Failure of parameter identification based on adaptive synchronization techniques

In this paper, several examples as well as their numerical simulations are provided to show some possible failures of parameter identification based on the so-called adaptive synchronization techniques. These failures might arise not only when the synchronized orbit produced by the driving system is designed to be either some kind of equilibrium or to be some kind of periodic orbit, but also when this orbit is deliberately designed to be chaotic. The reason for emergence of these failures is theoretically analyzed in the paper and the boundedness of all trajectories generated by the coupled systems is rigorously proved. Moreover, synchronization techniques are proposed to realize complete synchronization and unknown parameter identification in a class of systems where nonlinear terms are not globally Lipschitz. In addition, unknown parameter identification is studied in coupled systems with time delays.

Figure 1

(Color online) A successful complete synchronization between the Lorenz systems (4, 5) by means of the adaptive design coupling. Here, system (4) possesses an asymptotically stable equilibrium $E={(6.8313,6.8313,1.6667)}^T$ instead of the strange attractor. The variation of the driving signal with the response state are shown in (a) and the evolution of response state in the phase plane are depicted in (b). Here, $r_i=15$, ${\delta }_i=2$, and all the initial values are simply chosen as $x_i^0=y_i^0=10$, $q_i^0=1$, ${ϵ}_i^0=1$ $(i=1,2,3)$.

Figure 2

The variation of the error between the parameters $q_i$ and $p_i$ with time initiating from 0 to 60 with stepsize 0.01 $(i=1,2,3)$. In particular, $q_1$ fails to identify the accurate value of $p_1$. All the parameters and initial values for coupling systems are the same as those given in Fig. 1.

Figure 3

(Color online) The attractive periodic orbit generated by the original Chen’s system [system (6) when $\mathcal{Q}\equiv 0$]. The periodic orbit in the $x_1\text{−}{x}_2\text{−}{x}_3$ phase plane (a) and its projection in the $x_1\text{−}{x}_2$ plane (b).

Figure 4

(Color online) The variation of $q_i$ $(i=1,2)$ with time initiating from 0 to 10 with stepsize 0.01 when its initial value $q_{i\varrho }$ is taken differently $(\varrho =a,b,c,d,e)$. Here, $r_j=2$, ${\delta }_j=1$, and all the initial values are taken as $x_j^0=y_j^0=10$, ${ϵ}_j^0=1$, $j=1,2,3$.

Figure 5

(Color online) The strange attractor produced by system (8) are, respectively, plotted in the $x_1\text{−}{x}_2\text{−}{x}_3$ plane (a) and in the time-state-$x_4$ plane (b). The chaotic property is verified by the Lyapunov exponent portrait (c), where the largest Lyapunov exponent ${\lambda }_1$ is above zero.

Figure 6

(Color online) The variation of the error between the parameters $q_j$ and $p_j$ with time initiating from 0 to 25 with stepsize 0.01. Indeed, $q_{3,4}$ fails to identify the accurate value of $p_{3,4}$, respectively. Here, $r_j=15$, ${\delta }_j=2$, and all the initial values are taken as $x_j^0=1$, $y_1^0=6$, $y_2^0=30$, $y_3^0=10$, $y_4^0=1$, $q_j^0=0$, and ${ϵ}_j^0=1$ $(j=1,2,3,4)$.

Figure 7

(Color online) The variation of $x_3$ and $x_3(1+x_4^3)$ on the synchronized orbit ${\mathbf{x}}^{*}\left(t\right)$ with time, respectively.

Figure 8

(Color online) (a) Successful complete synchronization and parameter identification for chaotic driving signal when $a=-2$ and $b=4$; (b) failed parameter identification when $a=2$ and $b=1$. This failure is simply due to an approximate dependence between $f\mathbf{\left(}x\left(t\right)\mathbf{\right)}$ and $g\mathbf{\left(}x(t-\tau )\mathbf{\right)}$ in a macroscale. Here, both $f$ and $g$ are taken as sinusoid functions, time delays are taken as $\tau =10$, $\delta =2$, and time step size is 0.01.