biparametric investigation of the general standard map: multistability and crises

7/31/2019 Biparametric investigation of the general standard map: multistability and crises

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Introduction Conservative Map Dynamical Features Dissipative Map Dynamical Features Conclusion

Biparametric investigation of the general standard map:multistability and crises

Priscilla A. Sousa Silva, Maisa de Oliveira Terra

Departamento de Matemtica - Diviso de Cincias FudamentaisInstituto Tecnolgico de Aeronutica

Brazil

July 1st

ICNPAA 2010

Priscilla Silva, Maisa Terra (ITA) Biparametric investigation of the general standard map ICNPAA 2010 1 / 27


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Sumary

IntroductionMotivationMathematical ModelNumerical Techniques for Invariant Sets Detection

Conservative Map Dynamical FeaturesPhase space for different values of the forcing parameterInvariant Manifolds of the Origin and of a Period 3 UPO

Dissipative Map Dynamical FeaturesGeneral Bifucation StructureMultistability

Characterization of Abrupt Chaotic Attractors Phenomena

Conclusion



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Introduction

We investigate global bifurcations in the phase space of a biparametric two-dimensionalmap.

The mathematical model presents a wide assortment of dynamical phenomena, beingadequate for studying several fundamental features such as multistability, crises, and

chaotic transients. Starting with the conservative case of the map, we illustrate the mechanisms of cre-

ation, merging and destruction of typical chaotic attractors, as dissipation builds up.

Special attention is payed to the effects of the dissipative and the forcing parameters inthe feature of multistability.

The performed numerical study and characterization of the structures involved in theglobal transformations of the phase space are important to the comprehension of mul-

tistable systems and the observed phenomenology is common to a wide variety ofdynamical systems.



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The General Standard Map

2-D map derived from a model for the periodically

kicked mechanical rotor:

k+1 = k + pk (mod2)pk+1 = (1 )pk + f0sin(k + pk)

(1)

and p are the angular position and momentum ofthe mechanical pendulum.

Two parameters: f0 0 - kick amplitude (forcing applied in the continu-

ous time system at times kT, k = 1, 2,..., with T as a

constant period); (0, 1] - damping.

Ott, E. (1993), Chaos in Dynamical Systems, Cambridge University Press.



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Jacobian matrix:

M =

k+1/k k+1/pkpk+1/k pk+1/pk

M =

1 1

f0cos(k + pk) (1 ) + f0cos(k + pk)

(2)

det M = 1 (3)

When = 0, there is no friction at the pivot - the system is conservative.

When = 1, the map reduces to the 1-D circle map with zero rotation number.



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I d i C i M D i l F Di i i M D i l F C l i


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Computing Chaotic Saddles

Several algorithms for dissipative systems: sprinkler (Hsu, Ott andGrebogi, 1988), pim-triple (Nusse and Yorke, 1989), etc.

Sprinkler underlaying idea: given an attractor A one can define a

restraining region in the phase space R containing a chaotic saddle

C and no attractor. The trajectories of all initial conditions in R will

eventually leave this restraining region and go to the attractor, ex-

cept for those initial conditions lying exactly on the stable manifold

of C.

For general Hamiltonian systems, chaotic saddles must be com-

puted as the intersection of the stable and unstable invariant mani-

folds, unless when exit basins can be defined.

Hsu G.H., Ott, E. and Grebogi, C. (1988), Strange Saddles and the Dimensionsof their Invariant-Manifolds, Physics Letters A, Vol. 127, No. 4, pp. 199-204.

Kantz, H. and Grassberger, P. (1985), Reppelers, Semi-Attractors and Long-LivedChaotic Transients, Physica D, No. 1, pp. 75-86.

Nusse, H.E. and Yorke, J.A. (1989), A Procedure for Finding Numerical Trajecto-ries on Chaotic Saddles, Physica D, Vol. 36, pp. 137-156.

Silva, E.C., et al. (2002), Escape Patterns, Magnetic Footprints, and HomoclinicTangles due to Ergodic Magnetic Limiters, Physics of Plasmas, Vol. 9, No. 12, pp.4917-4928.


I t d ti C ti M D mi l F t Di i ti M D mi l F t C l i


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Conservative Map Dynamical Features

Chirikov-Taylor map.

= 0: dynamics is ruled by typical phenomenology of Hamiltoniansystems.

p can also be taken mod2: dynamics is located on a torus.

f0 = 0: system is integrable and p = p0 is a constant of the motion.

Orbits are given by k = 0 + kp0 (mod2), k = 1, 2,.... The orbit is periodic if p0/2 = m/n, m, n are integers and n= 0. If p0/2 is an irrational number, the orbit is quasiperiodic.

Ott, E. (1993), Chaos in Dynamical Systems, Cambridge University Press.

