Dynamical systems theory provides powerful tools to help understand problems related to the complex climate system. The TCD group engages in the development of the theory, such as bifurcation theory, Boolean delay equations, mixing and transport, as well as their applications. Other theoretical efforts include the stability of fluid flows and the dynamics of planetary orbits.

Delay equations for Boolean-valued variables (Boolean delay equations: BDEs) are an appropriate mathematical framework for such situations. The feedback thresholds results in the discrete, on-off character of the variables, and the interaction time scales of the feedbacks are expressed as delays. Systems of BDEs have been shown to exhibit complicated and fascinating types of behavior, similar to and also different from systems of ordinary or partial differential equations. The theory of BDEs, initiated about a decade ago, is still being actively developed.

The figure above shows the evolution of solutions, x_{1}=x_{1}(t) and x_{2}=x_{2}(t), for a system of two first-order BDEs that exhibits increase of complexity in time. As time progresses, the number of "jumps" per unit time increases according to a power law.

The figure at right shows a streak line around a pulsating dipole structure, called "thermon," in a two-layer model of oceanic flow, roughly indicating the mixing region. The figure at left shows the mixing rate between the dipoles interior and exterior regions as a function of the pulsation's amplitude.

The understanding of planetary orbits is based on Hamiltonian dynamics, averaging and perturbation theory. Due to their mutual gravitational attraction, planets perturb each other in various ways, on time scales of years and hundreds of years to thousands and millions of years. After averaging over the former, fast variations in their orbits, one obtains the so-called secular system. The dynamics of the linear part of this system can be described by its eigenmodes. One can make further simplifications by considering the phases of these modes relative to that of the largest one.

Our work on planetary motions includes topics such as Lie series, singularly weighted symplectic forms, the Great Inequality, secular resonances and the effects of dissipation on Hamiltonian dynamics. We also developed codes for the numerical integration of planetary orbits.

The latest results on Earth's orbital history show a particularly intriguing chaotic transition 65 million years ago which could be related to the KT impact.

**TCD Members:
**
M. Ghil,
K. Ide, A. Saunders, F. Varadi
References:
Dee, D., and M. Ghil, l984: Boolean difference equations, I: Formulation and dynamic behavior, *SIAM J. Appl. Math.*, **44**, lll-l26.
Ghil, M., and A. Mullhaupt, l985: Boolean delay equations: periodic and aperiodic solutions, *J. Stat. Phys.*, **41**, l25-l74.
Ghil, M., and G. Wolansky, 1992: Non-Hamiltonian perturbations of
integrable systems and resonance trapping, *SIAM J. Appl. Math.*,
**52**, 1148-1171.
Ghil, M., F. Varadi, and W. M. Kaula, 1996: On the secular motion of the Jovian planets, *Dynamics, Ephemerides and Astronomy in the Solar System*, S. Ferraz-Mello, B. Morando and J. E. Arlot (Eds.), IAU Symp. No. 172, pp. 57-60.
Ide, K., and M. Ghil, 1995: An analytical study of geophysical flow estimation and mixing using extended Kalman filtering for vortex models. In *Proc. 2nd WMO Ont'l Symp. on Assim. of Obs. in Meteor. & Oceanogr., Tokyo, March 1995*, WMO/TD-No. 651, PWPR Report Series No. 5, WMO, Geneva, Switzerland, Vol. I, pp. 159-163.
Sakuma, H., and M. Ghil, 1990: Stability of stationary barotropic modons by Lyapunov's direct method, *J. Fluid Mech.*, **211**, 393-416.
Sakuma, H., and M. Ghil, 1991: Stability of propagating modons for small-amplitude perturbations, *Phys. Fluids A*, **3**(3), 408-414.
Varadi, F., M. Ghil, and W. M. Kaula, 1995: The great inequality in a
Hamiltonian planetary theory, *From Newton to Chaos*, A. E. Roy and B. A. Stevens (Eds.), Plenum Press, NY, pp. 103-108.
Varadi, F., C. M. de la Barre, W. M. Kaula, and M. Ghil, 1995:
Singularly weighted symplectic forms and applications to asteroid motion, *Celest. Mech. Dyn. Astron.*, **62**, 23-41.
Wolansky, G., and M. Ghil, 1995: Stability of quasi-geostrophic flow
in a periodic channel, *Phys. Lett. A*, **202**, 111-116.
Wolansky, G., and M. Ghil, 1996: An extension of Arnol'd's second
stability theorem for the Euler equations, *Physica D*, **94**, 161-167.
Wolansky, G., M. Ghil and F. Varadi, 1996: Dynamical effects of the solar nebula's density gradient on asteroids in resonance with Jupiter,
*Icarus*, **132,** 137-150.
Varadi, F., M. Ghil and W. M. Kaula, 1998: Jupiter, Saturn and the edge of chaos, *Icarus*, **139**, 111-116.