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Slides NSC2016 - CHAOTIC PROPERTIES OF THE HÉNON MAP WITH A LINEAR FILTER

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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

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Outline

1

Introduction

2

Motivation: a bandlimited chaos-based communication system

3

H´ enon Map With Linear Filter

4

Conclusions

2 / 22

Outline

1

Introduction

2

Motivation: a bandlimited chaos-based communication system

3

H´ enon Map With Linear Filter

4

Conclusions

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...

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• 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.

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Outline

1

Introduction

2

Motivation: a bandlimited chaos-based communication system

3

H´ enon Map With Linear Filter

4

Conclusions

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)) AK×K and bK×1 constants. f(·) RK → RK 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 limn→∞ e(n) = 0 This happens only iff eigenvalues λi of A satisfy |λi| < 1, 1 ≤ i ≤ K

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The Communication System

( ) m n

( )

⋅ f

( )

c x m

• z

A b

1( )

x n ( ) n x

( )

⋅ f

( )

d y r

• z

A b

1( )

y n

( ) n y

ˆ( ) m n

( ) s n

( ) m n ′

• ( )

w n ( ) r n

Master x(n + 1) = Ax(n) + b + f(s(n)) s(n) = c(x1(n),m(n)) Slave y(n + 1) = Ay(n) + b + f(r(n)) m′(n) = d(y1(n),r(n)) ˆ m(n) = sign (m′(n))

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H´ enon map

Two-dimensional H´ enon Map: x(n + 1) = x1(n + 1) x2(n + 1)

• =

α − x2

1(n) + βx2(n)

x1(n)

• 10

20 30 40 50 60 70 80 90 100 −2 −1 1 2 x1 10 20 30 40 50 60 70 80 90 100 −2 −1 1 2 n x2

x1(n) and x2(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

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Introducing low-pass filters and channel

Filtered H´ enon map:   x1(n + 1) x2(n + 1) x3(n + 1)   =   α − x2

3(n) + βx2(n)

x1(n) NS−1

j=0

cjx1(n + 1 − j)   cj, 0 ≤ j ≤ NS − 1, filter coefficients

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Introducing low-pass filters and channel

m(n) s(n) ˆ m(n) n ω/π M(ω) S(ω) ˆ M(ω) −2 −2 −2 00 00 00 2 2 2 1 1 1 10 10 10 20 20 20 30 30 30 0.5 0.5 0.5 5180 5180 5180 5220 5220 5220 5240 5240 5240 5200 5200 5200

HS(ω) = 1 r(n) = s(n)

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H´ enon map

m(n) s(n) ˆ m(n) n ω/π M(ω) S(ω) ˆ M(ω) −2 −2 −2 00 00 00 2 2 2 1 1 1 10 10 10 20 20 20 30 30 30 0.5 0.5 0.5 5180 5180 5180 5220 5220 5220 5240 5240 5240 5200 5200 5200

HS(ω) = 1 ωc = 0.95π

m(n) s(n) ˆ m(n) n ω/π M(ω) S(ω) ˆ M(ω) −2 −2 −2 00 00 00 2 2 2 1 1 1 10 10 10 20 20 20 30 30 30 0.5 0.5 0.5 5180 5180 5180 5220 5220 5220 5240 5240 5240 5200 5200 5200

ωc = 0.95π ωs = 0.4π

Problem:

Does the transmitted signal remain chaotic?

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Outline

1

Introduction

2

Motivation: a bandlimited chaos-based communication system

3

H´ enon Map With Linear Filter

4

Conclusions

13 / 22

NS = 1, c0 = 1 - No filter

NS = 1: x(n + 1) =   α − x2

3(n) + βx2(n)

x1(n) c0

• α − x2

3(n) + βx2(n)

• 

 c0 = 1, x3(n) = x1(n) β = 0.3 α = 1.4 → Chaos

14 / 22

Ns = 1, c0 = 1

NS = 1: x(n + 1) =   α − x2

3(n) + βx2(n)

x1(n) c0

• α − x2

3(n) + βx2(n)

• 

 0 < c0 ≤ 1 β = 0.3, α = 1.4 c0 > 0.87 → Chaos

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Ns = 2

NS = 2: x(n+1) =   α − x2

3(n) + βx2(n)

x1(n) c0

• α − x2

3(n) + βx2(n)

• + c1x1(n)

  β = 0.3, α = 1.4 Purple - h < 0 Yellow - h > 0

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Low-pass Hamming filters

h ω/π α = 0.9, β = 0.3

NS = 5 NS = 7 NS = 10 NS = 20 NS = 50 NS = 100 NS = 200

0.2 0.4 0.6 0.8 1 −0.05 −0.04 −0.03 −0.02 −0.01 0.04 0.03 0.02 0.01

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Low-pass Hamming filters

50 150 h NS α = 0.9, β = 0.3

ωs = 0.25 ωs = 0.50 ωs = 0.75

100 200 −0.03 −0.02 −0.01 0.03 0.02 0.01

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Low-pass Hamming filters

x1(n) x1(n) NS ω/π NS = 10, ωS = 0.50, h < 0 NS = 20, ωS = 0.84, h < 0 X1(ω) X1(ω)

2 2 1 1 1 1 1 1 −2 −2 −1 −1 5000 5000 5100 5100 5200 5200 5300 5300 0.5 0.5

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Low-pass Hamming filters

x1(n) x1(n) NS ω/π NS = 50, ωS = 0.35, h > 0 NS = 100, ωS = 0.15, h > 0

X1(ω) X1(ω)

2 2 1 1 1 1 1 1 −2 −2 −1 −1 5000 5000 5100 5100 5200 5200 5300 5300 0.5 0.5

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Outline

1

Introduction

2

Motivation: a bandlimited chaos-based communication system

3

H´ enon Map With Linear Filter

4

Conclusions

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

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Acknowledgments

Thanks to CNPq and FAPESP for financial support

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