Slides NSC2016 - CHAOTIC PROPERTIES OF THE HNON MAP WITH A LINEAR - PDF document

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/303470659 Slides NSC2016 - CHAOTIC PROPERTIES OF THE HÉNON MAP WITH A LINEAR FILTER Data · May 2016 CITATIONS READS 0 15 2 authors: Rodrigo T Fontes Marcio Eisencraft University of São Paulo University of São Paulo 14 PUBLICATIONS 32 CITATIONS 152 PUBLICATIONS 578 CITATIONS SEE PROFILE SEE PROFILE Some of the authors of this publication are also working on these related projects: Practical Aspects of Chaos-Based Communication Systems View project All content following this page was uploaded by Marcio Eisencraft on 24 May 2016. The user has requested enhancement of the downloaded file.

CHAOTIC PROPERTIES OF THE H´ ENON MAP WITH A LINEAR FILTER Rodrigo T. Fontes and Marcio Eisencraft Escola Polit´ ecnica, University of S˜ ao Paulo 6th International Conference on Nonlinear Science and Complexity 1 / 22

Outline Introduction 1 Motivation: a bandlimited chaos-based communication system 2 H´ enon Map With Linear Filter 3 Conclusions 4 2 / 22

Outline Introduction 1 Motivation: a bandlimited chaos-based communication system 2 H´ enon Map With Linear Filter 3 Conclusions 4 3 / 22

1. Introduction Chaos-based digital communication: Message encoded by a chaotic signal Ultra-wideband applications Possible security improvement Spread spectrum properties Bandwidth problem: Chaotic signals are broadband in general Real-world channels are always bandlimited Chaotic synchronization is sensible to channel imperfections, like band limitations Problems are expected... 4 / 22

1. Introduction We have proposed a bandlimited digital chaos-based communication using discrete-time filters to limit the bandwidth of the chaotic signals EISENCRAFT, M.; FANGANIELLO, R. D. ; BACCAL´ A, L. A. . Synchronization of Discrete-Time Chaotic Systems in Bandlimited Channels. Mathematical Problems in Engineering (Print), v. 2009, p. 1-13, 2009. FONTES, R. T. ; Eisencraft, M. Noise Filtering in Bandlimited Digital Chaos-Based Communication Systems. In: 22nd European Signal Processing Conference (EUSIPCO 2014). We recently demonstrated that introducing the filters does not disturb synchronization FONTES, RODRIGO T. ; Eisencraft, Marcio . A Digital Bandlimited Chaos-based Communication System. Communications in Nonlinear Science & Numerical Simulation, v. 37, p. 374-385, 2016. However, there is no guarantee that the resulting signals are in fact chaotic! Objective Numerically analyze the highest Lyapunov exponent of the orbits of an H´ enon map added with a linear filter as a function of the filter coefficients. We would like to identify the conditions that the filter must satisfy in order to not disturb the chaotic properties of the original map. 5 / 22

Outline Introduction 1 Motivation: a bandlimited chaos-based communication system 2 H´ enon Map With Linear Filter 3 Conclusions 4 6 / 22

Discrete-time Wu e Chua Synchronization master : x ( n + 1) = Ax ( n ) + b + f ( x ( n )) slave : y ( n + 1) = Ay ( n ) + b + f ( x ( n )) A K × K and b K × 1 constants. f ( · ) R K → R K is nonlinear Synchronization error e ( n ) � y ( n ) − x ( n ) e ( n + 1) = y ( n + 1) − x ( n + 1) = A ( y ( n ) − x ( n )) = Ae ( n ) Master and slave are completely synchronized if lim n →∞ e ( n ) = 0 This happens only iff eigenvalues λ i of A satisfy | λ i | < 1 , 1 ≤ i ≤ K 7 / 22

The Communication System ′ ( ) m n ( ) m n ( ) ˆ( ) d y r m n ✄ , ( ) w n ( ) ( ) s n r n ( ) ( ) ( ) f f c x m ⋅ ⋅ ✁ , ������� 1 ( ) y n x n 1 ( ) b b y ( ) n x ( ) n A A z z ✂ � ✂ � ����������� �������� Master Slave x ( n + 1) = Ax ( n ) + b + f ( s ( n )) y ( n + 1) = Ay ( n ) + b + f ( r ( n )) s ( n ) = c ( x 1 ( n ) ,m ( n )) m ′ ( n ) = d ( y 1 ( n ) ,r ( n )) m ( n ) = sign ( m ′ ( n )) ˆ 8 / 22

H´ enon map Two-dimensional H´ enon Map: � x 1 ( n + 1) � α − x 2 � � 1 ( n ) + βx 2 ( n ) x ( n + 1) = = x 2 ( n + 1) x 1 ( n ) 2 1 x1 0 −1 −2 0 10 20 30 40 50 60 70 80 90 100 2 1 x2 0 −1 −2 0 10 20 30 40 50 60 70 80 90 100 n x 1 ( n ) and x 2 ( n ) for x (0) = [0 . 000 , 0 . 000] T (solid line) and x (0) = [0 . 000 , 0 . 001] T (dashed line), α = 1 . 4 and β = 0 . 3 9 / 22

