotic System Based on State Space Energy Feedback  Milan Stork  Dept. of Applied Electronics and Telecommunications stork@kae.zcu.cz Josef Hrusak Dept. of Applied Electronics and Telecommunications/RICE hrusak@kae.zcu.cz Daniel Mayer Dept. of Theory of Electrical Engineering mayer@kte.zcu.cz Abstract - Multi-wing chaotic attractors are highly complex nonlinear dynamical systems. A new four-scroll chaotic attractor was found by energy feedback controlling method in this paper. Spectral analysis shows that the system in the four-wing chaotic mode has very broad frequency bandwidth, verifying its random nature, and indicating the prospect for engineering applications such as secure communications, biology, etc. Creating a chaotic system with a more complicated topological structure such as a multi-scroll or multi-wing attractor, therefore, becomes a desirable task and sometimes a key issue for many engineering applications.  Keywords – state energy; nonlinear; error feedback;   stabilization; dissipation; chaotic I. INTRODUCTION Chaos is a field in mathematics which has found wide application around us. Chaos theory studies the behavior of dynamical systems which are nonlinear, highly initial condition sensitive, having deterministic (rather than probabilistic) underlying rules which every future state of the system must follow. Such systems exhibit aperiodic oscillations in the time series of state variables. It has a large or infinite number of unstable periodic patterns which is commonly termed as order in disorder. Long term prediction is almost impossible due to the sensitive dependence on initial conditions. Though such effect may seem quite unusual but it is however observed in very simple systems, for example, a ball placed at the crest of a hill might roll into different valleys depending on slight difference in the initial position. Most common chaotic phenomenon is observed in case of regular weather prediction. Other application of chaos theory is pervaded in many fields like geology, mathematics, biology, microbiology, computer science, economics, philosophy, politics, population dynamics, psychology, robotics etc. Some real world applications of chaotic time series are computer networks, data encryption, information processing, pattern recognition, economic forecasting, market prediction etc. In this paper it will shown that linear system controlled by a linear controller with energy feedback can generate required types of signals. Depending on parameters of the controller the generated output signal can be sinusoidal, quasi-periodic or even chaotic. II. THE STATE SPACE ENERGY APPROACH Let’s consider a class of finite dimensional nonlinear systems described in the following form   ( )( )x A x x B uy C x x= +=?                     (1) where the matrix C(x), defining the output measurement, is not a’'priori specified, and the structure of matrices A(x) and B reads  1 22 33110 0 0 0 10 0 0 0 00 0 . . .( ) ;  . . 0 .0 0 0 0 00 0 0 0 0 0nnn nnnA x Bα αα αααα αα−−−      −      −= =         −   −      ?? ?      (2) We start with presentation of some basic ideas of the state space energy based approach [1, 2, 3, 4, 5]. Let P0(t) denotes the output dissipation power of a zero input causal system with an informational output y(t) defined by:   0 0( ) ( ) , ( ) 0y t P t P t= ± ≥           (3) Let E(t) denotes the instantaneous value of the state space energy (stored in a state vector x(t)): 00 0( ) ( ) ,  :  tE t P d t t tτ τ∞= ∀ =∫             (4) The state space energy conservation principle holds                      0 ( )( ), ( ) P tdE x f xdtψ = −=           (5) where ψ  is the gradient vector of the state space energy potential field E, f is the state space velocity vector, and 〈.,.〉 denotes the operation of dual product. Because the choice of origin and that of the state space coordinate system is free we can define the gradientψ(x) of the energy E in its most simple form:    ( ) Tn2ii=1x x1E = x2ψ = → ∑                   (6) This research has been supported by the European Regional Development Fund and the Ministry of Education, Youth and Sports of the Czech Republic under the Regional Innovation Centre for Electrical Engineering, project No. CZ.1.05/2.1.00/03.0094  and by the internal project SGS-2015-002.  