Zoltán Toroczkai  

Nonlinear Dynamics and Chaos 
AgentBased Systems 





Leapfrogging vortex pairsThe advection problem of passive tracer particles in the timeperiodic velocity field of two leaprogging vortex rings (2 vortex pairs) is investigated in the context of chaotic scattering. We numerically determine a few basic unstable periodic orbits of the tracer dynamics, and the nonattracting chaotic set responsible for the motion of particles injected in front of the vortex system. The latter consists of two parts: a hyperbolic component based on strongly unstable periodic orbits, and a nonhyperbolic component that is close to KAM surfaces. The tracer dynamics has a single dimensionless parameter: the energy of the vortex system. As a new phenomenon, we point out the existence of stable bounded trajectories between the vortex pairs at sufficiently large energies. A quantitative characterization of the tracer dynamics in terms of the socalled free energy function is given and the multifractal spectrum of Lyapunov exponents, the escape rate and other characteristics of the transient chaotic motion are determined.


Two leapfrogging vortex rings (in the 2D cross section they appear as two vortex pairs) penetrating a series of differently colored layers. The fractal mixture of colors sits on the unstable manifold of the tracer dynamics. 



The nonattracting chaotic set of the two leapfrogging vortex pairs at four different instants within one period. 

Á. Péntek, T.
Té and Z. Toroczkai,
Chaotic
advection in the velocity field of leapfrogging vortex pairs, J. Phys.
A: Math. Gen. 28, 2191 (1995)
Á. Péntek, T. Té and Z. Toroczkai, Fractal tracer patterns in open hydrodynamical flows: the case of leapfrogging vortex pairs, Fractals, 3, 33 (1995) Á. Péntek, T. Té and Z. Toroczkai, Transient chaotic mixing in open hydrodynamical flows, Int. J. Bif. Chaos 6, 2619 (1996) 

Fractal basin boundaries: flow past a cylindrical obstacleWe introduce the concept of fractal boundaries in open
hydrodynamical flows based on two gedanken experiments carried out with passive
tracer particles colored differently. It is shown that the signature for the
presence of a chaotic saddle in the advection dynamics is a fractal boundary
between regions of different colors. The fractal parts of the boundaries found
in the two experiments contain either the stable or the unstable manifold of
this chaotic set. We point out that these boundaries coincide with streak lines
passing through appropriately chosen points. As an illustrative example, we
consider a model of the von Kármán vortex street, a time periodic
twodimensional flow of a viscous fluid around a cylinder. 

Á. Péntek, Z. Toroczkai, T. Té, C. Grebogi and J. A. Yorke Fractal boundaries in open hydrodynamical flows: Signatures of chaotic saddles, Phys.Rev.E 51, 4076 (1995)  
A marvel of topological chaos: Wada dye boundariesDyes of different colors advected by twodimensional flows which are asymptotically simple can form a fractal boundary that coincides with a chaotic saddle's unstable manifold. We show that such dye boundaries can have the Wada property: every boundary point of a given color on this fractal set is on the boundary of at least two other colors. The condition for this is the nonempty intersection of the saddle's stable manifold with at least three differently colored domains in the asymptotic inflow region. 

Z. Toroczkai, G. Károlyi, Á. Péntek, T. Té, C. Grebogi and J. A. Yorke Wada dye boundaries in open hydrodynamical flows, Physica A 239, 235 (1997)  
Chaos control Based on a simple geometrical construction, an
algorithm is given for stabilizing hyperbolic periodic orbits of twodimensional
maps, and threedimensional flows. The method does not require analytical
knowledge of the system's dynamics, only a rough location of the linearized
region around the periodic orbit to be stabilized (target region). No knowledge
is needed about the eigenvalues and eigenvectors of the periodic orbit. In case
of flows the perturbative velocity field which controls the system consists of
two dissipative memory terms using previous states of the system as feedback.
This method can thus be accessible to experiments. A single parameter is needed
which is easily measurable from four data points. A novel method to find the
target regions is also given. 

Z. Toroczkai,
Geometric
method for stabilizing unstable periodic orbits, Phys.
Lett. A 190,
71 (1994) B. Sass and Z. Toroczkai, Continuous extension of the geometric control method, J. Phys. A: Math. Gen. 29, 3545 (1996) 

A chaos control algorithm is developed to actively
stabilize unstable periodic orbits of higherdimensional systems. The method
assumes no knowledge of the model equations and a small number of experimentally
accessible parameters. General conditions for controllability are discussed. The
algorithm is applied to the Hamiltonian problem of point vortices inside a
circular cylinder with applications to an experimental plasma system. 

Á. Péntek, J. B. Kadtke and Z. Toroczkai, Stabilizing chaotic vortex trajectories: an example of highdimensional control, Phys. Lett. A 224, 85 (1996)  
Thermodynamic formalism for chaotic mapsIntermittent chaos is investigated by
means of an extended version of the statisticalmechanical
formalism developed by Sato and Honda [Phys. Rev. A 42, 3233
(1990)]. An exact criterion is given to classify intermittent
systems from the point of view of the generated chaotic phases
based on the probability distribution of the laminar lengths
which is an explicitly measurable quantity from the time series.
This criterion provides us with the generalization of the concept
of intermittency which broadens the class of critical phenomena
associated with the spectrum of dynamical entropies. It is shown
that, in contrast to general belief, the presence of the regular
chaos phase (i.e., vanishing Rényi entropies for inverse
temperatures q > 1) is not necessarily related to
intermittency. In fact, the absence of any phase transition or
the appearance of an anomalous chaos phase (i.e., infinite Rényi
entropies for q < 0) is also possible in intermittent systems.
We derive how the pressure, computed from a series of signals of
increasing length, approaches its asymptotic value in the regular
and anomalous phases. 

Z. Toroczkai and Á.
Péntek, Classification
criterion for dynamical systems in intermittent chaos, Phys.
Rev. E 48, 136 (1993)
Z. Toroczkai and Á. Péntek, Detecting phase transitions in intermittent systems by using the thermodynamical formalism, Z. Naturforsch. 49a, 1235 (1994) 

Thermodynamics of spin chains via the RuelleAraki transfer operatorThe Kac model, a spin chain with
exponentially decreasing longrange interactions, is investigated
by means of a simple functional representation of the transfer
operator. An analogy between the thermodynamics of spin chains
and of 1D chaotic maps allows us to use techniques developed for
generalized FrobeniusPerron equations to extract properties of
the spin system, such as free energy and the decay rate of the
correlation function. Although the Kac chain does not exhibit a
phase transition, we find that the correlation decay shows a
nonanalytic behavior at some finite temperature. We are also
interested in a generalized version of the Kac model where the
interaction still decays exponentially but in an oscillating
fashion. This leads to the appearance of complicated patterns in
the free energy caused by frustration which is a typical effect
for disordered systems. By working out the analogy with 1D
chaotic maps in more detail, we show how one can construct maps
with the same thermodynamics as the spin chain. The associated
maps turn out to be not smoothly differentiable, and their
derivatives exhibit fractal properties. 

Á. Péntek, Z. Toroczkai,
D.H. Mayer, and T. Té, Kac Model from a dynamical system's
point of view,
Phys. Rev. E 49, 2026 (1994)
Á. Péntek, Z. Toroczkai, D.H. Mayer, and T. Té, A generalized Kac model as a dynamical system, Z. Naturforsch. 49a, 1212 (1994) 
