We investigate the relationship between the Dynamical Systems analysis and the Lie Symmetry analysis of ordinary differential equations. We undertake this investigation by looking at a relativistic model of self-gravitating charged fluids. Specifically we look at the impact of specific parameters obtained from Lie Symmetries analysis on the qualitative behaviour of the model. Steady states, stability and possible bifurcations are explored. We show that, in some cases, the Lie analysis can help to simply the dynamical systems analysis.
Most of the real world problems (in biology, finance, economics, industry, etc.), and many fundamental laws of physics and chemistry are formulated in the form of differential (or difference) equations. Various methods for solving or analysing such equations have been developed. In the late 19th century, Marius Sophus Lie unified many of these methods by introducing the notion of (what has become known as) Lie groups [23]. The Lie theory of differential equations has been phenomenally successful in determining solutions to differential equations [3] and [21]. It is a useful tool that can be applied to either find solutions explicitly or it can be used to classify equations via equivalence transformations. A major hurdle has been the oftentimes tedious calculations involved in finding the symmetries. However, with the advent of very capable computer packages [11] and [6], this disadvantage has been overcome. Symmetries can be easily calculated for a variety of equations and then used to obtain solutions, if possible.
Another very useful approach to differential equations is that of dynamical systems analysis [26] in which the long-term behaviour of a system is investigated by focusing on linearization around equilibrium points. This approach has its genesis in Newtonian mechanics, and emphasizes on qualitative rather than quantitative questions [2], [17] and [22]. For example, it was eventually realized that equations describing the motion of the three-body problem (sun, earth and moon) were difficult to solve analytically [26]. Instead of focusing only of the exact positions of the planets at all times, people looked at their stability. Thus, the qualitative analysis is helpful, particularly when the exact solution of an equation cannot be found (and also can give more useful information even when exact solutions exist).
Though these two approaches look as they belong to different areas of mathematics, they have some structures in common. For instance, in a natural way, the equivariant bifurcation theory can be viewed as an application of Lie groups in symmetric systems [30]. Therefore, it is natural to consider the use of both approaches when analysing any differential equation of interest. In what follows, we present the results of applying both methods to an Emden–Fowler equation of index three. Such equations have a lengthy history [29], [19], [25], [10] and [14] but, we believe, have not been approached in this two-fold manner. We will show how each method can be used to obtain interesting information about the behaviour of the solutions.
In view of the foregoing, we observe that the constraints obtained by Kweyama et al. [15] in their quantitative study do not fully integrate with our qualitative study. The special values obtained via the Lie symmetry analysis are not special when it comes to the dynamical systems analysis, and with these values there is no bifurcation.
However, in Section 3.3 (where View the MathML source16f22+48g2M+12g2α2>0) it was difficult, even impossible, to pursue the stability analysis of the system, because the equilibrium points were complicated and there was no way to isolate the real values. In this case, we used the corresponding constraints obtained by Kweyama et al. in the Lie symmetry analysis to pursue our dynamical systems analysis and obtain the local behaviour of the system.
Therefore we can use the dynamical system analysis to test the relevance of symmetries, and of solutions (for example, special solutions or not) obtained from the Lie symmetry analysis. We can also predict the stability of the solutions, even possible bifurcations. On the other hand, in certain robust case of qualitative analysis, we can use Lie symmetry analysis to investigate constraints that will enable us to study the system locally. Thus we conclude and advise that it is quite important to utilize both techniques of analysis when dealing with differential equations.
