scholarly journals Global dynamics and bifurcation of planar piecewise smooth quadratic quasi-homogeneous differential systems

2018 ◽  
Vol 38 (4) ◽  
pp. 2029-2046
Author(s):  
Yilei Tang ◽  
◽  
Keyword(s):  
Piecewise Smooth ◽  
Global Dynamics ◽  
Differential Systems

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 On the Dynamics of the Unified Chaotic System Between Lorenz and Chen Systems

2015 ◽  
Vol 25 (09) ◽  
pp. 1550122 ◽  
Author(s):  
Jaume Llibre ◽  
Ana Rodrigues
Keyword(s):  
Chaotic System ◽  
Parameter Family ◽  
Algebraic Surfaces ◽  
Hopf Bifurcations ◽  
Global Dynamics ◽  
Differential Systems ◽  
Special Values ◽  
Unified Chaotic System ◽  
Invariant Algebraic Surfaces ◽  
The Family

A one-parameter family of differential systems that bridges the gap between the Lorenz and the Chen systems was proposed by Lu, Chen, Cheng and Celikovsy. The goal of this paper is to analyze what we can say using analytic tools about the dynamics of this one-parameter family of differential systems. We shall describe its global dynamics at infinity, and for two special values of the parameter a we can also describe the global dynamics in the whole ℝ3using the invariant algebraic surfaces of the family. Additionally we characterize the Hopf bifurcations of this family.

scholarly journals Global Dynamics of a Piecewise Smooth System for Brain Lactate Metabolism

2018 ◽  
Vol 18 (1) ◽  
pp. 315-332
Author(s):  
J.-P. Françoise ◽  
Hongjun Ji ◽  
Dongmei Xiao ◽  
Jiang Yu
Keyword(s):  
Piecewise Smooth ◽  
Global Dynamics ◽  
Lactate Metabolism ◽  
Piecewise Smooth System ◽  
Smooth System ◽  
Brain Lactate

ON THE SLIDING BIFURCATION OF A CLASS OF PLANAR FILIPPOV SYSTEMS

2013 ◽  
Vol 23 (03) ◽  
pp. 1350040 ◽  
Author(s):  
DINGHENG PI ◽  
JIANG YU ◽  
XIANG ZHANG
Keyword(s):  
Limit Cycles ◽  
Homoclinic Bifurcation ◽  
Qualitative Theory ◽  
Piecewise Smooth ◽  
The Other ◽  
Heteroclinic Cycle ◽  
Differential Systems ◽  
Sliding Bifurcation ◽  
Bifurcation Phenomena ◽  
Limit Cycle Bifurcation

In this paper, we study the sliding bifurcation phenomena of a class of planar piecewise smooth differential systems consisting of linear and quadratic subsystems. Using the differential inclusion and the qualitative theory of ordinary differential equations, we find some new interesting phenomena appearing in the piecewise smooth differential systems. In brief, we prove that the system may have sliding homoclinic bifurcation, sliding cycle bifurcation, semistable limit cycle bifurcation and heteroclinic cycle bifurcation. In addition, the mentioned systems can have at most two limit cycles, and the maximal number of limit cycles can be realized and central nested with one bifurcated from the sliding–crossing bifurcation of a sliding cycle and the other from the saddle homoclinic bifurcation. These two limit cycles collide and then both disappear. This novel scenario is verified by our systems.

Bifurcation Analysis of Planar Piecewise Smooth Systems with a Line of Discontinuity

2016 ◽  
Vol 26 (06) ◽  
pp. 1650104
Author(s):  
Dingheng Pi ◽  
Shihong Xu
Keyword(s):  
Hamiltonian System ◽  
Linear System ◽  
Qualitative Analysis ◽  
Bifurcation Analysis ◽  
Dynamical Behavior ◽  
Piecewise Smooth ◽  
Piecewise Smooth Systems ◽  
Differential Systems ◽  
Line Of Discontinuity ◽  
Smooth System

In this paper, we consider bifurcations of a class of planar piecewise smooth differential systems constituted by a general linear system and a quadratic Hamiltonian system. The linear system has four parameters. When the parameters vary in different regions, the left linear system can have a saddle, a node or a focus. For each case, we provide a completely qualitative analysis of the dynamical behavior for this piecewise smooth system. Our results generalize and improve the results in this direction.

Projective dynamics of homogeneous systems: local invariants, syzygies and the Global Residue Theorem

2012 ◽  
Vol 55 (3) ◽  
pp. 577-589 ◽  
Author(s):  
Z. Balanov ◽  
A. Kononovich ◽  
Y. Krasnov
Keyword(s):  
Explicit Formula ◽  
Homogeneous Polynomial ◽  
First Integrals ◽  
Global Dynamics ◽  
Bounded Solutions ◽  
Residue Theorem ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Homogeneous Systems ◽  
Local Invariants

AbstractWe give an explicit formula for the projective dynamics of planar homogeneous polynomial differential systems in terms of natural local invariants and we establish explicit algebraic connections (syzygies) between these invariants (leading to restrictions on possible global dynamics). We discuss multidimensional generalizations together with applications to the existence of first integrals and bounded solutions.

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