Organizing centers in discontinuous dynamical systems: bifurcations of higher codimension in theory and applications

• Applicant: Professor Dr. Viktor Avrutin
• Subject Area: Statistical Physics, Nonlinear Dynamics, Complex Systems, Soft and Fluid Matter, Biological Physics
• Term: from 2008 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 84580342

• Final Report Year: 2017

Final Report Abstract

Piecewise smooth dynamical systems are in the meanwhile well accepted as a mathematical tool which provides an adequate description for many applications originating from several areas, such as electronics (any kinds of switching circuits) and mechanics (impacting and friction dominated systems) as well as social sciences (systems including decision making) and economics (business cycles models). It is also well known that these systems demonstrate a large variety of bifurcation phenomena and in particular very complex bifurcation structures in multi-dimensional parameter spaces. The key idea behind this project was to contribute to the investigation of such structures in low-dimensional piecewise smooth maps by identification of their organizing centers. The focus of the project was on a special type of organizing centers, from which an infinite number of regions originate, associated with different asymptotic dynamics of the map. The regularities in the appearance of these dynamics determine the type of the organizing center. The challenges of the project were that it was neither clear in which classes of maps such organizing centers appear, nor under which conditions. Moreover, the bifurcation boundaries issuing from such organizing centers are typically associated with global bifurcations. Therefore, the normal form approach could not be applied to prove the abovementioned conditions. Within the scope of this project, we confirmed the initial hypothesis that the appearance of organizing centers is related to codimension-2 border collision bifurcations. For an arbitrary discontinuous piecewise monotone one-dimensional map, we provided a rigorous proof under which conditions a codimension-2 border collision bifurcation point represents an organizing centers of the type mentioned above. The obtained result applies to codimension-2 bifurcation points associated with border collision bifurcations of cycles of any periods and, moreover, to maps with any number of discontinuities. It has been shown that if the cycles undergoing border collision bifurcations at the codimension-2 bifurcation point are stable, then only two types of bifurcation structures may originate from the organizing center given by this point, namely the complete period adding and the complete period incrementing structures. If at least one of the cycles is unstable, then two further bifurcation structures may appear, namely bandcount adding and bandcount incrementing structures. However, in this case the results are less general since the appearing bifurcation structures may be incomplete. It has also been shown that the class of dynamical systems characterized by the presence of organizing centers of the type considered within the scope of this project is mainly restricted to piecewise monotone discontinuous one-dimensional maps. Extending the class of investigated maps to continuous one-dimensional maps as well as to twodimensional maps, one observes that the organizing centers appear in non-generic cases only, while in generic cases the regions become separated from the former organizing center and form more complex bifurcation structures.

Publications

• PhD thesis “Organizing Centers in One-Dimensional, Piecewise-Linear Maps”
  Björn Schenke
  
• Codimension-2 border collision bifurcations in onedimensional discontinuous piecewise smooth maps. Int. J. Bifurcat. Chaos, 24(2):1450024, 2014
  L. Gardini, V. Avrutin, and I. Sushko
  
• Cyclicity of chaotic attractors in one-dimensional discontinuous maps. Mathematics and Computers in Simulation (Special Issue “Discontinuous Dynamical Systems: Theory and Numerical Methods”), 95:126–136, 2014
  V. Avrutin, I. Sushko, and L. Gardini
  
• Bandcount adding structure and collapse of chaotic attractors in a piecewise linear bimodal map. Physica D, 309:37–56, 2015
  V. Avrutin, M. Clüver, and V. Mahout ana D. Fournier-Prunaret
  
• Calculation of homoclinic and heteroclinic orbits in 1D maps. Communications in Nonlinear Science and Numerical Simulation, 22(1-3):1201–1214, 2015
  V. Avrutin, B. Schenke, and L. Gardini
  
• Dynamics of a 2D piecewise linear Braess paradox model: Effect of the third partition. Int. J. Bifurcat. Chaos, 25(11):1530031, 2015
  V. Avrutin, Ch. Dibak, A. Dal Forno, and U. Merlone
  
• Bifurcation structure in the skew tent map and its application as a border collision normal form. J. of Diff. Equations and Applications, 22(8): 1040–1087, 2016
  I. Sushko, V. Avrutin, and L. Gardini
  
• Nonsmooth one-dimensional maps: Some basic concepts and definitions. J. of Diff. Equations and Applications, 22(12):1816–1870, 2016
  I. Sushko, L. Gardini, and V. Avrutin
  
• Bifurcation structures in a bimodal piecewise linear map. Front. Appl. Math. Stat., 3(7):1–21, 2017
  I. Sushko A. Panchuk and V. Avrutin
  
• Dangerous bifurcations revisited. Int. J. Bifurcat. Chaos, 26(14):1630040, 2017
  V. Avrutin, Zh.T. Zhusubaliyev, A. Saha, S. Banerjee, L. Gardini, and I. Sushko