Related papers: The strong unstable manifold and periodic solution…
We study a slow-fast system with two slow and one fast variables. We assume that the slow manifold of the system possesses a fold and there is an equilibrium of the system in a small neighbourhood of the fold. We derive a normal form for…
In this paper, we study the dynamical behaviors of neutral differential equations with small delays. We first establish the existence and smoothness of the global inertial manifolds for these equations. Then we further prove the smoothness…
We study dynamics and bifurcations of 2-dimensional reversible maps having a symmetric saddle fixed point with an asymmetric pair of nontransversal homoclinic orbits (a symmetric nontransversal homoclinic figure-8). We consider…
We present and implement an algorithm for computing the invariant circle and the corresponding stable manifolds for 2-dimensional maps. The algorithm is based on the parameterization method, and it is backed up by an a-posteriori theorem…
The objective of this paper is to derive the essential invariance and contraction properties for the geometric periodic systems, which can be formulated as a category of differential inclusions, and primarily rendered in the phase…
Partial differential equations with discrete (concentrated) state-dependent delays are studied. The existence and uniqueness of solutions with initial data from a wider linear space is proven first and then a subset of the space of…
Bifurcation equations, non-degeneracy and transversality conditions are obtained for the fold, transcritical, pitchfork and flip bifurcations for periodic points of one dimensional implicitly defined discrete dynamical systems. The backward…
A recent work by the authors on the existence of a periodic smooth finite-dimensional center manifold near a nonhyperbolic cycle in delay differential equations motivates the derivation of periodic normal forms. In this paper, we prove the…
Understanding how time delays impact the stability of a delay differential equation is important for modeling many natural and technological systems that experience time delays. Here we introduce a new stability criterion for…
We present the Constructive Methods of Invariant Manifolds for model reduction in physical and chemical kinetics, developed during last two decades. The problem of reduced description is studied as a problem of constructing the slow…
We revisit the box Minkowski model [Phys. Rev. Lett. 109, 221101 (2012)] and provide a strong argument that, subject to the Dirichlet boundary condition, it is unstable toward black hole formation for arbitrarily small generic…
Invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems. The theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems, and for…
A heterodimensional cycle is an invariant set of a dynamical system consisting of two hyperbolic periodic orbits with different dimensions of their unstable manifolds and a pair of orbits that connect them. For systems which are at least…
This paper considers a class of delay differential equations with unimodal feedback and describes the structure of certain unstable sets of stationary points and periodic orbits. These unstable sets consist of heteroclinic connections from…
For piecewise-linear maps the stable and unstable manifolds of hyperbolic periodic solutions are themselves piecewise-linear. Hence compact subsets of these manifolds can be represented using polytopes (i.e. polygons, in the case of…
In this paper, a method to construct topological template in terms of symbolic dynamics for the diamagnetic Kepler problem is proposed. To confirm the topological template, rotation numbers of invariant manifolds around unstable periodic…
Slow-fast dynamical systems, i.e., singularly or non-singularly perturbed dynamical systems possess slow invariant manifolds on which trajectories evolve slowly. Since the last century various methods have been developed for approximating…
The present paper investigates the structural stability of bidirectional cyclic negative feedback systems. To address this, we develop a generalized Floquet theory and construct nested invariant cones for the systems. Subsequently, we…
Two complementary analyses of a cyclic negative feedback system with delay are considered in this paper. The first analysis applies the work by Sontag, Angeli, Enciso and others regarding monotone control systems under negative feedback,…
A nonlinear cyclic system with delay and the overall negative feedback is considered. The characteristic equation of the linearized system is studied in detail. Sufficient conditions for the oscillation of all solutions and for the…