Exponential catastrophe

The above is known as the "exponential catastrophe of a resonant state."... 

The above is in contrast with the exact solution given by Eq. (117), which, as discussed above is free of any exponential catastrophe. 

Thermal explosions may be expected to develop whenever the rate of heat liberation in an exothermic reaction exceeds the rate of heat dissipation by conduction and convection. (An endothermic reaction can never cause a thermal explosion.) Because of the exponential dependence of the reaction rate on temperature, the rate increases rapidly as the temperature rises, until an explosion results. There is little difference, therefore, in the temporal behavior prior to explosion, between explosions that develop as a result of a thermal acceleration of the reaction rate, or those that occur by virtue of a catastrophic build-up of reactive reaction intermediates. 

This section concerns the Cauchy problem or initial value problem, where initial data at time t = 0 are given. It was noticed by Rutkevitch [6,7], and systematized by Joseph et al. [8], Joseph and Saut [9], and Dupret and Marchal [10] that Maxwell type models can present Hadamard instabilities, that is, instabilities to short waves. (See [11] for a recent discussion of more general models.) Then, the Cauchy problem is not well-posed in any good class but analytic. Highly oscillatory initial data will grow exponentially in space at any prescribed time. An ill-posed problem leads to catastrophic instabilities in numerical simulations. For example, even if one initiates the solution in a stable region, one could get arbitrarily close to an unstable one. 

Such a microcrack formation occurs as soon as the strain enhancement at any microfibril end is high enough for material separation. The higher the bulk strain the higher the number of such defects which are deformed so much that a microcrack can be opened. Their number increases almost exponentially with strain as can be concluded from the dependence of radical concentration on bulk strain. But the sample itself is still strong and will hold the load up to the point where the microcrack coalescence yields the first critical size crack which will start to grow catastrophically and will make the sample fail. 

No steady-state reaction is possible. The reaction rate, which is proportional to (X) also increases exponentially with time the autocatalysis reaches catastrophic proportions and explosion takes place. It must be stressed that the primary cause of the explosion is not the accumulation of heat in the system, as occurs in thermal explosions (see Chapter 7). The self-acceleration of the rate can take place isothermally. Naturally, as the reaction rate becomes very high, self-heating of the system may also take place and contribute to the explodon. 

The ISO 26262 standard has been developed to address the exponential growth in complexity of the software integrated into automotive systems, and the inherent potential for catastrophic failure. The standard aims to address these failures, by defining a safety lifecycle-process to ensure that safety is taken into account in the design of electronic systems in automotive applications. 

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