- Nonequlibrium Phenomena in General, Chaos, ..
- Coupled Map Lattices :
I introduced a class of models termed coupled map lattice (CML) about two decades ago
( Prog. Theo. Phys. 72 (1984) 480-486; (1985) 1033-1044). It has become one of the
standard tools to study pattern dynamics and spatiotemporal chaos.
Spatiotemporal intermittency (STI), which I reported in the above papers, turns out to be
a standard route to spatiotemporal chaos, while a different class of this STI is found in
K. Kaneko, Physica 34D (1989) 1-41.
Progresses in the study of CML are summarized in
Theory and Applications of Coupled Map Lattices} (Wiley, 1993, K.K. ed.) and
in Chapt.3 of
K. Kaneko and I. Tsuda Complex Systems: Chaos and Beyond
-----A Constructive Approach with Applications in Life Sciences
(Springer, 2000) .
- Globally Coupled Maps, Coupled Map Lattices, or generally coupled dynamical systems:
As a general system in globally coupled dynamical system, I introduced a model
called Globally Coupled Map (GCM). It now becomes one of the basic
models in high-dimensional chaos. The model shows a rich variety of novel phenomena,
in spite of its simplicity; clusterings, hierarchical clusterings, partial order in relationship
with spin glass, chaotic itinerancy, collcetive chaos,...
Some of them are rather well understood, while some are not yet fully clarified.
#### (K. Kaneko ``Clustering, Coding, Switching, Hierarchical Ordering, and Control in Network of Chaotic Elements", Physica 41 D (1990) 137-172; Chapt 4 of K. Kaneko and I. Tsuda Complex Systems: Chaos and Beyond -----A Constructive Approach with Applications in Life Sciences (Springer, 2000) )

Recent progress in collective dynamics is seen in#### T. Shibata, T. Chawanya and K. Kaneko ``Noiseless Collective Motion out of Noisy Chaos" Phys. Rev. Lett., 82 (1999) 4424-4427

#### T. Shibata and K. Kaneko ``Collective Chaos", Phys. Rev. Lett., 81 (1998) 4116-4119

An attempt to include the motion in elements is given by#### T. Shibata and K. Kaneko, Coupled Map Gas, Physica D 181(2003) 197-214

- Ubiquity of Chaotic Itinerancy

In high-dimensional chaos, the state often stays close to a low-dimensional motion, and then is replaced by high-dimensional chaotic motion, before it reaches another low-dimensional ordered motion. This itinerancy over ordereds states is repeated. I have found this chaotic itinerancy in the above GCM model, while similar phenomena are independently discovered by Tsuda (in nonequilibrium neural networks) and by Ikeda et al. (in optical turbulence). This chaotic itinerancy gives a novel a basic concept in high-dimensional dynamical systems, while its relevance to biological phenomena are also discussed. See for example#### K. Kaneko and I. Tsuda " Chaotic Itinerancy" Chaos, 13 (2003) 926-936

Relationship of Chaotic itinrenacy with Milnor attractors is also discussed.#### K. Kaneko ``On the Strength of Attractors in a High-dimensional System: Milnor Attractor Network, Robust Global Attraction, and Noise-induced Selection", Physica D, 124 (1998) 322-344

#### ``Dominance of Milnor Attractors and Noise-induced Selection in a Multi-attractor System", Phys. Rev. Lett., 78 (1997) 2736-2739

- Magic Number 7 in dynamical systems, and Dominance of Milnor Attractors:
How many dimensions are sufficient for `many' degrees of freedom? Is it related with
Combinatorial explosion in some structure in the phase space? Is the number predicted here
related with magic number 7 in pshycology? Does this also set a limit in control in high-dimensional chaos?
Also, we found that Milnor attractors are quite common in higgh-dimensional dynamical systems.
Is it related with the above questions? How is it relhiko Kaneko,
#### K. Kaneko. ``Magic Number 7 +- 2 in Globally Coupled Dynamical Systems" Phys. Rev. E, E.66, 055201(R) 2002

- Open Flow Dynamics:
Coupled dynamical systems with one-way coupling shows novel interesting behaviors,
associated with convective instability. Its relevance to signal transduction is also
discussed.
#### F.H. Willeboordse and K. Kaneko, `` Pattern Dynamics of a Coupled Map Lattice for Open Flow", Physica 86D (1995) 428-455

#### K. Fujimoto and K. Kaneko ``Noise Induced Boundary Dependence through Convective Instability", Physica D 129 (1999) 203-222

