Reliability of Numerical Solutions in Transient Chaos

Abstract: In dealing with nonlinear systems, it is common to use numerical solutions. Unlike the careful behavior towards the numerical results in chaotic regions, the validity of numerical results in regions of transient chaos might not always be taken into consideration. This article demonstrates that using numerical methods to solve systems undergoing transient chaos can be challenging and sometimes unreliable. To illustrate this issue, we use the Lorenz system [1] in the region of transient chaos as an example. We show how the result of the computation might completely change when using different mathematically equivalent expressions. This raises the question of which result should be relied on. To answer this question, we propose a method based on the Lyapunov exponent to determine the reliability of the numerical solution and apply it to the provided example. In fact, this method checks a necessary condition for the validity of the numerical solution. Then, by increasing the precision to the extent suggested by our method, we show that the result of our studied case passes this test. In the end, we briefly discuss the scope and limits of our method.