Shaping topologies of complex networks of chaotic mappings using mathematical circuits. René Lozi

Transcription

1 Shaping topologies of complex networks of chaotic mappings using mathematical circuits René Lozi Laboratory J. A. Dieudonné, UMR of CNRS 7351 University of Nice-Sophia Antipolis, Nice, France

2 Mathematical circuits Main idea: equations are bricks used to build "mathematical" machines x ( yx f ( x )), y x y z, z y x ( y x f ( x )), y x y z k1( y y ), 1 1 z y, x ( y x f ( x )), y x y z k2(y y ), 2 2 z y, x ( y x f ( x )), y x y z k 5( y y ), 5 5 z y, x ( yx f ( x )), y x y z, z y.

3 Bottom-Up versus Top-Down approach TOP Physical phenomenon Mathematical «circuit» UP mathematical modelling circuit engineering DOWN Equation model Set of equations BOTTOM

4 Top-Down approach TOP: Rayleigh-Bénard Problem : convection of a liquid heated from below DOWN: the Lorenz strange attractor

5 Edward Lorenz: born on May 23,1917, in West Hartford (Connecticut) dead on April 16, 2008, in Cambridge (Massachusset) Meteorologist: study of the Rayleigh-Bénard problem

6 From Rayleigh-Bénard to Lorenz equation Flow equations in a physical coordinate system (constant along y) ( ) (, ) 2 t ( x,z ) x (, ) t ( x,z ) x Rayleigh number, Prandtl number, (x,t,z) stream function, (x,t,z) temperature perturbation vs linear profile Discretization of equations: ( x,z,t ) m,n( t )sin( amx )sin( nz ) m,n m0 ( x,z,t ) m,n( t )cos( amx )sin( nz ) m,n m0

7 Lorenz Attractor (1963) Flow equations in a physical coordinate system (constant along y) x1 x1 x2 x x x x x x3 x1x2 x , 28, 8 3 "Butterfly effect"

8 Bottom-Up approach How to construct a chaotic system with any number of scrolls? (Professor Ron Chen s research) First, notice that the Lorenz system is invariant under the coordinate transformation: ( x, y, z) ( x, y, z) Second, therefore, one may add an identical system by copy and lift it up along the z-axis: z Z c c 0 Third, make some appropriate re-scaling: x K1X y K2Y z K3Z Finally, timely switch among different sub-systems

9 图 18(a) 图 18(b) 图 18(c) 图 18(d) 9

10 Bottom-Up approach in electrical engineering: BOTTOM: the Chua circuit dvc dt dvc dt di L L v C. 2 dt 1 C1 G( vc -v 2 C )- f ( v 1 C ), 1 2 C2 G( vc -v 1 C ) i 2 L, 1 f ( v ) m v ( m m ) v B v B 2 R 0 R 1 0 R p R p x ( yx f ( x )), y x y z, z y. 1 f ( x ) bx ( a b ) x1 x1 2

11 Leon O. Chua professor of electronics at the university of Berkeley and European Marie Curie Fellowship, Imperial College London, and, (snapshot during his lecture in Nice (electronic Chua s circuit, 1983) University, September 12, 2014) (memristor, 1971)

12 UP: Coupling of 5 Identical Chua s circuits

13 x ( y x f ( x )), y x y z k z x ( y x f ( x )), , ( y y ), y x y z k (y y ), z y y, x ( y x f ( x )), 1 5 y x y z k ( y y ), z y,

14 2. Chaotic vs random numbers Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions. Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for chaotic systems, rendering long-term prediction impossible in general. This property is popularly referred to as the butterfly effect. Random numbers are divided in pure random numbers produced only by physical devices and pseudo-random numbers (PRN). We are interested only in PRN. Chaotic numbers Pseudo-random numbers Pseudo-random numbers Random numbers However it is possible to generate the emergence of pseudo-random numbers from chaotic numbers by the mean of undersampling.

15 2.1 Continuous dynamical systems: Lorenz, Rössler, Chen, models and Chua s electronic circuit Lorenz Attractor (1963) Very popular, proven to be chaotic by W. Tucker only in Tucker, W.: The Lorenz attractor exists, PhD thesis, Uppsala University, Sweden, (1999). Strong dissipativity non reliable numerical computation x1 x1 x2 x x x x x x3 x1x2 x , 28, 3

16 Rössler attractor (1976) In 1976, O. E. Rössler followed a different direction of research to obtain a chaotic model. Considering that, due to extreme simplification used by Lorenz in order to obtain his equation, there is no actual link between this equation and the Rayleigh-Benard problem from which it is originate; he followed a new way in the study of a chemical multi-vibrator. x1 x2 x 3, x2 x1 ax 2, x3 b x 3( x1 c ), a 0.2,b 0.2,c 5.7

