We deal with analytic three-dimensional symmetric systems whose origin is a Hopf-zero singularity. Once it is not completely analytically integrable, we provide criteria on the existence of at least one functionally independent analytic first integral. In the generic case, we characterize the analytic partially integrable systems by using orbitally equivalent normal forms. We also solve the problem through the existence of a class of formal inverse Jacobi multiplier of the system.

In this paper we consider the unfolding of saddle-node

\[ X= \frac{1}{xU_a(x,y)}\Big(x(x^{\mu}-\varepsilon)\partial_x-V_a(x)y\partial_y\Big), \]

$(\varepsilon,\,a)$

$\varepsilon \approx 0$

$a$

$A$

$ {\mathbb {R}}^{\alpha },$

$\mathcal {T}(s;\varepsilon,\,a)$

$\partial _s\mathcal {T}(s;\varepsilon,\,a)$

$-\infty$

$(s,\,\varepsilon )\to (0^{+},\,0)$

$A.$

This article studies propagating traveling waves in a class of reaction-diffusion systems which model isothermal autocatalytic chemical reactions as well as microbial growth and competition in a flow reactor. In the context of isothermal autocatalytic systems, two different cases will be studied. The first is autocatalytic chemical reaction of order m without decay. The second is chemical reaction of order m with a decay of order n, where m and n are positive integers and m>n≥1. A typical system in autocatalysis is A+2B→3B and B→C involving two chemical species, a reactant A and an auto-catalyst B and C an inert chemical species.

The numerical computation gives more accurate estimates on minimum speed of traveling waves for autocatalytic reaction without decay, providing useful insight in the study of stability of traveling waves.

For autocatalytic reaction of order m = 2 with linear decay n = 1, which has a particular important role in chemical waves, it is shown numerically that there exist multiple traveling waves with 1, 2 and 3 peaks with certain choices of parameters.

We consider systems of quasi-periodic linear differential equations associated to a ‘resonant’ frequency vector ${\it\omega}$, namely, a vector whose coordinates are not linearly independent over $\mathbb{Z}$. We give sufficient conditions that ensure that a small analytic perturbation of a constant system is analytically conjugate to a ‘resonant cocycle’. We also apply our results to the non-resonant case: we obtain sufficient conditions for reducibility.

In this paper, we give a formal algebraic notion of exponents for linear differential systems at any singularity as eigenvalues of the residue of a regular connection on a maximal lattice (that we call ‘Levelt's lattice’). This allows us to establish upper and lower bounds for the sum of these exponents for differential systems on ${\mathbb P}^{1}(\mathbb{C})$.

The equi-attraction properties concerning the global attractors $\a_\lam$ of dynamical systems $S_\lam(t)$ with parameter $\lam\in\Lam$, where $\Lam$ is a compact metric space, are investigated. In particular, under appropriate conditions, it is shown that the equi-attraction of the family $\{\a_\lam\}$ is equivalent to the continuity of $\a_\lam$ in $\lam$ with respect to the Hausdorff distance.

We discuss the Monge problem for under-determined systems of ordinary differential equations with an arbitrary degree of freedom and give a sufficient condition, in terms of truncated multi-flag systems, for the Monge property to hold. This condition extends in a natural way the Cartan criterion valid for systems with one degree of freedom.

Soit ${{H}_{0}}\,=\,\frac{{{x}^{2}}+{{y}^{2}}}{2}$ un hamiltonien isochrone du plan ${{\mathbb{R}}^{2}}$. On met en évidence une classe d’hamiltoniens isochrones qui sont des perturbations polynomiales de ${{H}_{0}}$. On obtient alors une condition nécessaire d’isochronisme, et un critère de choix pour les hamiltoniens isochrones. On voit ce résultat comme étant une généralisation du caractère isochrone des perturbations hamiltoniennes homogènes considérées dans $\left[ \text{L} \right],\,\left[ \text{P} \right],\,\left[ \text{S} \right]$.

In a vicinity of a stationary solution we consider a real analytic system of ODE of order four, depending on a small parameter. We look for families of periodic solutions which contract to the stationary solution, when the parameter tends to zero. We apply the general methods developed in [2] for the study of complex bifurcations and in [4] for local resolutions of singularities.

We will prove the Theorem of Hartman-Grobman in a very general form. It states the topological equivalence of the flow of a nonlinear non-autonomous differential or difference equation with critical component to the flow of a partially linearized equation. The critical spectrum has not necessearily to be contained in the imaginary axis or the unit circle respectively. Further on we will employ the socalled calculus on measure chains within dynamical systems theory. Within this calculus the usual one dimensional time scales can be replaced by measure chains which are essentially closed subsets of R. The paper can be understood without knowledge of this calculus.

So our main theorem will be valid even for equations defined on very strange time scales such as sequences of closed intervals. This is especially interesting for applications within the theory of differential-difference equations or within numerical analysis of qualitative phenomena of dynamical systems.