#### ``Sensitive boundary condition dependence of noise-sustained structure" Phys. Rev. E., 63 (2001) 036218-222

- Relevance of discreteness in (molecule) number to (reaction) dynamics
We often adopt rate equation to study the change of concentrations
in chemical reaction dynamics. However, in a cell, often the number of each molecule species
is not large eonough to validiate this continuum limit approcimation.
The effect of this smallness in numbers is usually calculated as fluctuations to the
rate equation, say by Langevin equation. When the number is much smaller, however,
the discreteness 0,1,2,... may be important. We found that such discreteness induces a new type
of transitions (symmetry breaking) in the distribution of molecule numbers.
Relevance of the result to complex reaction networks is also under investigation.
#### Y. Togashi and K. Kaneko `` Transitions Induced by the Discreteness of Molecules in a Small Autocatalytic System'' Phys. Rev. Lett. , 86 (2001) 2459-2462; ``Amplification of Chemical Concentrations by Discreteness Induced Transitions in Small Autocatalytic Systems", J. Phys. Soc. Jpn, 72 (2003)62-68 ; Y. Togashi and K. Kaneko" Novel Steady State Induced by Discreteness of Molecules in Reaction-Diffusion Systems; submitted to Phys. Rev. E

- Dynamical Systems with memory
We try to understand 'memory' as a response to external operation
into a dynamical system. An example is given, with the use of
uni-directionally coupled dynamical system:
#### S. Ishihara and K Kaneko Simple dynamical system model of history dependent phenomena J. Phys. Soc. Jpn.

In relation, we are interested in how fast dynamics are `embeded' into slow dynamics (which may look like the converse to `slaving principle').#### K. Fujimoto and K. Kaneko `How fast elements influence slow elements', Physica D.

#### K. Fujimoto and K. Kaneko ; Convective Instability with Time Scale Translation of the Transmitted Disturbance " Physica D, in press

- Function Dynamics
Is it possible to go beyond the framework using separation of
rule and states, taht is generally assimed in dynamical systems? This question
is pursued to set up a general framework for a biological system,
and is answered in the affirmative.
#### N. Kataoka and K. Kaneko, Entangled Network in Function Dynamics, Physica D, 2003

#### Y. Takahashi, N. Kataoka, K. Kaneko, and T. Namiki ``Function Dynamics", Jap J. Appl. Math., 18 (2001) 405-423

#### N. Kataoka and K. Kaneko ``Functional Dynamics II", Physica D 149 (2001) 174-196 ``Functional Dynamics I", Physica D (2000); ``Functional Dynamics for Natural Language", BioSystems, 57 (2000) 1-11

- Dynamics of Plastic Networks
#### Junji Ito and Kunihiko Kaneko `` Spontaneous structure formation in a network of chaotic units with variable connection strengths", Phys. Rev. Lett., 88 (2002) 028701-1

#### `Self-organized hierarchical structure in a plastic network of chaotic units" Neural Networks 13 (2000) 275-281

- Dynamics of Internal Degrees of Freddom, to make Robust Energy conversion possible
#### N. Nakagawa and K. Kaneko, ``Relaxation, the Boltzmann-Jeans Conjecture and Chaos" Phys. Rev. E 64., (2001) 055205(R)-209; J. Phys. Soc. Japan, 69 (2000) 36-39

#### N, Nakagawa, K. Kaneko and T.S. Komatsu, ``Long-term Relaxation of a Composite System in Partial Contact with a Heat Bath", J. Phys. Soc. Jpn, 69 (2000) 3214-3222

#### N. Nakagawa and K. Kaneko " Autonomous Energy Transducer: Proposition, Example, Basic Characteristics Physica A, in press

- Dynamics of Computation
#### A. Saito and K. Kaneko, ``Inaccessibility and Undecidability in Computation, Geometry and Dynamical Systems", Physica D, 155 (2001) 1-33

- Dynamical Systems Game Theory
#### E. Akiyama and K. Kaneko ``Dynamical Systems Game Theory and Dynamics of Games,'' Physica D, 147, (2000) 221-258.

#### ``Dynamical Systems Game Theory II'' Physica D. (2002)

- Dynamics of Differentiation
#### H. Takagi and K. Kaneko ``Differentiation and Replication of Spots in a Reaction Diffusion System with Many Chemicals", Int. J. Bifurcation and Chaos, in press Europhys. Lett., 56 (2001) 145-151.

#### K.Kaneko, ``Coupled Maps with Growth and Death: An Approach to Cell Differentiation", Physica 103 D(1997) 505-527