A Lyapunov function for pullback attractors of nonautonomous differential equations.
A polynomial class of Markus-Yamabe counterexamples.
In the paper [CEGHM] a polynomial counterexample to the Markus-Yamabe Conjecture and to the discrete Markus-Yamabe Question in dimension n ≥ 3 are given. In the present paper we explain a way for obtaining a family of polynomial counterexamples containing the above ones. Finally we study the global dynamics of the examples given in [CEGHM].
A proof of the two-dimensional Markus-Yamabe Stability Conjecture and a generalization
The following problem of Markus and Yamabe is answered affirmatively: Let f be a local diffeomorphism of the euclidean plane whose jacobian matrix has negative trace everywhere. If f(0) = 0, is it true that 0 is a global attractor of the ODE dx/dt = f(x)? An old result of Olech states that this is equivalent to the question if such an f is injective. Here the problem is treated in the latter form by means of an investigation of the behaviour of f near infinity.
Asymptotic behaviour and Hopf bifurcation of a three-dimensional nonlinear autonomous system.
Asymptotic behaviour of positive solutions of the model which describes cell differentiation.
Asymptotic stability of switching systems.
Attractors for nonautonomous multivalued evolution systems generated by time-dependent subdifferentials.
Bifurcation diagram of a cubic three-parameter autonomous system.
Cone invariance and squeezing properties for inertial manifolds for nonautonomous evolution equations
In this paper we summarize an abstract approach to inertial manifolds for nonautonomous dynamical systems. Our result on the existence of inertial manifolds requires only two geometrical assumptions, called cone invariance and squeezing property, and some additional technical assumptions like boundedness or smoothing properties. We apply this result to processes (two-parameter semiflows) generated by nonautonomous semilinear parabolic evolution equations.
Ecuaciiones disipativas, conjuntos absorbentes y atractores
Example of the Smooth Skew Product in the Plane with the One-dimensional Ramified Continuum as the Global Attractor*
The example is constructed of the C1-smooth skew product of interval maps possessing the one-dimensional ramified continuum (containing no arcs homeomorphic to the circle) with an infinite set of ramification points as the global attractor.
Global attractivity without stability for Liénard type systems.
Global attractor of a differentiable autonomous system on the plane
We study the structure of a differentiable autonomous system on the plane with non-positive divergence outside a bounded set. It is shown that under certain conditions such a system has a global attractor. The main result here can be seen as an improvement of the results of Olech and Meisters in [7,9] concerning the global asymptotic stability conjecture of Markus and Yamabe and the Jacobian Conjecture.
Global attractors of non-autonomous quasi-homogeneous dynamical systems.
Gradient systems of closed operators
A classical result on the existence of global attractors for gradient systems is extended to the case of a semigroup S(t) lacking strong continuity, but satisfying the weaker property of being a closed map for every fixed t ≥ 0.
Inertial manifolds for nonautonomous dynamical systems and for nonautonomous evolution equations
Invariant manifolds and almost automorphic solutions of second-order monotone equations
Nonautonomous attractors of skew-product flows with digitized driving systems.
On peaks in carrying simplices
A necessary and sufficient condition is given for the carrying simplex of a dissipative totally competitive system of three ordinary differential equations to have a peak singularity at an axial equilibrium. For systems of Lotka-Volterra type that result translates into a simple condition on the coefficients.
On singularly perturbed ordinary differential equations with measure-valued limits