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In this paper, we will derive new understandings of some of the problems of mathematics and physics dealt with in the introduction of the concept of blown‐ups and the study of related traditional thinking logic and establishment of mathematical models. The concept of blown‐ups not only explains clearly the mystery of nonlinearity, but also suggests a change to our traditional way of thinking. Blown‐ups embody a leap from specified local confusions to a general whole evolution.

Blown up theory is very important in modern forecasting science, and will result in revolution not only in forecasting theories but also in applied theories and applied methods. Moreover, the blown‐up theory will involve re‐thinking and re‐formulation of some concepts in traditional theories. This article is a record of dialogue between Professor OuYang and the author on some important issues. It is believed that this record will not only benefit us greatly, but also be inductive for young generations in developing their way of thinking and research directions.

In this article, the historical victories and shortfalls of calculus and calculus‐based methods are briefly described so that a need of developing the theory of blown‐ups is given. Listed are several challenges of the theory of blown‐ups to the traditional thinking logic of continuity and characteristics of the theory of blown‐ups.

Based on the theory of blown‐ups, as described by OuYang in 1995 and 1994, for nonlinear dynamic systems on general pansystems transformation, optimization and panderivatives, etc., in this paper, we employ the method, developed in OuYang (1994), to show that the blown‐ups of the nonlinear heat conductive equation is similar to the evolution of observable ground temperature “currents”. According to this analysis, we conduct a simulation based on the historical data of Tang Shan Earthquake in 1976. The simulation results show that: the blown‐ups of the ground temperature “currents” around the earthquake area can be applied to predict forthcoming earthquakes. As for the most disastrous Tang Shan Earthquake, the prediction time is about five months in advance. If the ground temperature “current” that embodies the earth’s crust movement satisfies the unintegrable panderivative equation, we can with enough information demonstrate blown‐up mechanism and forecast relevant earthquakes.

In this paper, we apply the concept of blown‐ups to discuss the laser evolution model. Our first conclusion implies that: after a certain amount of photons have been absorbed under certain conditions, a blown‐up will occur and so some pulse photons will be emitted, whose stability depends on the value of the parameter. Our second implies that: single model laser equation contains no “chaotic” behaviors, which agrees with laboratory experiments and shows why slaving principle becomes invalid.

This paper contains an application of the study on shallow water currents. With the employment of the tool of the first and the second lines of defense of the blown‐up graphs, and based on the test forecasting done in 1994 using the software developed for the blown‐up graphs, we will analyze the characteristics of the blown‐ups of 1994 torrential rains in Southern China, and pose a new theory on the formation of the rains.

The nonhomogeneous information is applied to the study of sudden and severe weather processes in this paper, based on the theory of blown‐ups (OuYang, 1994). Severe weather processes, such as torrential rain and severe convection, are clarified first by using the nonhomogeneous parameters in the spring in Hubei Province, and second by combining with the severe convection forecasting criteria. We can distinguish the severe convective weather processes and forecast the short‐range severe convective weather by using this theory in the spring in Hubei province. After practical applications from March 15 to May 15 in 1995, it showed that the forecast accuracy was satisfactory.

In this article, a review of the past history of open problems and challenges facing the theory of blown‐ups is given. It expresses the hope that the open problems and challenges will be further studied by our readers in the years to come.

The discontinuity problem of general non‐linear evolutionary equations and the chaos of complex models of poly‐order are studied. The concept of “blown‐up” of Navier‐Stokes equation is discussed, which was introduced by OuYang in 1994 and 1995. The results show that the blown‐up of the model appears during the whole evolution of general non‐linear equations; and that the concept of “chaos” of Lorenz’s model should be restudied questioningly.

Based on the theory of blown‐ups, in this paper, we will analyze chemical reaction problems, such as the rate of chemical reactions, gaseous chemical reactions, liquid chemical reactions, Schlogl’s model and the reaction diffusion equation. This study not only applies the theory of blown‐ups, but also offers a new method and a new thinking logic for nonlinear chemical dynamics. And, in a certain sense, this study brings new insight to the chemical dynamics and related research in laboratories.