where n is the order of the system representation. In some situations it may be useful to consider the integral of E(x) as an additional concept of the state space hyper-energy J, which divided by the length of interval T=[t0,t1], defines a mean value of  the E(x). III. THE STATE SPACE ENERGY FEEDBACK  In this part we try to attack the “problem of oscillations” not from the standard “observation of reality point of view”, but from the opposite direction, i.e. we intend to develop a consistent approach to the real-world situations from a “generation of controlled oscillations point of view.” The objective is to stabilize the state space energy E(t) on any prescribed value E* by means of a linear  controller, but under the assumption that instead of the measured output signal y(t), the information about the actual value of the state space energy E[x(t)] is assumed to be available to the controller.  At least the following three interpretations are natural: ∑ either the state space energy is continuously measured, and a standard linear “output error“ controller is used, where not the informational output y(t}, but the integrated output dissipation power is used in the feedback informational channel [2], [3], ∑ or the actual state vector x(t) is continuously measured and used in the feedback informational channel; then the corresponding actual value of the state space energy is computed and a standard linear “output error“ controller is used, ∑  or the actual values of the input u(t) and that of the output y(t) are continuously measured and used in the feedback informational channel, including a state reconstructor [3] and a state energy error controller. The structure for single input-single output system with nonlinear part of controller is shown in the Fig.1.     Linear partof controller S*( )E t 0( )x t[ ( )]E x t( )u t ( )y t( )x t   Nonlinear partof controller   STATE VECTORFEEDBACK  NONLINEAR CONTROLLER  Fig.1. Linear system with nonlinear part of controller and the state vector measurement Let's start with analysis of the state representation eq. (1) where the information about the system structure is contained in the triple of matrices (A, B, C), e.g. as defined by eq. (2), where the diagonal elements of the matrix A represent the dissipation parameters, the off-diagonal elements of represent the internal interaction parameters between both the first order subsystems, and the elements of the matrices B and C represent the parameters of external interactions. Two typical solutions of the state energy error control problem are illustrated by simulation results: harmonic oscillations and generation of chaotic oscillations. The block diagram of linear quadrature oscillator with dissipation is shown in Fig. 2. This system can be described by as second order system with the same dissipation α1 and frequency proportional of α2. 1 1 1 2 22 2 1 1 2x x xx x xα αα α= − += − −?                (7) Because of dissipativity, without control, the oscillation in this system vanishes after short time which depends on initial conditions and values of dissipative coefficient. Using of different controllers is possible for amplitude stabilization of oscillations [6, 7]. In this paper the new principle based on state space energy was selected. The block diagram of linear quadrature oscillator with proportional-integral (PI) controller is displayed.  1x 2x∑ ∫2α1α−∑ ∫1α−2α− Fig. 2. Linear quadrature oscillator with dissipation  Σ ∫2-α2α1x? 1xΣ∫2x 2x?1-αu3-αX 1ux+∑ ∫ikpk∑2(.) 2(.)1x2x21x22x*( )E t Linear system0.50.5 Linear controlleru Fig. 3. The linear quadrature oscillator controlled with a state energy error PI-controller  The whole system - oscillator with PI controller with nonlinear feedback is represented by                 ( )( )( )1 1 1 2 2 12 2 1 3 23*2 211 22* 2 211 1 2 32( )( )( ) ( )( )( ) ( )IIPx x x x u tx x xx k E tE t k E E tE t x xu t x k E x x xα αα α= − + += − −== −= + = − + + ????      (8)   The first equation can be rewritten as 1 1 1 2 2( ( ) )x x u t xα α= − +?                 (9) From this equation can be seen that system can be dissipative if  u(t)-α1<0                           (10) or anti-dissipative if  u(t)-α1>0                          (11) therefore with appropriate control this system can hold desired energy. Based on previous results, the dissipativity/anti-dissipativity is controlled by controller which must hold prescribed energy E* where state space energy of oscillator is given as  2 21 21[ ( )] ( )2E x t x x= +                 (12)  Results depend on proportional and integral gains of PI controller. It will show that for some gain values of PI controller the system can be chaotic but holds (in average) the prescribed energy. The system can be therefore simply switched as sinusoidal oscillator or system with chaotic oscillations. IV. RESULTS OF COMPUTER SIMULATIONS The system according Fig. 3, described by eq. (8) was simulated (System 1) for different values of gains of PI controller. It is important to note that prescribed energy was E1*=1.5 (from t=0 to 150) and E2*=1.4 (from 150 to 300) with initial conditions x1(0)=0.1; x2(0)=0; x3(0)=0.   