Evaluation of dynamical models: Dissipative synchronization and other techniques

Some recent developments for the validation of nonlinear models built from data are reviewed. Besides giving an overall view of the field, a procedure is proposed and investigated based on the concept of dissipative synchronization between the data and the model, which is very useful in validating models that should reproduce dominant dynamical features, like bifurcations, of the original system. In order to assess the discriminating power of the procedure, four well-known benchmarks have been used: namely, Duffing-Ueda, Duffing-Holmes, and van der Pol oscillators, plus the Hénon map. The procedure, developed for discrete-time systems, is focused on the dynamical properties of the model, rather than on statistical issues. For all the systems investigated, it is shown that the discriminating power of the procedure is similar to that of bifurcation diagrams—which in turn is much greater than, say, that of correlation dimension—but at a much lower computational cost.

Figure 1

Graphical interpretation of (a) consistent prediction and (b) inconsistent prediction [54]. In (a) the shaded region indicates that there is a subset of forecasts that are indistinguishable from the “true” state $\left({\mathit{x}}_{i+1}\right)$ within observational uncertainty.

Figure 2

Identification data for the Duffing-Ueda oscillator (a) input and (b) output. These data are used only for building models.

Figure 3

(a) Maximum normalized synchronization error, ${ϵ}_{\mathrm{m}}\left(c\right)$ and (b) cost of synchronization for Duffing-Ueda models: (dot-dashed line) ${\mathcal{M}}_1$, (dashed line) ${\mathcal{M}}_2$, (solid line) ${\mathcal{M}}_3$, and (solid line, boldface) ${\mathcal{M}}_4$.

Figure 4

Correlation dimension curves for the models. The $x$ axis is, as usual, ${\log }_{10}ϵ$. Each plot shows the curves for different realizations of the attractor at $A=11$, obtained initializing the models at different initial conditions. The thick dashed line is the curve that corresponds to the data produced by the differential equation (8): (a) ${\mathcal{M}}_1$, (b) ${\mathcal{M}}_2$, (c) ${\mathcal{M}}_3$, and (d) ${\mathcal{M}}_4$.

Figure 5

Bifurcation diagrams with $u\left(t\right)=A\cos \left(t\right)$ and $4.5\le A\le 12$ for (a) ${\mathcal{M}}_1$, (b) ${\mathcal{M}}_2$, (c) ${\mathcal{M}}_3$, (d) ${\mathcal{M}}_4$, and (e) original equation (8). ${\mathcal{M}}_4$ seems to be the best model in what concerns the sequence of bifurcations.

Figure 6

Maximum normalized synchronization error ${ϵ}_{\mathrm{m}}\left(c\right)$ against the cost of synchronization, $J_{\mathrm{rms}}\left(c\right)$, for Duffing-Holmes models: (dot-dashed line) ${\mathcal{M}}_1$, (dashed line) ${\mathcal{M}}_2$, (solid line) ${\mathcal{M}}_3$, and (solid line, boldface) ${\mathcal{M}}_4$. Each point on the lines corresponds to a particular value of $c$.

Figure 7

Bifurcation diagrams with $u\left(t\right)=A\cos \left(t\right)$ and $0.22\le A\le 0.35$ for (a) ${\mathcal{M}}_1$, (b) ${\mathcal{M}}_2$, (c) ${\mathcal{M}}_3$, (d) ${\mathcal{M}}_4$, and (e) original equation (9). Based on this figure, ${\mathcal{M}}_3$ seems to be the best model.

Figure 8

Maximum normalized synchronization error ${ϵ}_{\mathrm{m}}\left(c\right)$ against the cost of synchronization, $J_{\mathrm{rms}}\left(c\right)$, for van der Pol models: (dot-dashed line) ${\mathcal{M}}_1$, (dashed line) ${\mathcal{M}}_2$, and (solid line) ${\mathcal{M}}_3$. Each point on the lines corresponds to a particular value of $c$ which was varied in the range $0\le c\le 0.9$.

Figure 9

Bifurcation diagrams with $u\left(t\right)=A\cos \left(4t\right)$ and $4\le A\le 25$ for (a) ${\mathcal{M}}_1$, (b) ${\mathcal{M}}_2$, (c) ${\mathcal{M}}_3$, (d), and original equation (10). ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ display several coexisting attractors. Based on this figure, ${\mathcal{M}}_3$ seems to have the diagram which is closest to that of the original oscillator.

Figure 10

Bifurcation diagrams for the example with the Hénon map: (a) ${\mathcal{M}}_1$, (b) ${\mathcal{M}}_2$, (c) ${\mathcal{M}}_3$, and (d) original map. All the maps use $y(k-1{)}^2$ as an independent variable (i.e., in the domain of the map). The associated parameter has been labeled ${\theta }_{11}$. For the original map ${\theta }_{11}=\beta$.

Figure 11

Maximum normalized synchronization error ${ϵ}_{\mathrm{m}}\left(c\right)$ against the cost of synchronization, $J_{\mathrm{rms}}\left(c\right)$, for Hénon models: (dashed line) ${\mathcal{M}}_1$, (dot-dashed line) ${\mathcal{M}}_2$, and (solid line) ${\mathcal{M}}_3$. Each point on the lines corresponds to a particular value of $c$ which was varied in the range $0\le c\le 0.9$.