Bifurcations and catastrophes in a nonlinear dynamical model of the western Pacific subtropical high ridge line index and its evolution mechanism

Abstract

Despite much previous effort, the establishment of an accurate model of the western Pacific subtropical high (WPSH) and analysis of its chaotic behavior has proved to be difficult. Based on a phase-space technique, a nonlinear dynamical model of the WPSH ridge line and summer monsoon factors is constructed here from 50 years of data. Using a genetic algorithm, model inversion and parameter optimization are performed. The Lyapunov spectrum, phase portraits, time history, and Poincaré surface of section of the model are analyzed and an initial-value sensitivity test is performed, showing that the model and data have similar phase portraits and that the model is robust. Based on equilibrium stability criteria, four types of equilibria of the model are analyzed. Bifurcations and catastrophes of the equilibria are studied and related to the physical mechanism and actual weather phenomena. The results show that the onset and enhancement of the Somali low-level jet and the latent heat flux of the Indian monsoon are among the most important reasons for the appearance and maintenance of the double-ridge phenomenon. Violent breakout and enhancement of the Mascarene cold high will cause the WPSH to jump northward, resulting in the “empty plum” phenomenon. In the context of bifurcation and catastrophe in the dynamical system, the influence of the factors considered here on the WPSH has theoretical and practical significance. This work also opens the way to new lines of research on the interaction between the WPSH and the summer monsoon system.

Appendix: Reconstruction idea of the dynamical model

Suppose that the physical law of a nonlinear system evolving with time can be expressed as follows:

$$ \frac{d{q}_i}{dt}={f}_i\left({q}_1,{q}_2,\dots, {q}_i,\dots, {q}_N\right),i=1,2,......N $$

(7)

where f _i is the generalized nonlinear function of q ₁ , q ₂,.  .  .  , q _i  ,  .  .  .  , q _N and N is the number of state variables. Ncan generally be determined by the complexity of dynamical system and measured by calculating its fractal dimensions. The difference form of Eq. 7 can be written as follows:

$$ \frac{q_i^{\left(j+1\right)\varDelta t}-{q}_i^{\left(j-1\right)\varDelta t}}{2\varDelta t}={f}_i\left({q}_1^{j\varDelta t},{q}_2^{j\varDelta t},\dots, {q}_i^{j\varDelta t},\dots, {q}_N^{j\varDelta t}\right)\kern0.5em j=2,3,\dots ..M-1 $$

(8)

where Mis the length of time series of observed data. The model parameters and the system structure can be gained by inversion algorithm based on the observed data.${f}_i\left({q}_1^{j\varDelta t},{q}_2^{j\varDelta t},\dots, {q}_i^{j\varDelta t},\dots, {q}_N^{j\varDelta t}\right)$is an unknown nonlinear function and we assume that ${f}_i\left({q}_1^{j\varDelta t},{q}_2^{j\varDelta t},\dots, {q}_i^{j\varDelta t},\dots, {q}_N^{j\varDelta t}\right)$ contains two parts: G _jk representing the expanding items containing variable q _i and P _ik just representing corresponding parameters which are real numbers ( i = 1 , 2 ,  .  .  . N,j = 1 , 2 ,  .  .  . M, k = 1 , 2 ,  .  .  .  , K). It can be supposed that ${f}_i\left({q}_1,{q}_2,\dots, {q}_n\right)={\displaystyle \sum_{k=1}^K{G}_{jk}{P}_{ik}}$(8). The matrix form of Eq. 8 is D = GP, in which

$$ D=\left\{\begin{array}{c}\hfill {d}_1\hfill \\ {}\hfill {d}_2\hfill \\ {}\hfill \dots \hfill \\ {}\hfill {d}_M\hfill \end{array}\right\}=\left\{\begin{array}{c}\hfill \frac{q_i^{3\varDelta t}-{q}_i^{\varDelta t}}{2\varDelta t}\hfill \\ {}\hfill \frac{q_i^{4\varDelta t}-{q}_i^{2\varDelta t}}{2\varDelta t}\hfill \\ {}\hfill \dots \hfill \\ {}\hfill \frac{q_i^{M\varDelta t}-{q}_i^{\left(M-2\right)\varDelta t}}{2\varDelta t}\hfill \end{array}\right\}\kern0.5em ,G=\left\{\begin{array}{c}\hfill {G}_{11},{G}_{12},......{G}_{1K}\hfill \\ {}\hfill {G}_{21},{G}_{22},......{G}_{2,K}\hfill \\ {}\hfill \dots \hfill \\ {}\hfill {G}_{M1},{G}_{M2},.....{G}_{M,K}\hfill \end{array}\right\},P=\left\{\begin{array}{c}\hfill {P}_{i1}\hfill \\ {}\hfill {P}_{i2}\hfill \\ {}\hfill \dots \hfill \\ {}\hfill {P}_{iK}\hfill \end{array}\right\} $$

Coefficients of the above-generalized unknown equation can be identified through inverting the observed data. Given a vector D, the vector P can be solved to satisfy the above equation. It is a nonlinear system with respect to q; however, it is a linear system with respect to P(assume P is unknown). So the classical least square method can be introduced to estimate the equation and the regular equation G ^T GP = G ^T D can be derived by making the residual sum of squares S = (D − GP)^T(D − GP) minimum.

As G ^T G is usually a singular matrix, its eigenvalues and eigenvectors can be solved easily. After removing those with 0 value, the remaining components are K numbers of λ ₁ , λ ₂,.  .  .  , λ _i which can make up a diagonal matrix Λ _k , and the corresponding K numbers eigenvectors which can form the diagnostic matrix U _L .

with ${V}_L=\frac{G{U}_i}{\lambda_i}$, $H={U}_L{\varLambda}^{-1}{V}_L^T$, equation P = HD can be solved, and the parameter Pcan be obtained.

Based on the above approach, coefficients of the nonlinear dynamical systems can be determined and the nonlinear dynamical equations of observed data can be established.