17 Otto Rössler ( ), german chimist (Rössler s attractor, 1976)

18 Rössler s chimical multivibrator (from Christophe Letellier and Valérie Messager (2010))

19 Chua attractor (1983) x ( y ( x )) y x y z z y , 28.58, m 0, m ( x ) x g( x ) m1x ( m0 m 1 )[ x 1 x 1 ] 2

20 Chen Attractor (1999) Yet another chaotic attractor, International Journal of Bifurcation and Chaos, 9(7), , 1999 x a( y x ) y ( c a )x xz cy z xy bz a 35,b 3,c 28

21 Chen Attractor (1999) Guangron (Ron) Chen professor City University of Hong Kong, visiting Nice University, July 7, 2014

22 Template of the Chen attractor (Martin Rosalie PhD Thesis Rouen, November 2014)

23 2.2 Two-Dimensional discrete dynamical systems: Hénon mapping (1976) Simplest model of Poincaré map of Lorenz equation. H : a,b 2 2 : y bx x 2 H y 1 ax a,b a 1.4, b 0.3 Associated dynamical system x y 1ax 2 n1 n n yn1 bxn with initial value: Linearized version in 1978 x y 0 0 x y1a x L a,b : y bx a 1.7, b 0.5

24 Michel Hénon ( ), french astronom (observatory of Nice, France) (Hénon mapping, 1976) Michel Hénon s office City of Nice Nice Observatory (under snow!)

25 2.3 One-Dimensional discrete dynamical systems: Logistic and Tent Map Maps of some interval included in the real line f: f : 0,1 0,1 f : 1,1 1,1 Logistic map symmetric tent map f ( x ) rx( 1 x ) r f ( x ) 1 2 x Associated dynamical system xn1 rx ( 1 x ) n n Associated dynamical system xn1 1 2 x n

26 Symmetric tent map f : 1,1 1,1 f ( x ) 1a x a a 2 Invariant measure = Lebesgue measure However numerical instability leads to the collapse of solutions to the unstable fixed point x = ,5 0-0,5-1 -1,0-0,5 0,0 0,5 1,0 symmetric tent map G. Yuan & J. A. Yorke, Collapsing of chaos in one dimensional maps, Physica D, 136,18-30 (2000).

27 Shaping topologies of complex networks of chaotic mappings

28 2.4 The route from chaos to pseudo-randomness via chaotic or mixing undersampling

29 Step 1: Ultra-weak coupling f ( x ) 1 2 x xn1 1 2 x n xn 1 (1 ε 1) f ( xn ) ε 1 f ( yn) yn1 ε 2 f ( xn ) (1 ε 2) f ( yn) Ultra-weak coupling means i 10-7 for floating points or i for double précision numbers Ultra-weak coupling is efficient in order to restore numerically the chaotic properties of chaotic mappings, avoiding any numerical collapse

30 1 X n 1 F X n A xn f ( X n ) X n p xn f( X ) n 1 f( xn ) p f( xn ) A j p =1- ε ε ε ε 1,1 1, j 1,2 1,p-1 1,p j2 j p ε =1- ε ε ε 2,1 2,2 1, j 2,p-1 2,p j1, j2 jp1 ε ε =1- ε p,1 p,p-1 p, p 1, j j1

31 Step 2: Chaotic and mixing under sampling Example in 4-D: Let be three thresholds using directly the sequences One mixes and samples 1 4 n n 2 4 q n n 3 4 n n x iff x T, T 1 2 x x iff x T, T 2 3 x iff x T3,1 instead of x,,,,,, and 0 x1 x2 x n x n x,,,,,, 1 0 x1 x2 x n x n 1 to obtain using x, x1, x2,, x q, x q 1, 0-1 < T 1 < T 2 < T 3 < 1 x, x, x,, x, x, i i i i i n n1 x, x, x 1 1,, x n, x n 1,

32 2.5 Geometric undersampling Step 1: Ring coupling of several tent maps Instead of using one single tent maps f : 1,1 1,1 several (up to 10 or 20) tent maps coupled in a ring way. Moreover we restrain de new p-dimensional map to the torus:, we use simultaneously 1,1 p

33 Ring coupling of 4 tent maps Example with 4 coupled symmetric tent maps: with k i = +1 or f: x 1 2 x k x x 1 2 x k x x 1 2 x k x x 1 2 x k x n1 n 1 n n1 n 2 n n1 n 3 n n1 n 4 n In order to confine the vector X n+1 on the torus 1,1 p, we modify the components in the following way: if ( x 1) add 2 j n1 if ( x 1) substract 2 j n1

34 Ring coupling of p tent maps In the p-dimensional case the mapping is defined by: xn1 1 2 xn k1xn m m m1 xn1 1 2 xn kmxn x 1 2 x k x p p 1 xn1 1 2 xn k pxn p1 p1 p n1 n p1 n with if ( x 1) add 2 j n1 if ( x 1) substract 2 j n1

