Generally, Collet-Eckmann maps require unimodality and multimodality. The inverse is not true. In this paper, we will prove that S-unimodal maps are Collet-Eckmann maps in a specific range of their bifurcation parameters. The proof is based on the fact that the family of robustly chaotic unimodal maps known in the literature are all topologically conjugate to one another and the fact that if tw...

The dynamics of mechanical systems with dry friction is affected by non-smooth bifurcations, which have been recently partially classified as ‘sliding bifurcations’. In applied science a bifurcation is usually seen as the point in which the number of fixed points and/or (quasi-)periodic solutions changes. The paper describes with several detailed examples that ‘sliding bifurcations’ do not alwa...

Motivated by the importance and universal character of phase singularities which are clarified recently, we study the local structure of equi-phase loci near the dislocation locus of complex valued planar and spatial waves, from the viewpoint of singularity theory of differentiable mappings, initiated by H. Whitney and R. Thom. The classification of phase-singularities are reduced to the classi...

We develop and analyze a mathematical model, in the form of a system of ordinary differential equations (ODEs), governing cancer growth on a cell population level with combination immune, vaccine and chemotherapy treatments. We characterize the ODE system dynamics by locating equilibrium points, determining stability properties, performing a bifurcation analysis, and identifying basins of attra...

In this expository and resources chapter we review selected aspects of the mathematics of dynamical systems, stability, and chaos, within a historical framework that draws together two threads of its early development: celestial mechanics and control theory, and focussing on qualitative theory. From this perspective we show how concepts of stability enable us to classify dynamical equations and...

Rotating rings of tetrahedra are well known from recreational mathematics. Rings of N tetrahedra with N even are analyzed by symmetry-adapted versions of classical counting rules of mechanism analysis. For N ≥ 6 a single state of selfstress is found, together with N − 5 symmetry-distinct mechanisms, which include the eponymous rotating mechanism. For N = 4 in a generic configuration, a single m...

Motivated by mirror symmetry, we consider a Lagrangian fibration X → B and Lagrangian maps f : L →֒ X → B, when L has dimension 2, exhibiting an unstable singularity, and study how their caustic changes, in a neighbourhood of the unstable singularity, when slightly perturbed. The integral curves of ∇fx, for x ∈ B, where fx(y) = f(y)−x ·y, called “gradient lines”, are then introduced, and a study...

Motivated by mirror symmetry, we consider the Lagrangian fibration R → R and Lagrangian maps f : L →֒ R → R, exhibiting an unstable singularity, and study how the bifurcation locus of gradient lines, the integral curves of ∇fx, for x ∈ B, where fx(y) = f(y) − x · y, changes when f is slightly perturbed. We consider the cases when f is the germ of a fold, of a cusp and, particularly, of an ellipt...

DOMAINS OF INVARIANT CURVES 1 Rafael de la Llave 2 3 and Stathis Tompaidis 2 4 Department of Mathematics The University of Texas at Austin Austin, TX 78712-1082 Abstract. We present theoretical arguments (based on in nite dimensional bifurcation theory) and numerical evidence (based on non-perturbative methods) that the boundaries of analyticity of invariant curves can be described as an accumu...

A mathematical model describing the dynamics  of a  delayed  stage structure prey - predator  system  with  prey  refuge  is  considered.  The  existence,  uniqueness  and bounded- ness  of  the  solution  are  discussed.    All  the  feasibl e  equilibrium  points  are determined.  The   stability  analysis  of  them  are  investigated.  By  employ ing  the time delay as the bifurcation parame...