Analysis of discontinuous bifurcations for elasto-plastic geomaterials with effect of damage

2002 ◽  
Vol 159 (1-4) ◽  
pp. 65-76 ◽  
Author(s):  
Y. Q. Zhang ◽  
H. Hao ◽  
M. H. Yu
Keyword(s):  
Discontinuous Bifurcations

New Discontinuity-Induced Bifurcations in Chua's Circuit

2015 ◽  
Vol 25 (06) ◽  
pp. 1550090 ◽  
Author(s):  
Shihui Fu ◽  
Qishao Lu ◽  
Xiangying Meng
Keyword(s):  
Dynamical Systems ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Chua’S Circuit ◽  
Jacobian Matrices ◽  
Chua's Circuit ◽  
Before And After ◽  
Boundary Equilibrium ◽  
Discontinuous Bifurcations ◽  
Nonsmooth Dynamical Systems

Chua's circuit, an archetypal example of nonsmooth dynamical systems, exhibits mostly discontinuous bifurcations. More complex dynamical phenomena of Chua's circuit are presented here due to discontinuity-induced bifurcations. Some new kinds of classical bifurcations are revealed and analyzed, including the coexistence of two classical bifurcations and bifurcations of equilibrium manifolds. The local dynamical behavior of the boundary equilibrium points located on switch boundaries is found to be determined jointly by the Jacobian matrices evaluated before and after switching. Some new discontinuous bifurcations are also observed, such as the coexistence of two discontinuous and one classical bifurcation.

A HYPERCHAOTIC CRISIS

2010 ◽  
Vol 20 (04) ◽  
pp. 1193-1200 ◽  
Author(s):  
LING HONG ◽  
YINGWU ZHANG ◽  
JUN JIANG
Keyword(s):  
Chaotic Systems ◽  
Basin Of Attraction ◽  
Cell Mapping ◽  
Fractal Boundary ◽  
Generalized Cell Mapping ◽  
Complicated Pattern ◽  
Formation And Evolution ◽  
Discontinuous Bifurcations ◽  
Boundary Crisis ◽  
Chaotic Saddle

A crisis is investigated in high dimensional chaotic systems by means of generalized cell mapping digraph (GCMD) method. The crisis happens when a hyperchaotic attractor collides with a chaotic saddle in its fractal boundary, and is called a hyperchaotic boundary crisis. In such a case, the hyperchaotic attractor together with its basin of attraction is suddenly destroyed as a control parameter passes through a critical value, leaving behind a hyperchaotic saddle in the place of the original hyperchaotic attractor in phase space after the crisis, namely, the hyperchaotic attractor is converted into an incremental portion of the hyperchaotic saddle after the collision. This hyperchaotic saddle is an invariant and nonattracting hyperchaotic set. In the hyperchaotic boundary crisis, the chaotic saddle in the boundary has a complicated pattern and plays an extremely important role. We also investigate the formation and evolution of the chaotic saddle in the fractal boundary, particularly concentrating on its discontinuous bifurcations (metamorphoses). We demonstrate that the saddle in the boundary undergoes an abrupt enlargement in its size by a collision between two saddles in basin interior and boundary. Two examples of such a hyperchaotic crisis are given in Kawakami map.

Unfolding a chaotic bifurcation

1990 ◽  
Vol 431 (1882) ◽  
pp. 371-383 ◽  
Keyword(s):  
Digital Simulation ◽  
Chaotic Attractors ◽  
Cusp Catastrophe ◽  
Control Phase ◽  
Codimension Two ◽  
Small Constant ◽  
Constant Bias ◽  
Discontinuous Bifurcations ◽  
Forced Pendulum ◽  
Codimension Two Bifurcation

The differential equation of the sinusoidally forced pendulum is studied by digital simulation in a régime where two simple, symmetrically related chaotic attractors grow and merge continuously as the forcing amplitude is increased. By introducing a small constant bias in the forcing to break the symmetry, two discontinuous bifurcations unfold from the single merging event. Considering both forcing amplitude and bias together as controls, a codimension two bifurcation of chaotic attractors is defined, whose geometric structure in control-phase space is closely related to the elementary cusp catastrophe. The chaotic bifurcations are explained in terms of homoclinic structures (Smale cycles) in the Poincaré map.

Discontinuous Bifurcations in Stick-Slip Mechanical Systems

2001 ◽  
Author(s):  
Ugo Galvanetto
Keyword(s):  
Dynamical Systems ◽  
Dry Friction ◽  
Scientific Literature ◽  
Mechanical Systems ◽  
Stick Slip ◽  
Floquet Multipliers ◽  
Discontinuous Bifurcations ◽  
Active Research ◽  
Grazing Bifurcations

Abstract Non-smooth dynamical systems exhibit continuous and discontinuous bifurcations. Continuous bifurcations are well understood and described in many textbooks, whereas discontinuous bifurcations are still the object of active research. Grazing bifurcations, C-bifurcations and other types of bifurcations characterised by jumps of the relevant Floquet multipliers have been described in the scientific literature. This paper deals with two discontinuous bifurcations found in mechanical systems affected by dry friction.

Discontinuous bifurcations in a nonassociated Mohr material

1991 ◽  
Vol 12 (3-4) ◽  
pp. 255-265 ◽  
Author(s):  
Niels Saabye Ottosen ◽  
Kenneth Runesson
Keyword(s):  
Discontinuous Bifurcations

Discontinuous Bifurcation of a Soft-Impact System

2020 ◽  
Vol 30 (09) ◽  
pp. 2050132
Author(s):  
Shihui Fu ◽  
Yuan Liu ◽  
Joseph Páez Chávez
Keyword(s):  
Numerical Simulation ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Tangency Point ◽  
Impact Zone ◽  
Impact System ◽  
Discontinuous Bifurcation ◽  
Discontinuous Bifurcations ◽  
The Impact ◽  
Narrow Bands

In this paper, we investigate discontinuous bifurcations of a soft-impact system, which is nonsmooth at the switching boundary consisting of two parts. We find that there are no periodic orbits located only in the impact zone, and when grazing bifurcation on one part of the switching boundary occurs, the tangency point changes may occur for different bifurcation parameters, which is also verified by numerical simulation. In particular, we discover degenerate inner and external corner bifurcations, which can produce chaotic behavior, for example, period-doubling cascades and a degenerate inner corner bifurcation that induce chaotic responses. In this way, our investigation reveals the presence of narrow bands of chaotic motion induced by the afore mentioned dynamical phenomena.

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