Local and global bifurcations in 3D piecewise smooth discontinuous maps

2021 ◽  
Vol 31 (1) ◽  
pp. 013126
Author(s):  
Mahashweta Patra ◽  
Sayan Gupta ◽  
Soumitro Banerjee
Keyword(s):  
Piecewise Smooth ◽  
Global Bifurcations ◽  
Local And Global Bifurcations

LOCAL AND GLOBAL BIFURCATIONS IN THREE-DIMENSIONAL, CONTINUOUS, PIECEWISE SMOOTH MAPS

2011 ◽  
Vol 21 (06) ◽  
pp. 1617-1636 ◽  
Author(s):  
SOMA DE ◽  
PARTHA SHARATHI DUTTA ◽  
SOUMITRO BANERJEE ◽  
AKHIL RANJAN ROY
Keyword(s):  
Three Dimensional ◽  
Global Bifurcation ◽  
Period Doubling ◽  
Piecewise Smooth ◽  
Numerical Continuation ◽  
Homoclinic Tangency ◽  
Global Dynamics ◽  
Global Bifurcations ◽  
Border Collision Bifurcation ◽  
Smooth Map

In this work, we study the dynamics of a three-dimensional, continuous, piecewise smooth map. Much of the nontrivial dynamics of this map occur when its fixed point or periodic orbit hits the switching manifold resulting in the so-called border collision bifurcation. We study the local and global bifurcation phenomena resulting from such borderline collisions. The conditions for the occurrence of nonsmooth period-doubling, saddle-node, and Neimark–Sacker bifurcations are derived. We show that dangerous border collision bifurcation can also occur in this map. Global bifurcations arise in connection with the occurrence of nonsmooth Neimark–Sacker bifurcation by which a spiral attractor turns into a saddle focus. The global dynamics are systematically explored through the computation of resonance tongues and numerical continuation of mode-locked invariant circles. We demonstrate the transition to chaos through the breakdown of mode-locked torus by degenerate period-doubling bifurcation, homoclinic tangency, etc. We show that in this map a mode-locked torus can be transformed into a quasiperiodic torus if there is no global bifurcation.

scholarly journals Bifurcations of Nonconstant Solutions of the Ginzburg-Landau Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-19
Author(s):  
Norimichi Hirano ◽  
Sławomir Rybicki
Keyword(s):  
Landau Equation ◽  
Conley Index ◽  
Global Bifurcations ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation ◽  
Local And Global Bifurcations

We study local and global bifurcations of nonconstant solutions of the Ginzburg-Landau equation from the families of constant ones. As the topological tools we use the equivariant Conley index and the degree for equivariant gradient maps.

Codimension ${\bi n}$ flips

1998 ◽  
Vol 18 (5) ◽  
pp. 1115-1137 ◽  
Author(s):  
JAQUES GHEINER
Keyword(s):  
Global Bifurcations ◽  
Local And Global Bifurcations

The generic unfolding of codimension $n$ flips (one eigenvalue $-1$ and the others with norm different from 1) embedded in a Morse–Smale diffeomorphism is analyzed. Local and global bifurcations are described.

Response Scenario and Nonsmooth Features in the Nonlinear Dynamics of an Impacting Inverted Pendulum

2005 ◽  
Vol 1 (1) ◽  
pp. 56-64 ◽  
Author(s):  
Stefano Lenci ◽  
Lucio Demeio ◽  
Milena Petrini
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Investigation ◽  
Inverted Pendulum ◽  
Numerical Results ◽  
Dynamical Response ◽  
Global Bifurcations ◽  
Bifurcation Diagrams ◽  
Local And Global Bifurcations ◽  
And Behavior

In this work, we perform a systematic numerical investigation of the nonlinear dynamics of an inverted pendulum between lateral rebounding barriers. Three different families of considerably variable attractors—periodic, chaotic, and rest positions with subsequent chattering—are found. All of them are investigated, in detail, and the response scenario is determined by both bifurcation diagrams and behavior charts of single attractors, and overall maps. Attention is focused on local and global bifurcations that lead to the attractor-basin metamorphoses. Numerical results show the extreme richness of the dynamical response of the system, which is deemed to be of interest also in view of prospective mechanical applications.

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