We report the exact analytical expression of the surface W4(a,b;λ)=0 defining stability domains for period-4 motions in the Hénon map, valid for arbitrary eigenvalues λ and parameters a and b. For λ=+1 (fold bifurcations) the expression reproduces all previous results and gives a new one. For...

We introduce a class of models composed by lattices of coupled complex-amplitude oscillators which preserve the norm. These models are particularly well adapted to investigate phenomena described by the nonlinear Schrödinger equation. The coupling between oscillators is parameterized by the...

We prove a theorem establishing a direct link between macroscopically observed periodic motions and certain subsets of intrinsically discrete orbits which are selected naturally by the dynamics from the skeleton of unstable periodic orbits (UPOs) underlying classical and quantum dynamics. As a...

We report exact analytical expressions locating the 0→1, 1→2 and 2→4 bifurcation curves for a prototypical system of two linearly coupled quadratic maps. Of interest is the precise location of the parameter sets where Naimark–Sacker bifurcations occur, starting from a non-diagonal...

We show that dissipative dynamical systems with constant Jacobian allow one to recover numerical values of control parameters under which the system is operating. This is done by performing measurements on self-similar (fractal) structures in phase space. We illustrate parameter recovery...

We present a method for investigating the simultaneous movement of all zeros of equations of motions defined by discrete mappings. The method is used to show that knowledge of the interplay of all zeros is of fundamental importance for establishing periodicities and relative stability properties...

In 1963 Myrberg determined a period-doubling cascade of the quadratic map to accumulate at 1.401155189… As found later, the geometric way with which model parameters approach this value has universal behavior and several characteristic exponents associated with it. In the present paper we...

This paper describes how certain shrimp-like clusters of stability organize themselves in the parameter space of dynamical systems. Clusters are composed of an infinite affine-similar repetition of a basic elementary cell containing two primay noble points, a head and a tail, defining an axis of...

In number theory, “units” are very special numbers characterized by having their norm equal to unity. So, in the real quadratic field Z (√3) the number of −2 + √3 ≅ −0.2679491924… is a unit because (−2 + √3) (−2 - √3) = 1. In this paper we determine precisely the...

We simulate a 2D coupled map lattice formed by individual units consisting of a multi-attractor quartic map. We show that the interesting recently discovered non-trivial collective behaviors (where macroscopic quantities show well-defined, usually regular, temporal evolution in spite of the...