The non-piecewise-linear autonomous system. II. The complex bifurcation structure

2002 ◽  
Author(s):  
Yu Zhiping ◽  
Zhao Jing
Keyword(s):  
Autonomous System ◽  
Piecewise Linear ◽  
Bifurcation Structure ◽  
Linear Autonomous System

GENERATING 3-D SCROLL GRID ATTRACTORS OF FRACTIONAL DIFFERENTIAL SYSTEMS VIA STAIR FUNCTION

2007 ◽  
Vol 17 (11) ◽  
pp. 3965-3983 ◽  
Author(s):  
WEIHUA DENG
Keyword(s):  
Differential System ◽  
Autonomous System ◽  
Function Approach ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential System ◽  
Linear Autonomous System ◽  
Fractional Differential ◽  
The One ◽  
Number Of Equilibria

This paper discusses the stair function approach for the generation of scroll grid attractors of fractional differential systems. The one-directional (1-D) n-grid scroll, two-directional (2-D) (n × m)-grid scroll and three-directional (3-D) (n × m × l)-grid scroll attractors are created from a fractional linear autonomous system with a simple stair function controller. Being similar to the scroll grid attractors of classical differential systems, the scrolls of 1-D n-grid scroll, 2-D (n × m)-grid scroll and 3-D (n × m × l)-grid scroll attractors are located around the equilibria of fractional differential system on a line, on a plane or in 3D, respectively and the number of scrolls is equal to the corresponding number of equilibria.

GENERATING CHAOS VIA A DYNAMICAL CONTROLLER

2001 ◽  
Vol 11 (03) ◽  
pp. 865-869 ◽  
Author(s):  
GUO-QUN ZHONG ◽  
KIM F. MAN ◽  
GUANRONG CHEN
Keyword(s):  
Computer Simulation ◽  
Autonomous System ◽  
Quadratic Function ◽  
Feedback Controller ◽  
Nonlinear Feedback ◽  
Circuit Implementation ◽  
Linear Autonomous System

This Letter studies the generation of chaos from a linear autonomous system by employing a dynamical nonlinear feedback controller. The system setup is quite simple, and the only nonlinearity is a piecewise-quadratic function in the form of x|x|. Both computer simulation and circuit implementation are given to verify the chaos generated by this mechanism.

Bifurcation Structures in a Bimodal Piecewise Linear Map: Chaotic Dynamics

2015 ◽  
Vol 25 (03) ◽  
pp. 1530006 ◽  
Author(s):  
Anastasiia Panchuk ◽  
Iryna Sushko ◽  
Viktor Avrutin
Keyword(s):  
Parameter Space ◽  
Chaotic Dynamics ◽  
Piecewise Linear ◽  
Chaotic Attractors ◽  
Bifurcation Structure ◽  
Tent Map ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Replacement Technique ◽  
Bifurcation Structures

In this work, we investigate the bifurcation structure of the parameter space of a generic 1D continuous piecewise linear bimodal map focusing on the regions associated with chaotic attractors (cyclic chaotic intervals). The boundaries of these regions corresponding to chaotic attractors with different number of intervals are identified. The results are obtained analytically using the skew tent map and the map replacement technique.

scholarly journals A Chaotic Three Dimensional Non-Linear Autonomous System Beyond Lorenz Type Systems

2012 ◽  
Vol 36 (2) ◽  
pp. 159-170
Author(s):  
Md Shariful Islam Khan ◽  
Md Shahidul Islam
Keyword(s):  
Autonomous System ◽  
Three Dimensional ◽  
Type Systems ◽  
Control Parameters ◽  
Fundamental Properties ◽  
Non Linear ◽  
Linear Autonomous System ◽  
Non Linear System ◽  
Academy Of Sciences ◽  
The Relationship

Some fundamental properties of a chaotic three-dimensional non-linear system of the Lorenz type systems were studied. The invariance, dissipation, bifurcation and the strange attractors were investigated and analyzed one 1-scroll, two 2-scroll and two 4-scroll attractors by adding control   parameters to this system. The relationship and connecting function for the 2-scroll attractor of this system were also explored. DOI: http://dx.doi.org/10.3329/jbas.v36i2.12959 Journal of Bangladesh Academy of Sciences, Vol. 36, No. 2, 159-170, 2012  

scholarly journals Bifurcation Structure in a Bimodal Piecewise Linear Business Cycle Model

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Viktor Avrutin ◽  
Iryna Sushko ◽  
Fabio Tramontana
Keyword(s):  
Business Cycle ◽  
Parameter Space ◽  
Piecewise Linear ◽  
Cycle Model ◽  
Bifurcation Structure ◽  
Analytical Expression ◽  
Business Cycle Model

We study the bifurcation structure of the parameter space of a 1D continuous piecewise linear bimodal map which describes dynamics of a business cycle model introduced by Day-Shafer. In particular, we obtain the analytical expression of the boundaries of several periodicity regions associated with attracting cycles of the map (principal cycles and related fin structure). By crossing these boundaries the map displays robust chaos.

Existence and stability of periodic solutions of a third-order non-linear autonomous system simulating immune response in animals

1977 ◽  
Vol 77 (1-2) ◽  
pp. 163-175 ◽  
Author(s):  
In-Ding Hsü ◽  
Nicholas D. Kazarinoff
Keyword(s):  
Immune Response ◽  
Steady State ◽  
Periodic Solutions ◽  
Stability Criterion ◽  
Autonomous System ◽  
Third Order ◽  
Non Linear ◽  
Existence And Stability ◽  
Linear Autonomous System ◽  
Non Linear System

SynopsisA 3 × 3 autonomous, non-linear system of ordinary differential equations modelling the immune response in animals to invasion by active self-replicating antigens has been introduced by G. I. Bell and studied by G. H. Pimbley Jr. Using Hopf's theorem on bifurcating periodic solutions and a stability criterion of Hsu and Kazarinoff, we obtain existence of a family of unstable periodic solutions bifurcating from one steady state of a reduced 2×2 form of the 3×3 system. We show that no periodic solutions bifurcate from the other steady state. We also prove existence and exhibit a stability criterion for families of periodic solutions of the full 3×3 system. We provide two numerical examples. The second shows existence of orbitally stable families of periodic solutions of the 3×3 system.

On a bifurcation structure mimicking period adding

2011 ◽  
Vol 467 (2129) ◽  
pp. 1503-1518 ◽  
Author(s):  
Björn Schenke ◽  
Viktor Avrutin ◽  
Michael Schanz
Keyword(s):  
Periodic Orbits ◽  
Parameter Space ◽  
Piecewise Linear ◽  
The Other ◽  
Asymptotically Stable ◽  
Periodic Domain ◽  
Bifurcation Structure ◽  
Symbolic Sequences ◽  
Stable Periodic ◽  
The One

In this work, we investigate a piecewise-linear discontinuous scalar map defined on three partitions. This map is specifically constructed in such a way that it shows a recently discovered bifurcation scenario in its pure form. Owing to its structure on the one hand and the similarities to the nested period-adding scenario on the other hand, we denoted the new bifurcation scenario as nested period-incrementing bifurcation scenario. The new bifurcation scenario occurs in several physical and electronical systems but usually not isolated, which makes the description complicated. By isolating the scenario and using a suitable symbolic description for the asymptotically stable periodic orbits, we derive a set of rules in the space of symbolic sequences that explain the structure of the stable periodic domain in the parameter space entirely. Hence, the presented work is a necessary step for the understanding of the more complicated bifurcation scenarios mentioned above.

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