Is there any suggestion to solve a system of ODEs with Mathematica that exhibits singularities?
Solving a problem with the aid of Mathematica does not change the nature of the mathematical problem to be solved. In the case at hand, if by "singularities" you are referring to the singular points of the ODE, then you are to solve the problem as you would do analytically: apply the transformations one ordinarily applies in determining the series solutions of ODEs with regular-singular and/or singular points, solve the transformed problem, then apply the inverse transformation to arrive at the sought-after solution. Any textbook on ODEs has a chapter on the series solution of these equations. For instance, see Chapter 4 in the book Mathematical Methods for the Physical Sciences, by K.F. Riley (Cambridge University Press, Cambridge, 1974).
Solving a problem with the aid of Mathematica does not change the nature of the mathematical problem to be solved. In the case at hand, if by "singularities" you are referring to the singular points of the ODE, then you are to solve the problem as you would do analytically: apply the transformations one ordinarily applies in determining the series solutions of ODEs with regular-singular and/or singular points, solve the transformed problem, then apply the inverse transformation to arrive at the sought-after solution. Any textbook on ODEs has a chapter on the series solution of these equations. For instance, see Chapter 4 in the book Mathematical Methods for the Physical Sciences, by K.F. Riley (Cambridge University Press, Cambridge, 1974).
Behnam's answer has captured the essentials issues that one must deal with when trying to solve ODE's with singularities using Mathematica. For more details on how to do this with Mathematica, check out the following notebooks on singular points on my web site: http://www.ekayasolutions.com/UCDMath/ODEMath.php. PDF versions are also available for download.
A nice application of how to address singular points in the numerical solution of ODEs using Mathematica is the attached PDF on the Lane-Emden equation which describes a model for stellar structures as hydrostatic self-gravitating masses using a polytropic equation of state.
In general, you can apply numerical continuation methods. There is e.g. a nice Matlab-toolbox called "MatCont". I'm not sure, something similar exists for Mathematica.
If you want to stop integration before error-messages would appear use event-control in NDSolve (WhenEvent "StopIntegration")
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