Over recent years, a great deal of experimental studies and modeling
simulations have been directed toward the identification of various
dynamical and structural invariants to serve as key signatures uniting often diverse nonlinear systems into a single class.

One such class of low order dissipative systems has been identified to
possess one common, easily recognizable pattern involving spiral structures in a biparametric phase space. Such patterns turn out to be ubiquitously alike in both time-discrete and time-continuous systems.
 
Despite the overwhelming number of studies reporting the occurrence of spiral structures, there is still little known about the fine construction details and underlying bifurcation scenarios for these patterns. In this talk we study the genesis of the spiral structures in several low order systems and reveal the generality of underlying global bifurcations. We will start with the Rossler model and demonstrate that such parametric patterns are the key feature of systems with homoclinic connections involving saddle-foci
meeting a single Shilnikov condition. Besides, we show that the
organizing center for spiral structures in the Rossler model with the
Shilnikov saddle-focus is related to the change of the topology of the
attractor transitioning between the spiral and screw-like types. The
structure skeleton is formed by saddle-node bifurcation curves originating from a codimension-two Belyakov point corresponding to the transition to the saddle-focus from a simple saddle. The occurrence of this bifurcation causing complex dynamics is common for a plethora of dissipative systems, describing (electro)chemical reactions, population dynamics, electronic circuits, including the Chua circuit.
 
The other group of spiral structures is made of models with the Lorenz
attractor. Here, under consideration is the iconic Lorenz equation. For
thorough explorations of the dynamics of Lorenz-like models we have proposed the algorithmically easy, though powerful toolkit based on the symbolic description of a single trajectory -- an unstable separatrix of the saddle singularity of the model. A new computational technique based on the symbolic kneading invariant description for examining dynamical chaos and parametric chaos in systems with Lorenz-like attractors is proposed and tested. This technique uncovers the stunning complexity and universality of spiral structures in the iconic Lorenz equation, a normal formal and a laser model from nonlinear optics. Note that this new technique permits to locate, among other structures, the T-points in the parametric phase space.
 
A key tool in all these studies is the new free software TIDES based on
the Taylor series method.
