Codimension-2 Border Collision, Bifurcations in One-Dimensional, Discontinuous Piecewise Smooth Maps

We consider a two-parametric family of one-dimensional piecewise smooth maps with one discontinuity point. The bifurcation structures in a parameter plane of the map are investigated, related to co...

[1]  Procaccia,et al.  New universal scenarios for the onset of chaos in Lorenz-type flows. , 1986, Physical review letters.

[2]  Robert Ghrist,et al.  Resonant Gluing bifurcations , 2000, Int. J. Bifurc. Chaos.

[3]  P. Holmes,et al.  Knotting Within the Gluing Bifurcation , 2022 .

[4]  Paul Glendinning,et al.  The gluing bifurcation: I. Symbolic dynamics of closed curves , 1988 .

[5]  M. Schanz,et al.  Breaking the continuity of a piecewise linear map , 2012 .

[6]  Godfrey H. Hardy,et al.  An introduction to the theory of numbers (5. ed.) , 1995 .

[7]  Michael A. Zaks,et al.  Universal scenarios of transitions to chaos via homoclinic bifurcations , 1989 .

[8]  Michael Schanz,et al.  Period Adding in Piecewise Linear Maps with Two Discontinuities , 2012, Int. J. Bifurc. Chaos.

[9]  Joan S. Birman,et al.  Knotted periodic orbits in dynamical systems—I: Lorenz's equation , 1983 .

[10]  C. Moreira,et al.  BIFURCATION OF THE ESSENTIAL DYNAMICS OF LORENZ MAPS ON THE REAL LINE AND THE BIFURCATION SCENARIO FOR LORENZ LIKE FLOWS: THE CONTRACTING CASE , 2010 .

[11]  M. Martens,et al.  Universal models for Lorenz maps , 1996, Ergodic Theory and Dynamical Systems.

[12]  C. Sparrow.  The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors , 1982 .

[13]  Stephen John Hogan,et al.  Local Analysis of C-bifurcations in n-dimensional piecewise smooth dynamical systems , 1999 .

[14]  J. Keener.  Chaotic behavior in piecewise continuous difference equations , 1980 .

[15]  Michael Schanz,et al.  Border-Collision bifurcations in 1D Piecewise-Linear Maps and Leonov's Approach , 2010, Int. J. Bifurc. Chaos.

[16]  Steve Abbott,et al.  Modular functions and Dirichlet series in number theory , 2nd edition, by T. M. Apostol. Pp 204. DM98. 1990. ISBN 3-540-97127-0 (Springer) , 1991, The Mathematical Gazette.

[17]  Scaling properties and renormalization invariants for the “homoclinic quasiperiodicity” , 1993 .

[18]  Procaccia,et al.  First-return maps as a unified renormalization scheme for dynamical systems. , 1987, Physical review. A, General physics.

[19]  Abbas Golmakani,et al.  Lorenz attractors in unfoldings of homoclinic-flip bifurcations , 2011 .

[20]  Laura Gardini,et al.  Border Collision bifurcations in 1D PWL Map with One Discontinuity and Negative Jump: Use of the First Return Map , 2010, Int. J. Bifurc. Chaos.

[21]  Michael Schanz,et al.  Codimension-three bifurcations: explanation of the complex one-, two-, and three-dimensional bifurcation structures in nonsmooth maps. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  T. Apostol.  Modular Functions and Dirichlet Series in Number Theory , 1976 .

[23]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[24]  Michael Schanz,et al.  On multi-parametric bifurcations in a scalar piecewise-linear map , 2006 .

[25]  R. Labarca,et al.  A characterization of the kneading sequences associated to Lorenz maps of the interval , 2012 .

[26]  Essential Dynamics for Lorenz maps on the real line and the Lexicographical World ? ? Partially supp , 2006 .

[27]  Ale Jan Homburg,et al.  Some global aspects of homoclinic bifurcations of vector fields , 1996 .

[28]  Laura Gardini,et al.  Border collision bifurcation curves and their classification in a family of 1D discontinuous maps , 2011 .

[29]  R. F. Williams,et al.  Structural stability of Lorenz attractors , 1979 .

[30]  D. R. Heath-Brown,et al.  An Introduction to the Theory of Numbers, Sixth Edition , 2008 .

[31]  Michael Schanz,et al.  Sufficient conditions for a period incrementing big bang bifurcation in one-dimensional maps , 2011 .

[32]  P. Holmes,et al.  Knots and Orbit Genealogies in Three Dimensional Flows , 1993 .

[33]  Michael Schanz,et al.  Calculation of bifurcation Curves by Map Replacement , 2010, Int. J. Bifurc. Chaos.

[34]  New Directions in Dynamical Systems: Global Bifurcations in Flows , 1988 .

[35]  James A. Yorke,et al.  Border-collision bifurcations including “period two to period three” for piecewise smooth systems , 1992 .

[36]  C. Moreira,et al.  Bifurcation of the essential dynamics of Lorenz maps and applications to Lorenz-like flows: Contributions to the study of the expanding case , 2001 .

[37]  E. T..  An Introduction to the Theory of Numbers , 1946, Nature.

[38]  M. Schanz,et al.  Organizing centers in parameter space of discontinuous 1D maps. The case of increasing/decreasing branches , 2012 .

[39]  B. Winckler.  Renormalization of Lorenz Maps , 2011 .

[40]  Michael Schanz,et al.  Multi-parametric bifurcations in a piecewise–linear discontinuous map , 2006 .