Chaos in Fractional Standard

Map and Fractional Dissipative

Standard Map

Mark Edelman

Stern College for Women at Yeshiva University,

Courant Institute of Mathematical Sciences at NYU

Dresden, Germany, 2011

Outline

• Motivation

• Fractional Maps.

• Fractional attractors.

• Summary.

Chaos in Fractional Standard

Map and Fractional Dissipative

Standard Map

books published in 2010-2011:

Physics:

1. V. E. Tarasov Dynamics: Application of Fractional Calculus to Dynamics of

Particles, Fields and Media. (Springer) 2011.;

2. V. E. Tarasov, Theoretical Physics Models with Integro-Differentiation of

Fractional Order. (IKI, RCD) 2011 (in Russian)

3. R. Herrmann, Fractional Calculus: An Introduction for Physicists. (World Scientific)

2011;

Modeling and Control:

4. R. Caponetto, G. Dongola, and L. Fortuna, Fractional Order Systems: Modeling

and Control Applications. (World Scientific ) 2010;

5. I. Petras, Fractional-Order Nonlinear Systems. (Springer) 2011.

Viscoelasticity:

6. F. MainardiG. M. , Fractional Calculus and Waves in Linear Viscoelasticity: An

Introduction to Mathematical Models. (Imperial College Press) 2010;

Systems with Long-Rang Interaction:

7. A. C. J. Luo and V. Afraimovich (Eds.), Long-range Interaction, Stochasticity and

Fractional Dynamics. (Springer) 2010;

8. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics. (Oxford University

Press) 2005.

Motivation

• Recognition of the importance of non-linearity in fractional dynamics: references 1, 5 ,and 7 above;

A. Wineman (2007 and 2009) in viscoelasticity, V. Gafiychuk et al. (2008, 2008, 2009, 2010) in reaction - diffusion systems;

and in systems (biological) with memory that can be described by the Volterraintegral equations of the second kind : F. Hoppensteadt, A nonlinear renewal equation with periodic and chaotic solutions. SIAM-AMS Proc., 10 1976.

• Important role of maps in regular dynamics (easy for theoretical analysis and computer simulations).

• New types of attractors and stability discovered as a result of initial

investigation of fractional maps:

1. V.E. Tarasov, G.M. Zaslavsky, Fractional equations of kicked systems and

discrete maps. J. Phys. A 41 2008,

2. M. Edelman, V. E. Tarasov, Fractional standard map. Phys. Let. A 374 2009;

3. V. E. Tarasov, M. Edelman, Fractional dissipative standard map. Chaos 20

2010.

4 M. Edelman, Fractional Standard Map: Riemann-Liouville vs. Caputo. Com.

Nonlin. Sci. Num. Sim., 16 2011.

Motivation

Fractional Attractor

What are the conditions for the existence of different types of

fractional attractors and cascade of bifurcation type

trajectories in particular?

Problem we are trying to address

Fractional Standard Map

)2 (mod

)2 (mod sin

11

1

nnn

nnn

PXX

XKPP

0 where

0)(sin0

ε

nT

txKx

n

Standard Map (SM) also known

as “Chirikov standard map”

Fractional Riemann-Liouville Standard Map.

.)1()( where

),2 (mod )1(1

sin

0

1

11

1

sss

n

i

in

nnn

mmmV

inVPΓ(α)

X

XKPP

Derived from Derived from

.0where

0)(sin0

0

ε

nT

txKxD

n

t

)21( ,)(

)(

)2(

1

0 12

2

0

t

tt

dx

dt

dxD

Fractional Standard MapsFractional Riemann-Liouville

Standard Map

).()( where

),2 (mod )1(1

sin

1

0

0

1

11

1

tXDntPP

inVPΓ(α)

X

XKPP

nn

n

i

in

nnn

Derived from Derived from

0)sin(0

0

n

t nT

txKxD

Fractional Caputo

Standard Map

With the initial conditions

.0 requires and giveswhich

,0)0)(( and )0)((

010

2

01

1

0

Xp P

xDpxD tt

torusaon considered becan , and both, where

,sin11

sinsin11

0

1

01

1

0

2

1

PX

)(X)i(nVΓ(α)

P XX

X)(X)i(nV)Γ(α

KPP

i

n

i

αnn

ni

n

i

αnn

0)sin(0

0

n

t

C nT

txKxD

With the initial conditions

;)0( ,)0)(()0)(()0( 00

11

0 xxpxDxDp tt

C

and momentum P defined as .xP

(Tarasov, Nov.2009)

Fractional Dissipatice Standard

Map (Fractional Zaslavsky Map)

),2 (mod P

sin

1nn1

1

XX

XKPeP

n

nnn

From the equation

1 ,21 where,)sin()()(0

00 α-βαR,qntxtxDqtxDn

tt

The dissipative standard map

can be derived in the form

0

)sin(n

ntXXqX

. ,

1)/q,-( where

qΓεμeK

e

q

q

Two forms of the fractional dissipative standard map where derived in Tarasov &

Edelman, Chaos, Jun. 2010. Here we consider one of them derived from the equation

in the form

function. Gamma incomplete theis and

,)1(,1(),1(),( where

),2 (mod 11

sin

)1(1

0

11

1

Γ(a,b)

baabeabaW

) (q,k-n-WP)Γ(α

μ X

XKPeP

ba

α

n

k

αkn

nnn

Fractional Attractors for K>Kc (example K=4.5)

)sin(2

)(

)(2 ),sin(

)(2

11

111

ll

l

clll

xK

p

VKxV

Kx

)sin(2

)(

)( ,

)()sin(

22

22

ll

l

c

l

l

xK

p

VK

KVx

ncomputatiodirect and

)1( ,)1(

n) large(for sassumption From

. , 11

l

n

nl

n

n

nnnn

ppxx

ppxx

ncomputatiodirect and

1

])1(1[2

n) large(for sassumption From

. ,

1

11

-α

l

n

n

n

ln

nnnn

Anp)(-p

xx

ppxx

CBTT

Evolution of the cascade of

bifurcation type trajectories

with the change in α.

