I'm sorry, but I can see no solution to this problem due to these factors, combined:

* Chaotic system, which implies extreme sensitivity to initial conditions

* Noisy transmission of x1, which implies no precision in the knowledge of initial condition

* No transmission of x2 and x3, implies a very good model and initial conditions well known would be needed to estimate system evolution

Together, these three points to me disprove the possibility of reconstructing x2 and x3 however well you can model the system.

Avinash, Olof's comments should be taken seriously. Nevertheless, we can probably imagine a system whose "chaoticity" is contained entirely in x1, so that, given the direct measurements of x1, we are able to reconstruct x2 and x3 "deterministically". This is our only chance to solve the problem.

In contrast, the issue with EKF you mentioned is purely technical. The Kalman gain is a 3 x 1 matrix. If the filter model is properly designed, this matrix contains three non-zero weights, with which the single measurement y = x1 is distributed among the estimates of x1, x2, and x3.

Anyway, the EKF is based on a linearization technique and therefore not very well suitable for highly nonlinear systems.

Olof Dahlberg and Vasiliy Tereshkov , thank you for looking into the problem.

Actually this problem has already been dealt using EKF in the paper : http://www.sciencedirect.com/science/article/pii/S1007570406001481?np=y

And the results they have published are pretty accurate despite using random initial conditions at receiver (EKF) end.

Transmitter end initial conditions :[-1.0032  2. 3545 - 0.087]

EKF initial conditions : [40 17 17]

I have tried to do this by EKF,UKF, Generic particle filter, Auxiliary Particle filter, Regularized particle filter. But the results are not really satisfactory unless I choose receiver initial conditions close to the transmitter ones.