数理解析研究所講究録
第 1713 巻 2010 年 1-3

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Origin of Intermittency in Economic andSolar-Terrestrial Systems
Abraham C.-L. ChianNational Institute for Space Research (INPE), Brazil
Research Institute for Mathematical Sciences (RIMS), Kyoto
University, Japan
E-mail: achian.dge.inpe.br

Abstract
Characterization of the complex dynamics of economic cycles, byidentifying regular and irregular patterns and regime switching between different dynamic phases in the economic time series, is thekey to improve economic forecasting. Statistical analysis of stock
markets and foreign exchange markets have demonstrated the intermittent nature of nonlinear economic time series, which exhibits
non-Gaussian behavior in the probability distribution function ofprice changes and power-law dependence on frequency in the spectral density. Nonlinear deterministic models of economic dynamics
are capable of simulating intermittent time series due to a transition from order to chaos, or from weak chaos to strong chaos,
which can explain the origin and nature of intermittency observedin economic systems.
We discuss the complex systems approach to intermittency basedon a forced van der Pol oscillator of business cycles and a model ofAlfv\’en chaos in the solar-terrestrial environment. The techniqueof numerical modeling is applied to characterize the fundamental
properties of complex economic and solar-terrestrial systems which

2exhibit multiscale and multistability behaviors, as well as coexistence of order and chaos. In particular, we focus on the dynamics
and structure of unstable periodic orbits and chaotic saddles withina periodic window of the bifurcation diagram, at the onset of asaddle-node bifurcation and at the onset of a crisis, as well as inthe chaotic regions associated with type-I intermittency and crisisinduced intermittency. Inside a periodic window, chaotic saddles
are responsible for the transient motion preceding convergence toa periodic or a chaotic attractor. The links between chaotic saddles, crisis and intermittency in complex systems are discussed. We
show that a chaotic attractor is composed of chaotic saddles andunstable periodic orbits located in the gap regions of chaotic saddles. Both type-I intermittency and crisis-induced intermittency
are the results of the occurrence of explosion at the onset of bifurcation, whereby the gap regions of chaotic saddles are filled by new
unstable periodic orbits are created by the explosion.
Nonlinear modeling of chaotic saddle, crisis and intermittency
can improve our understanding of the dynamics of economic and financial intermittency observed in business cycles and financial markets, as well as the dynamics of climate change and space weather.
In view of the universal mathematical nature of chaotic systems,
the results obtained from our simple prototype model of economic
dynamics can in fact be applied to more complicated economic
scenarios, including spatiotemporal economic systems. Characterization of the complex dynamics of economic systems provides anefficient guide for pattern recognition and forecasting of businessand financial cycles, as well as for optimization of management
strategy and decision technology.

3References
Chian, A. C.-L. Complex Systems Approach to Economic Dyanmics. Lecture Notes in Economics and Mathematical Systems.
Springer, Berlin (2007).

Chian, A. C.-L., Santana, W. M. , Rempel, E. L., Borotto, F. A.,
Hada, T., Kamide, Y. Chaos in driven Alfv\’en systems: unstable
periodic orbits and chaotic saddles. Nonlinear Processes in Geophysics 14, 17 (2007).
Kamide, Y., Chian, A. C.-L. Handbook of Solar-Terrestrial Environment. Springer, Berlin (2007).