Naimark–Sacker bifurcations in linearly coupled quadratic maps

Naimark-Sacker Bifurcations in Linearly Coupled Quadratic Maps

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 period-2 orbit. This result is the key to understand the onset of synchronization in networks of quadratic maps.

Bifurcations in Globally Coupled Chaotic Maps

We propose a new method to investigate collective behavior in a network of globally coupled chaotic elements generated by a tent map. In the limit of large system size, the dynamics is described with the nonlinear FrobeniusPerron equation. This equation can be transformed into a simple form by making use of the piecewise linear nature of the individual map. Our method is applied successfully to...

Quadratic Volume-Preserving Maps: Invariant Circles and Bifurcations

We study the dynamics of the five-parameter quadratic family of volume-preserving diffeomorphisms of R. This family is the unfolded normal form for a bifurcation of a fixed point with a triple-one multiplier and is also the general form of a quadratic three-dimensional map with a quadratic inverse. Much of the nontrivial dynamics of this map occurs when its two fixed points are saddle-foci with...

Topological Properties of Linearly Coupled Expanding Maps Lattices

From synchronization to multistability in two coupled quadratic maps

The phenomenology of a system of two coupled quadratic maps is studied both analytically and numerically. Conditions for synchronization are given and the bifurcations of periodic orbits from this regime are identified. In addition, we show that an arbitrarily large number of distinct stable periodic orbits may be obtained when the maps parameter is at the Feigenbaum period-doubling accumulatio...