Parameter estimation of dynamical systems via a chaotic ant swarm

Through the construction of suitable objective function, the parameter estimation of the dynamical system could be converted to the problem of parameter optimization. Based on the chaotic ant swarm optimization approach, we investigate the problem of parameter optimization for the dynamical systems in the presence of noise. We systematically analyze the basic relationships among the complexity of objective function, the length of time series, and the performance of the searching algorithm. Furthermore, we consider the effect of measurable additive noise on the objective function. Numerical simulations are also provided to show the effectiveness and feasibility of the proposed methods.

Figure 1

The graph of the objective function for the time series of stable system (8) at $\theta =0.5,x_0=0.3,h=0.001$: (a) $W=2$; (b) $W=20$.

Figure 2

The graphs of the objective function for the time series of stable system (8) with Gaussian noise of zero mean, 0.01 variance, and 0.1 amplitude and here $\theta =0.5,x_0=0.3,h=0.001$: (a) $W=20$; (b) $W=2000$.

Figure 3

The graphs of the objective function for the time series of incipiently unstable system (8) at $\theta =1.5,x_0=0.3,h=0.001$: (a) $W=2$; (b) $W=20$.

Figure 4

The graphs of the objective function for the time series of incipiently unstable system (8) with Gaussian noise of zero mean, 0.01 variance, and 0.1 amplitude and here $\theta =1.5,x_0=0.3,h=0.001$: (a) $W=2$; (b) $W=20$.

Figure 5

The graphs of the objective function for the time series of chaotic system (9) at $\theta =1.85,x_0=0.3,h=0.001$: (a) $W=5$; (b) $W=20$.

Figure 6

The graphs of the objective function for the time series of chaotic system (9) with Gaussian noise of zero mean, 0.01 variance, and 0.1 amplitude and here $\theta =1.85,x_0=0.3,h=0.001$: (a) $W=5$, (b) $W=10$, (c) $W=20$, and (d) $W=100$.

Figure 7

Estimate of parameter $\theta$ in one-dimensional stable linear system, and here, we plot search values of all the ants in order to observe nonlinear dynamical searching process of the ant swarm as a whole.

Figure 8

(Color) The error bars of the estimation results in Tables , where the small red dots are data point, the bigger blue and green dots represent the data mean (M), the bars with blue line show the range (R), and the bars with green line show the standard error (SE). (a) Error bars of the results in Table ; (b) error bars of the results in Table ; (c) error bars of the results in Table ; (d) error bars of the results in Table .

Figure 9

(Color) The error bars of the estimation results in Tables , where the small red dots are data point, the bigger blue and green dots represent the data mean (M), the bars with blue line show the range (R), and the bars with green line show the standard error (SE). (a) Error bars of the results in Table ; (b) error bars of the results in Table .

Figure 10

Estimate of parameter ${\theta }_1$ of Lorenz chaotic system, and here, we plot search values of all the ants in order to observe nonlinear dynamical searching process of the ant swarm as a whole.

Figure 11

Estimate of parameter ${\theta }_2$ of Lorenz chaotic system, and here, we plot search values of all the ants in order to observe nonlinear dynamical searching process of the ant swarm as a whole.

Figure 12

Estimate of parameter ${\theta }_3$ of Lorenz chaotic system, and here, we plot search values of all the ants in order to observe nonlinear dynamical searching process of the ant swarm as a whole.

Figure 13

The chaotic time series of the system $x\left(t+1\right)=x\left(t\right)e^{\left[3-x\left(t\right)\right]}$.