Fig. 4. Phase portrait of system 1 described by eq. (8), for  ki=0.885; kp=0.099; α1=1; α2=1.4; α3=1.4;   Fig. 5. Time evolution of system 1 described by eq. (8), for ki=0.885; kp=0.099; α1=1; α2=1.4; α3=1.4;  The results are shown in following figures: Phase portrait of x1 and x2 - Fig. 4, time evolutions of state space variables - Fig. 5,  prescribed energy of the control systems (dash line) and true energy (solid line) - Fig. 6. The frequency spectrum of the state space variable x1 is shown in Fig. 7.  Fig. 6. Time evaluation of energy (system 1) according eq. (12)). Dash line is prescribed energy, solid is state space energy.   Fig. 7. The frequency spectrum of the state space variable x1 of system described by eq. (8). System 1 In the next part the same system was used, but the sign of 2 coefficient α2 and -α2 was exchanged in time t=150, therefore "rotation" of the system was reversed. The simulation results leads to 4 wing chaotic system (System 2), see phase portrait in Fig. 8 and the next figures 9, 10, 11, time evolution of state space variables, state space energy and frequency spectrum respectively [8 - 11]. Initial conditions are x1(0)=0.1; x2(0)=0.01; x3(0)=0  Fig. 8. Phase portrait of system 2 with reversed rotation, ki=0.885; kp=0.099; α1=1; α2=1.4; α3=1.4;  E1=1.50; E2=1.4   Fig. 9. Time evolution of system 2 with reversed rotation, ki=0.885; kp=0.099; α1=1; α2=1.4; α3=1.4;  E1=1.50; E2=1.4  Fig. 10. Energy of system 2 with reversed rotation, ki=0.885; kp=0.099; α1=1; α2=1.4; α3=1.4;  E1=1.50; E2=1.4  Fig. 11.The frequency spectrum of system 2 with reversed rotation, ki=0.885; kp=0.099; α1=1; α2=1.4; α3=1.4;  E1=1.50; E2=1.4     TABLE I.  SYSTEMS PROPERTIES.  System 1 System 2  x1 x2 x1 x2 Lyap. Exponent 0.031 0.037 0.038 0.048 Hurst exponent 0.71 0.798 0.713 0.80 Capacity Dim. 1.32 1.29 1.36 1.27 Correl. Dim. 1.67 1.71 1.66 1.56  In Tab. I some important characteristic of the booth systems are presented – Largest Lyapunov exponent, Hurst exponent, Capacity dimension and Correlation dimension.  CONCLUSION In this paper the chaotic system based on linear system, linear controller but nonlinear state space feedback was presented and simulated. The theory of state space energy approach was used. Depending on parameters of the controller the system can generate sinusoidal, quasi-periodic or even chaotic signal.  REFERENCES [1] D. J. Hill and P. J. Moylan: “Dissipative dynamical systems: basic input-output and state properties”, Journal of the Franklin Institute, vol. 309, no. 5, pp. 327–357, May 1980.  [2] M. Stork, J. Hrusak, D. Mayer: "Energy based State Space Approach of Nonlinear Systems Simulation and Construction by means of electronic circuits", Recent Researches in Circuits and Systems, Proceedings of the 16th WSEAS International Conference on Systems ISBN: 978-1-61804-108-1, pp.78-84, 2012 [3] D. Mayer, J. Hrusak, M. Stork, "On state-space energy based generalization of Brayton–Moser topological approach to electrical network decomposition," Springer, Computing, 2013. [4] A. Astolfi, R. Ortega, and R. Sepulchre: “Stabilization and disturbance attenuation of nonlinear systems using dissipativity theory”, European Journal of Control, vol. 8, no. 5, pp. 408–433, 2002.  [5] D. Jeltsema: “Modeling and control of nonlinear networks: a power–based perspective”, Ph.D. dissertation, Delft University of Technology, The Netherlands, May 2005. [6] R. Mancini and R. Palmer, “Sine-Wave Oscillator,” Application Report SLOA060 - March 2001, Texas Instruments   [7] Lindberg, E. Oscillators - an Approach for a Better Understanding, Invited tutorial, Proc. of the Europian Conference on Circuit Theory and Design 2003, ECCTD`03, Cracow, Poland, 1-4  September, 2003.  [8] J. Lu, T. Zhou, G. Chen, and X. Yang, “Generating chaos with a switching piecewise-linear controller,”Chaos, vol. 12, no. 2, pp. 344–349, 2002.  [9] W. Liu, W. K. S. Tang, and G. Chen, “2 × 2-scroll attractors generated in a three-dimensional smooth autonomous system,” International Journal of Bifurcation and Chaos, vol. 17, no. 11, pp. 4153–4157, 2007.  [10] J. Lu and G. Chen, “Generating multiscroll chaotic attractors: theories, methods and applications,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 16, no. 4, pp. 775–858, 2006.  [11] S. Dadras and H. R. Momeni, “A novel three-dimensional autonomous chaotic system generating two, three and four-scroll attractors,” Physics Letters A, vol. 373, no. 40, pp. 3637–3642, 2009.  

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A new chaotic system based on state space energy feedback

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A new chaotic system based on state space energy feedback

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