We investigate a dynamical system having a special structure namely a codimension-one invariant manifold that is preserved under the variation of parameters. We derive conditions such that bifurcations of codimension-one and of codimension-two occur in the system. The normal forms of these bifurcations are derived explicitly. Both local and global bifurcations are analyzed and yield the transcritical bifurcation as the codimension-one bifurcation while the saddle-node–transcritical interaction and the Hopf–transcritical interactions as the codimension-two bifurcations. The unfolding of this degeneracy is also analyzed and reveal global bifurcations such as homoclinic and heteroclinic bifurcations. We apply our results to a modified Lotka–Volterra model and to an infection model in HIV diseases.

Abstract. In this paper we developed an integrating factor matrix method to derive conditions for the existence of first integrals. We use this novel method to obtain first integrals, along with the conditions for their existence, for two and three dimensional Lotka-Volterra systems with constant terms. The results are compared to previous results obtained by other methods.

We study the interaction of saddle-node and transcritical bifurcations in a Lotka-Volterra model with a constant term representing harvesting or migration. Because some of the equilibria of the model lie on an invariant coordinate axis, both the saddle-node and the transcritical bifurcations are of codimension one. Their interaction can be associated with either a single or a double zero eigenvalue. We show that in the former case, the local bifurcation diagram is given by a nonversal unfolding of the cusp bifurcation whereas in the latter case it is a nonversal unfolding of a degenerate Bogdanov-Takens bifurcation. We present a simple model for each of the two cases to illustrate the possible unfoldings. We analyse the consequences of the generic phase portraits for the Lotka-Volterra system.

In this paper we present an analysis of a three-dimensional singularly perturbed conservative system. We add a constant vector in the vector field to remove one of the symmetries in the system. Using the geometric argument, and a theorem which is derived from the implicit function theorem, we prove the existence of equilibria in the system and also derive some local bifurcations of these equilibria, i.e. saddle-node bifurcations. We also show that although we have two saddle-nodes in the system, the codimension-2 bifurcation called the cusp bifurcation is not possible. More sophisticated bifurcations, such as the Hopf bifurcation and the bifurcation of the created periodic solution are derived by using the numerical continuation software MATCONT. Following the periodic solution, we found a sequence of period-doubling bifurcations which leads to the existence of infinitely many periodic solutions. The coexistence of two attractors is an interesting phenomenon which is observed in this paper. Using asymptotic analysis, we discuss the dynamics in the neighborhood of a particular line in phase space at which the competition between these attractors takes place.

We generalise the concept of duality to systems of ordinary difference equations (or maps). We propose a procedure to construct a chain of systems of equations which are dual, with respect to an integral H, to the given system, by exploiting the integral relation, defined by the upshifted version and the original version of H. When the numerator of the integral relation is biquadratic or multi-linear, we point out conditions where a dual fails to exists. The procedure is applied to several two-component systems obtained as periodic reductions of 2D lattice equations, including the nonlinear Schrödinger system, the two-component potential Korteweg-De Vries equation, the scalar modified Korteweg-De Vries equation, and a modified Boussinesq system.