Invariant Algebraic Surfaces in Three Dimensional Vector Fields
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This work is devoted to investigating the behaviour of invariant algebraic curves for the two dimensional Lotka-Volterra systems and examining almost a geometrical approach for finding invariant algebraic surfaces in three dimensional Lotka-Volterra systems. We consider the twenty three cases of invariant algebraic curves found in Ollagnier (2001) of the two dimensional Lotka-Volterra system in the complex plane and then we explain the geometric nature of each curve, especially at the critical points of the mentioned system. We also investigate the local integrability of two dimensional Lotka-Volterra systems at its critical points using the monodromy method which we extend to use the behaviour of some of the invariant algebraic curves mentioned above. Finally, we investigate invariant algebraic surfaces in three dimensional Lotka- Volterra systems by a geometrical method related with the intersection multiplicity of algebraic surfaces with the axes including the lines at infinity. We will classify both linear and quadratic invariant algebraic surfaces under some assumptions and commence a study of the cubic surfaces.
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