Two standard methods for numerically estimating Lyapunov exponents are reviewed and it is noted that a numerical integration scheme that preserves orthonormality is required. A procedure is introduced for modifying arbitrary rth order numerical schemes to preserve orthonormality. Convergence is shown for the particular case when explicit Euler's method is taken as the arbitrary method. This mot...

We address the issue of how to identify the equations of a largely unknown chaotic system from knowledge about its state evolution. The technique can be applied to the estimation of parameters that drift slowly with time. To accomplish this, we propose an adaptive strategy that aims at synchronizing the unknown real system with another system whose parameters are adaptively evolved to converge ...

To study the effect of parameter mismatch on the stability in a general fashion, we derive variational equations to analyze the stability of synchronization for coupled near-identical oscillators. We define master stability equations and associated master stability functions, which are independent of the network structure. In particular, we present several examples of coupled near-identical Lor...

Geophysical Fluid Dynamics (GFD) came of age in the 1950s and 1960s, with the mathematically inspired work of Charney, Lorenz, Stommel, Veronis and von Neumann, among others. In the last twenty years or so, mathematical, statistical and numerical developments for the Navier-Stokes equations and basic geophysical fluid dynamics models are stimulating a deepening and broadening of its central con...

Oscillations described by autonomous three-dimensional differential equations display multiple periodicities and chaos at critical parameter values. Regardless of the subsequent scenario the key instability is often an initial bifurcation from a single period oscillation to either its subharmonic of period two, or a symmetry breaking bifurcation. A generalized third-order nonlinear differential...