LMI-based estimation of the domain of attraction of equilibrium points for nonlinear non-polynomial dynamical systems

2009 ◽  
Author(s):  
Graziano Chesi
Keyword(s):  
Dynamical Systems ◽  
Equilibrium Points ◽  
Domain Of Attraction ◽  
Polynomial Dynamical Systems

Mathematical Modeling of the λ Switch

2010 ◽  
pp. 573-603
Author(s):  
Dmitriy Laschov ◽  
Michael Margaliot
Keyword(s):  
Mathematical Modeling ◽  
Differential Equation ◽  
Mathematical Model ◽  
Gene Regulation ◽  
Equilibrium Points ◽  
Domain Of Attraction ◽  
Verbal Description ◽  
Quantitative Understanding ◽  
Living Organisms ◽  
Scientific Challenge

Gene regulation plays a central role in the development and functioning of living organisms. Developing a deeper qualitative and quantitative understanding of gene regulation is an important scientific challenge. The Lambda switch is commonly used as a paradigm of gene regulation. Verbal descriptions of the structure and functioning of the Lambda switch have appeared in biological textbooks. We apply fuzzy modeling to transform one such verbal description into a well-defined mathematical model. The resulting model is a piecewise-quadratic, second-order differential equation. It demonstrates functional fidelity with known results while being simple enough to allow a rather detailed analysis. Properties such as the number, location, and domain of attraction of equilibrium points can be studied analytically. Furthermore, the model provides a rigorous explanation for the so-called stability puzzle of the Lambda switch.

A Fuzzy Nonlinear Modeling of a McPherson Suspension System

2005 ◽  
Author(s):  
Ali Tavassoli ◽  
Hamed Jafarian ◽  
Mohammad Eghtesad
Keyword(s):  
Dynamical Systems ◽  
Initial Conditions ◽  
Nonlinear Modeling ◽  
Fuzzy Model ◽  
Domain Of Attraction ◽  
Suspension System ◽  
Reliable Model ◽  
Sugeno Model ◽  
Takagi Sugeno

The Takagi-Sugeno fuzzy model (TSfm) is a universal approximation of continuous real functions that are defined in a closed and bounded subset of Rn. This strong property of the TSfm can find several applications in modeling of dynamical systems that are described by differential equations. In this paper, we consider Takagi-Sugeno fuzzy model for a McPherson suspension system. One advantage of TSfm is its wide domain of attraction in compare with the other methods. To apply TSf modeling, one must precisely choose the nonlinear terms of the system governing equations. For each nonlinear term, there should be selected some linear subsystems that together represent the equivalent of the original nonlinear suspension system. This equivalence, for our case study, is illustrated by simulation results for various road disturbances and initial conditions which show the Takagi-Sugeno model can give a realistic and reliable model for dynamical systems.

The number of equilibrium points of perturbed nonlinear positive dynamical systems

Automatica ◽  
2020 ◽  
Vol 112 ◽  
pp. 108732 ◽  
Author(s):  
Cameron McBride ◽  
Domitilla Del Vecchio
Keyword(s):  
Dynamical Systems ◽  
Equilibrium Points

scholarly journals Global Dynamical Systems Involving Generalized -Projection Operators and Set-Valued Perturbation in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Yun-zhi Zou ◽  
Xi Li ◽  
Nan-jing Huang ◽  
Chang-yin Sun
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Banach Spaces ◽  
Equilibrium Points ◽  
Projection Operators ◽  
Initial Value ◽  
New Class ◽  
The Fixed Point Theorem ◽  
Generalized Projection Operators ◽  
Generalized Dynamical Systems

A new class of generalized dynamical systems involving generalizedf-projection operators is introduced and studied in Banach spaces. By using the fixed-point theorem due to Nadler, the equilibrium points set of this class of generalized global dynamical systems is proved to be nonempty and closed under some suitable conditions. Moreover, the solutions set of the systems with set-valued perturbation is showed to be continuous with respect to the initial value.

STABILITY BOUNDARY CHARACTERIZATION OF NONLINEAR AUTONOMOUS DYNAMICAL SYSTEMS IN THE PRESENCE OF A SUPERCRITICAL HOPF EQUILIBRIUM POINT

2013 ◽  
Vol 23 (12) ◽  
pp. 1350196 ◽  
Author(s):  
JOSAPHAT R. R. GOUVEIA ◽  
FABÍOLO MORAES AMARAL ◽  
LUÍS F. C. ALBERTO
Keyword(s):  
Dynamical Systems ◽  
Equilibrium Point ◽  
Equilibrium Points ◽  
Stability Boundary ◽  
Complete Characterization ◽  
Center Manifolds ◽  
The Stability ◽  
Hyperbolic Equilibrium ◽  
Autonomous Dynamical Systems

A complete characterization of the boundary of the stability region (or area of attraction) of nonlinear autonomous dynamical systems is developed admitting the existence of a particular type of nonhyperbolic equilibrium point on the stability boundary, the supercritical Hopf equilibrium point. Under a condition of transversality, it is shown that the stability boundary is comprised of all stable manifolds of the hyperbolic equilibrium points lying on the stability boundary union with the center-stable and\or center manifolds of the type-k, k ≥ 1, supercritical Hopf equilibrium points on the stability boundary.

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.

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