Lichtenberg and Lieberman (1992), Regular and Chaotic Dynamics, Springer-Verlag.




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Conservative Map Dynamical Features

f0 > 0: integrability is broken. Phase space consists of chaotic seainterspersed with stability islands.

KAM tori restrict the chaotic sea spreading.

The last torus encircling the (, p) cilinder disappears at fc0 0.97(Greene, 1979).

For f0 > fc0

the energy of the rotor can increase without bound.

Chirikov (1979): no discernable torus for f0 > 889

.

See Ott (1993) and references therein.



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Phase space for different values of the forcing parameter

f0

= 1.0, = 0.0




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t oduct o Co se at e ap y a ca eatu es ss pat e ap y a ca eatu es Co c us o


f0

= 2.5, = 0.0



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f0 = 9.0, = 0.0




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Invariant Manifolds

Stable (a) and unstable (b) manifolds of the saddle point at the origin for f0 = 1.0, = 0.0




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Invariant Manifolds

Stable (a) and unstable (b) manifolds of a period 3 UPO for f0 = 4.5, = 0.0




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Invariant Manifolds

Stable (a) and unstable (b) manifolds of the saddle point at the origin for f0 = 4.5, = 0.0




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Dissipative Map Dynamical Features

For > 0 there is a bounded cylinder [, ] [ymax, ymax] that contains all theattractors, with

ymax =f0

.

Bifurcation structure reveals:

Elliptic orbits of the conservative system become sinks when dissi-

pation is introduced.

Multistability.

Long transients.

Dynamical behavior is dominated by appearance and disappear-

ance of periodic attractors of different periods.

The complexity of the bifurcation structure depends strongly on f0and .




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Bifurcation diagram for a set of 20 fixed initial conditions for f0 = 4.5.




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Very Small Dissipation

Multiple asymptotic states for (a)

= 0.002 and (b)

= 0

.02 both withf0 = 4.5.

Feudel et al. (1996): more than 100 co-existing low period attractors.

The number of coexisting

attractors can be made arbitrarily largeas 0.

Basins of attraction are closelyinterwoven extremely lowpredictability of the final state.

Feudel et al. (1996), Map with More than 100 coexisting low-period periodic at-tractors, Physical Review E, Vol. 54, No. 1, pp. 71-81.

Feudel and Grebogi (2003), Why Are Chaotic Attractors Rare in Multistable Sys-tems?, Physical Review Letters, Vol. 91, No. 13, 134102.




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Moderate Dissipation

Chaotic attractor (black) and three 2P attracting orbits(red,blue,green) for f0 = 4.5 and = 0.47330. The

basins are closely interwoven and the structure of thebasin boundary is fractal.

Basins of attraction of the 2P attracting orbits (white, cyan and blue) and ofthe chaotic attractor (magenta).




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Boundary crisis

Left side: stable (blue) and unstable (red) manifolds of the 18P UPO and chaotic attractor (black) for = 0.47333695 andf0 = 4.5. Right side: magnification of the lower part of the phase space shown in left side along with nine points (green crosses)

of the 18P UPO.




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Bifurcation Diagram for Moderate Dissipation

Bifurcation diagram constructed following a stable period-2 solution and using the final state for a given value as the initialcondition for the next , for f0 = 4.5.




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Merging crisis

Left side: as increases the chaotic attractors (black and red) grow until they touch ( = 0.676935) a 2P UPO (green) at theboundary that separates their basins of attraction. The stable manifold of this UPO is shown in blue.

Right side: a single double-banded chaotic attractor results from the crisis.




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Fixed point at the origin

Double banded attractor (black), stable (blue) and unstable (red) manifolds of the fixed point (, p) = (0, 0).



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Conclusion

We have investigated the multistable behavior of the Standard Map when dissipation isintroduced.

For low and moderate to high f0 the phase space is characterized by the presence of amultitude of periodic attractors.

Their basins are highly interwoven and the basin boundaries are fractal. For moderate f0 roughly three different regions in parameter space can be defined as goes

from 0 to 1. As increases, chaotic attractors persist for larger intervals in parameter space.

We have numerically followed and characterized a boundary crisis, as well as a mergingand an interior crises, depicting the role of the invariant structures involved, for the caseof intermediate f0.

Specifically, in the case of the merging and the interior crises last reported, both thestable period-two solution from which a sequence of bifurcations originates the chaoticattractor and the unstable periodic orbit embedded at the chaotic saddle to which theattractor collides are solutions that already exist in the Hamiltonian case.

The performed numerical study and characterization of the structures involved in theglobal transformations of the phase space is important to the comprehension of mul-tistable systems and the observed phenomenology is common to a wide variety ofdynamical systems.


biparametric investigation of the general standard map: multistability and crises

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