Weak Chaos in CBTT

Cascade of bifurcation type trajectory for α=1.1, K=3.5.

Separation of two trajectories with initial separation Δp=10-6.

FDSM: x0=0, p0=0.7

α=1.2;

K=10 on the right,

K=9.1 on the left

Fractional Attractors (FDSM)

Structure of the chaotic attractor for K =12.93 and different values of α

Fractional Chaotic Attractors (FDSM)

(Γ=5, Ω=0).

105 iterations on each trajectory.

Summary

• The FDSs in general, are more stable than corresponding

systems of higher integer order without memory. They demonstrate

asymptotically attracting fixed and periodic points (sinks),

attracting trajectories, strange attractors, cascade of bifurcation

type trajectories.

• Fractional attractors are different from the regular attractors in

the following: a) they exist in the asymptotic sense; b) the

attracting trajectories may intersect and the strange attractors may

overlap; c) there can be more than one way to approach the same

attractor; d) existence of the cascade of bifurcation type attractors

is a general property of the FDSs.

• In order to find CBTTs in the physical systems or models described

by FDEs it is important to investigate areas of parameters where in

the corresponding integer systems a transition from regular motion

to chaos through the period-doubling bifurcations occurs.

α=1.9 in all cases

a) and d): K=3;

b) K=2; c) K=0.6 .

a), b), and d): n=200;

c): n=500;

N=105 in all cases.

Fractional Attractors (FSMRL phase space, K<Kc)

Fixed Points and

Attracting Slow

Diverging Trajectories

(ASDT)

b) FSMRL; K=2, α=1.4; p0=0.3 - fast; p0=5.3

ASCT; 50000 iterations;

c) FSMRL; K=2, α=1.4; p0=0.3;

d) FSMRL; K=2, α=1.4; p0=5.3 (ASCT);

e) FSMC; K=2, α=1.4; p0=0.3;

f) FSMC; K=3, α=1.9; p0=1.6+0.002i, 0≤i<50

Fractional Attractors (convergence, K<Kc)

FSMRL

Trajectories from the basin of attraction converge

to the stable fixed point fast:

xn~n-1-α , pn~n-α ;

from outside of the basin, slow (attracting slow

converging trajectories - ASCT):

xn~n-α , pn~n-α+1.

FSMC

All trajectories converge at the same slow rate:

xn~n-α+1 , pn~n-α+1.

Attracting Slow Diverging Trajectories (ASDT)

Numerical evaluation of ASDTs

(Fig. a) reveals that pn=Cn2-α .

Then it is possible to deduce

analytically that in ASDTs with

xlim=0 for α≥1.8

Fractional Attractors (FSMRL, K<Kc)

.)1(

)()2(2

,1

)(2

1

2

0

K

nMx

nMpp

n

n

α=1.9 in all cases.

a) FSMRL; K=3, p0=1.6+0.002i,

0≤i<50; SM (α=2) has only one central

island;

b) FSMC; K=3, α=1.9, p0=1.7+0.002i,

0≤i<50;

c) FSMRL; K=2, p0=4+0.08i,

0≤i<125; SM has T=4 islands;

d) FSMC; K=2, p0=-3.14+0.0314i,

0≤i<200;

e) FSMRL; K=0.6, p0=2+0.04i,

0≤i<50; SM has T=2 and T=3 islands;

f) FSMC; K=0.6, p0=-3.14+0.0314i,

0≤i<200;

Fractional Attractors(K<Kc)

FSMRL

Fractional Attractors for K>Kc (K=4.5 attractors)

FSMC

a) xn+1=-xn

pn+1=-pn;

b) two sets

xn+1=xn-π

pn+1=-pn;

c,d,e)a single

trajectory

with po=0.3;

f) FSMRL:

7 disjoint

chaotic

attractors;

FSMC: two

overlaping

attractors

a) α=1.8; b) α=1.71; c,d) α=1.65

e) α=1.45; f) α=1.02

On the left: FSMRL

• K=4.5; α=1.65

• K=3.5; α=1.1

On the right: Caputo

with the same

parameters as in the

case a).

Fractional Attractors

Attracting Cascade of

Bifurcations Type

Trajectories (CBTT)

FSMC p0=2, K=4, α=1.5 FSMC p0=5, K=5.2, α=1.5

Fractional Attractors

Cascade of Bifurcations

Type Trajectories (CBTT)

Vs. Inverse CBTT.

Fractional AttractorsFSM with Caputo derivative.

Fractional Chaotic Attractor: case K=3.5, α=1.1

iterations 500 200,n0

,200/2 ,0 00

npx

iterations 20000 5,n0

,5/2 ,0 00

npx

iterations 50000

,20 ,0 00 px

Fractional chaotic attractors are similar to the chaotic attractors of the dissipative

systems but they depend on the initial conditions. Different attractors may overlap.

Phase Space (K>2π)Riemann-Liouville Fractional Standard Map (FSMRL).

a) K=6.908745; α=1.999 b) K=6.908745; α=1.9