35 The NIST tests

36 NIST Test for p=10 Example: K 1, K 2, K 3, K 4, K 5, K 6, K 7, K 8, K 9, K 10 = 1 (joint work with Ina Taralova and Andrea Espinel-Rojas) Conclusion: each stream generates pseudo-random numbers

37 Step 2: Ring coupling of 2 tent maps Example with 2 coupled symmetric tent maps: the vector X n+1 is confined on the torus The coefficients k i are set to +1. 1,1 p identified to a square. x 1 2 x x x 1 2 x x n1 n n n1 n n with if ( x 1) add 2 j n1 if ( x 1) substract 2 j n1 It is possible to define critical lines forming a partition of the square.

38 Invariant partition of the square First quadrant (I) Second quadrant (II) Fourth quadrant (IV) Third quadrant (III)

39 Invariant partition of quadrant I First quadrant (I) Image of the first quadrant f(i)

40 Comparison with numerical iterations

41 Geometric subsampling We first select the iterated points belonging to a subsquare of the lozenge Then we enlarge it to the initial square

42 3. Mathematical circuits Electrical circuits: they are composed of individual electronic components (generators, resistors, transitors, capacitors, inductors, diodes) connected by conductives wires through which electric current can flow. Mathematical circuits: they are composed of individual mathematical components (generators, couplers, samplers, mixers, reducers, shapers, ) connected through streams of data. The combination of such mathematical components leads to several new applications (chaotic cryptography, chaotic optimization, evolutionnary algorithms, )

43 Graphical symbols in electrical circuits There are several national and international standards for graphical symbols in circuit diagrams: IEC 60617, ANSI standard Y32 Continuous current discrete data (kown as) IEEE Std 315.

44 3.1 Graphical symbols in «mathematical circuits»: generators 3-D Continuous chaotic generator: Chua equation x ( yx f ( x )), y x y z z y 1 f ( x ) bx ( a b ) x1 x1 2 parameters Electric battery a b 7 7 continuous signal x( t ) ( x( t ),y( t ),z( t ))

45 Discrete generators 2-D Discrete chaotic generator: Hénon mapping x 2 H y 1 ax a,b : y bx xn1 y 1ax yn1bxn parameters a 1.4, b 0.3, discrete signal 1-D generator: Logistic map xn1 4x n( 1 x n ) 2 n ( x y n, n )

46 COUPLER

47 Couplers (Ring coupler, full coupler) Ring coupler x ( y x f ( x )), y x x y z k ( y y ), z y, y x y z k z x ( y x f ( x z ( y x f ( x )), y, 3 2 ( y y ), )), 1 5 y x y z k( y y ), y, y 5 y 4 y 1 y 3 y 2

48 Full coupler It has been shown that the ultra-weak coupling of symmetric tent maps allows the production of series of chaotic numbers equally ditributed on the interval [-1,1]. f ( x ) 1 2 x, x f ( x ) n1 n X F X A f ( X ) n1 n n X n f ( Xn ) x x 1 n p n 1 f ( xn ) p f ( xn ) A j p 1,1=1- ε1, j ε1,2 ε1,p-1 ε1,p j2 j p ε 2,1 2,2=1- ε2, j ε2,p-1 ε 2,p j1, j2 jp1 εp,1 ε p,p-1 p, p=1- εp, j j1

49 Symbol of a full coupler Full coupler

50 Reducer Producing random number from chaotic mapping is possible using special coupling of several such mappings on the torus [-1,1] P. The reducer allows the iterates to stay on the torus xn112 xn k1xn m m m1 xn112 xn kmxn x 12 x k x p p 1 xn112 xn k px n p1 p1 p n1 n p1 n j if( x n1 1 ) add 2 j if ( x n1 1 ) substract 2

51 Chaotic multi-stream pseudorandom number generators (Cms-PRNG) Using together ring coupler, full coupler and reducer, it is possible to build a Pseudo Random Number Generator p p j xn112 xn k1 1 1,j xn 1, jxn j3 j3 x 1 2 x k 1 x x p p m m m1 j n1 n m m, j n m, j n j1, jm;m1 j1, jm;m1 p2 p2 p1 p1 p j xn112 xn k p1 1 p1, j xn p1, jxn j1 j1 x p 1 p 1 p p 1 j n x n k p 1 p, j x n p, j x n j2 j2

52 Mathematical circuit of Cms-PRNG Cms-PRNG producing three streams of uncorrelated pseudo-random numbers, with coding keys.

53 Example of method of chaotic cryptography E. Cherrier and R. Lozi (ICITST 2011) The noise-resisting ciphering method proposed is based on a Chaotic Multi-stream Pseudo Random Number Generator Coding and transmitting principle

54 This method is noise-resisting. Decoding

55 Thank you for your attention