Introducing low-pass filters and channel Filtered H´ enon map: c j , 0 ≤ j ≤ N S − 1 , α − x 2    3 ( n ) + βx 2 ( n )  x 1 ( n + 1) filter coefficients  = x 2 ( n + 1) x 1 ( n )    � N S − 1 x 3 ( n + 1) c j x 1 ( n + 1 − j ) j =0 10 / 22

Introducing low-pass filters and channel 2 30 M ( ω ) 20 m ( n ) 0 10 − 2 00 5180 5200 5220 5240 0 . 5 1 2 30 20 s ( n ) S ( ω ) 0 10 − 2 00 5180 5200 5220 5240 0 . 5 1 2 30 M ( ω ) 20 m ( n ) 0 ˆ ˆ 10 − 2 00 1 5180 5200 5220 5240 0 . 5 n ω/π H S ( ω ) = 1 r ( n ) = s ( n ) 11 / 22

H´ enon map 2 30 2 30 M ( ω ) M ( ω ) m ( n ) 20 m ( n ) 20 0 0 10 10 − 2 00 − 2 00 5180 5200 5220 5240 0 . 5 1 5180 5200 5220 5240 0 . 5 1 2 2 30 30 20 20 S ( ω ) S ( ω ) s ( n ) s ( n ) 0 0 10 10 − 2 − 2 00 00 5180 5200 5220 5240 0 . 5 1 5180 5200 5220 5240 0 . 5 1 30 2 30 2 M ( ω ) M ( ω ) m ( n ) 20 m ( n ) 20 0 0 ˆ ˆ ˆ 10 ˆ 10 − 2 00 − 2 00 5180 5200 5220 5240 0 . 5 1 5180 5200 5220 5240 0 . 5 1 n ω/π n ω/π H S ( ω ) = 1 ω c = 0 . 95 π ω c = 0 . 95 π ω s = 0 . 4 π Problem: Does the transmitted signal remain chaotic? 12 / 22

Outline Introduction 1 Motivation: a bandlimited chaos-based communication system 2 H´ enon Map With Linear Filter 3 Conclusions 4 13 / 22

N S = 1 , c 0 = 1 - No filter N S = 1 : c 0 = 1 , x 3 ( n ) = x 1 ( n )  α − x 2  3 ( n ) + βx 2 ( n ) β = 0 . 3 x ( n + 1) = x 1 ( n )   α = 1 . 4 → Chaos � α − x 2 � c 0 3 ( n ) + βx 2 ( n ) 14 / 22

N s = 1 , c 0 � = 1 N S = 1 : 0 < c 0 ≤ 1  α − x 2  3 ( n ) + βx 2 ( n ) β = 0 . 3 , α = 1 . 4 x ( n + 1) = x 1 ( n )   c 0 > 0 . 87 → Chaos � α − x 2 � c 0 3 ( n ) + βx 2 ( n ) 15 / 22

N s = 2 N S = 2 : β = 0 . 3 , α = 1 . 4  α − x 2  3 ( n ) + βx 2 ( n ) Purple - h < 0 x ( n +1) = x 1 ( n )   Yellow - h > 0 � α − x 2 � c 0 3 ( n ) + βx 2 ( n ) + c 1 x 1 ( n ) 16 / 22

Low-pass Hamming filters α = 0 . 9 , β = 0 . 3 0 . 04 0 . 03 0 . 02 0 . 01 0 h − 0 . 01 − 0 . 02 N S = 5 N S = 7 N S = 10 − 0 . 03 N S = 20 N S = 50 − 0 . 04 N S = 100 N S = 200 − 0 . 05 0 0 . 2 0 . 4 0 . 6 0 . 8 1 ω/π 17 / 22

Low-pass Hamming filters α = 0 . 9 , β = 0 . 3 0 . 03 ω s = 0 . 25 ω s = 0 . 50 ω s = 0 . 75 0 . 02 0 . 01 0 h − 0 . 01 − 0 . 02 − 0 . 03 0 50 100 150 200 N S 18 / 22

Low-pass Hamming filters N S = 10 , ω S = 0 . 50 , h < 0 2 1 1 X 1 ( ω ) x 1 ( n ) 0 − 1 − 2 5000 5100 5200 5300 0 0 . 5 1 N S = 20 , ω S = 0 . 84 , h < 0 2 1 1 X 1 ( ω ) x 1 ( n ) 0 − 1 − 2 5000 5100 5200 5300 0 0 . 5 1 N S ω/π 19 / 22

Low-pass Hamming filters N S = 50 , ω S = 0 . 35 , h > 0 2 1 1 X 1 ( ω ) x 1 ( n ) 0 − 1 − 2 5000 5100 5200 5300 0 0 . 5 1 N S = 100 , ω S = 0 . 15 , h > 0 2 1 1 X 1 ( ω ) x 1 ( n ) 0 − 1 − 2 5000 5100 5200 5300 0 0 . 5 1 N S ω/π 20 / 22

Outline Introduction 1 Motivation: a bandlimited chaos-based communication system 2 H´ enon Map With Linear Filter 3 Conclusions 4 21 / 22

4. Conclusions Filtering a H´ enon map can modify chaotic regions Bandlimited chaos-based communications systems must be carefully projected to transmit chaotic signals More general and systematic results are under investigation 22 / 22

Acknowledgments Thanks to CNPq and FAPESP for financial support View publication stats View publication stats 23 / 22