Estimation of initial conditions from a scalar time series

TL;DR

This paper presents a novel method to estimate initial conditions of multivariable dynamical systems from a single scalar time series, enabling efficient synchronization even in chaotic regimes.

Contribution

The authors introduce a modified multidimensional Newton-Raphson method that incorporates system evolution, allowing accurate initial condition estimation from minimal scalar data.

Findings

Works for periodic and chaotic systems

Effective even with positive Lyapunov exponents

Enables trivial synchronization of chaotic signals

Abstract

We introduce a method to estimate the initial conditions of a mutivariable dynamical system from a scalar signal. The method is based on a modified multidimensional Newton-Raphson method which includes the time evolution of the system. The method can estimate initial conditions of periodic and chaotic systems and the required length of scalar signal is very small. Also, the method works even when the conditional Lyapunov exponent is positive. An important application of our method is that synchronization of two chaotic signals using a scalar signal becomes trivial and instantaneous.