A solution (u(s),v(s)) of the differential system u′=v,v′=−cv−u(u−a)(1−u)+w,w′=−(ɛ/c)(u−γw).with a,c,ɛ∈R such that (u(s),v(s))→(0,0) when s→±∞ is a traveling pulse of the FitzHugh–Nagumo equation. ...The limit of this differential system when ɛ→0 gives rise to the polynomial differential system u′=v,v′=−cv−u(u−a)(1−u)+w,where now a,c,w∈R. We give the complete description of its phase portraits in the Poincaré disc (i.e. in the compactification of R2 adding the circle S1 of the infinity) modulo topological equivalence.
Let x˙=P(x,y), y˙=Q(x,y) be a differential system with P and Q real polynomials, and let d=max{degP,degQ}. A singular point p of this differential system is a global center if R2∖{p} is filled with ...periodic orbits. We prove that if d is even then the polynomial differential systems have no global centers.
We study the global dynamics of the classic May-Leonard model in $\mathbb{R}^3$. Such model depends on two real parameters and its global dynamics is known when the system is completely integrable. ...Using the Poincare compactification on $\mathbb R^3$ we obtain the global dynamics of the classical May-Leonard differential system in $\mathbb{R}^3$ when $\beta =-1-\alpha$. In this case, the system is non-integrable and it admits a Darboux invariant. We provide the global phase portrait in each octant and in the Poincar\'e ball, that is, the compactification of $\mathbb R^3$ in the sphere $\mathbb{S}^2$ at infinity. We also describe the $\omega$-limit and $\alpha$-limit of each of the orbits. For some values of the parameter $\alpha$ we find a separatrix cycle $F$ formed by orbits connecting the finite singular points on the boundary of the first octant and every orbit on this octant has $F$ as the $\omega$-limit. The same holds for the sixth and eighth octants.
We characterize the bounded polynomial vector fields in R2. Additionally we provide a necessary condition but not sufficient which must be satisfied by bounded polynomial vector fields in Rn.
In this paper we classify the phase portrait and the limit sets of a special class of piecewise smooth vector fields in the plane with different configurations of straight lines as switching ...manifolds. We study the behavior at infinity through a Poincaré compactification as well as the relations between canonical regions and vector fields which are defined over the switching regions. Results addressing the global behavior of trajectories, tangency points, vector fields over the switching manifolds, and equilibrium and pseudo-equilibrium points at the finite and the infinite part are stated. In particular, we prove the existence of 123 distinct phase portraits for the class of piecewise smooth vector fields that we consider.
The gravitational Szekeres differential system is completely integrable with two rational first integrals and an additional analytical first integral. We describe the dynamics of the Szekeres system ...when one of these two rational first integrals is negative, showing that all the orbits come from the infinity of R4 and go to infinity.
•The 4D Szekeres system provides exact solutions of a kind of Einstein equations.•This system has two rational first integrals F and H.•We prove that if F<0 then all the orbits come from and go to infinity.
We completely characterize the global phase portraits in the Poincaré disk for all planar Hamiltonian vector fields with linear plus cubic homogeneous terms having a nilpotent saddle at the origin.
•The Abel differential equation has been applied to modelize problems from ecology, control theory, electrical circuits, cosmology,...•The interest for understanding the dynamics of this equation is ...proved by the hundreds of papers dedicated to it in MathSciNet.•In this paper we characterize the phase portraits of the complex Abel polynomial differential equation z′=(z−a)(z−b)(z−c).•The real version of this equation depends on six parameters and consequently the classification of their phase portraits needs some work.
In this paper we characterize the phase portraits of the complex Abel polynomial differential equationsz˙=(z−a)(z−b)(z−c),with z∈C, a,b,c∈C. We give the complete description of their topological phase portraits in the Poincaré disc, i.e. in the compactification of R2 adding the circle S1 of the infinity.
