Limit cycle attractor simulation :

• Implementation using  Coupled Map Lattice (CML)  :

Shown on the right plot the 3D projection of the 11-D attractor of the transmitter state. Both authorized and unauthorized receiver can not observe the transmitter state and need to reconstruct the attractor in a reconstructed embedding space using delays of the coupling signal  y(n) which is transmitted from the transmitter to the receiver (Left figure).

• 3D  projection of  the  transmitter  11D  attractor  :

The system trajectory converges to an attractor which takes the form of a limit cycle :

                     

• Changing the receiver dynamics changes the position of the attractor  :

                 


In order to enhance security of DDE the dynamics of the receiver is altered, preferably at the beginning of each transmitted bit. Changing the transmitter dynamics results in a change in position of the attractor, and prevents an attempt to reconstruct the position of the attractor by monitoring a large number of  trajectories. The attractor shown in the above figure  has different dynamics and therefore different position than the previous attractor   .

• Changing the receiver dynamics changes the position of the attractor  :

At the beginning of each transmitted bit, the transmitter state is set to a random initial value. Shown below is the superposition of  multiple trajectories that starts at a random initial state and converge to the system attractor . The trajectories of the same transmitted bit  converge to the sytsem's  attractor , however the transient depends on the initial state.


                   

• Reconstructed embedding state space of the transmitter :

                   

Since neither authorized nor unauthorized receiver can observe the transmitter secret state, both need to reconstruct the converging trajectories and the attractor in a reconstructed embedding space. Shown in the above figure is a reconstruction of the  attractor  in a  embedding space, E[n],  created using delays of the coupling signal  y :     E[n] =   [ y[n], y[n-1], y[n-2] ]  .