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.

scholarly journals Asymptotics for some discretizations of dynamical systems, application to second order systems with non-local nonlinearities

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Thierry Horsin ◽  
Mohamed Ali Jendoubi
Keyword(s):  
Asymptotic Behavior ◽  
Dynamical Systems ◽  
Numerical Simulations ◽  
Equilibrium Points ◽  
Infinite Dimensional ◽  
Infinite Dimensional Dynamical Systems ◽  
Finite Dimensional ◽  
Angle Condition ◽  
Non Local ◽  
Second Order Systems

<p style='text-indent:20px;'>In the present paper we study the asymptotic behavior of discretized finite dimensional dynamical systems. We prove that under some discrete angle condition and under a Lojasiewicz's inequality condition, the solutions to an implicit scheme converge to equilibrium points. We also present some numerical simulations suggesting that our results may be extended under weaker assumptions or to infinite dimensional dynamical systems.</p>

Saddle-node equilibrium points on the stability boundary of nonlinear autonomous dynamical systems

2017 ◽  
Vol 33 (1) ◽  
pp. 113-135 ◽  
Author(s):  
Fabíolo Moraes Amaral ◽  
Luís Fernando C. Alberto ◽  
Josaphat R. R. Gouveia
Keyword(s):  
Dynamical Systems ◽  
Equilibrium Points ◽  
Stability Boundary ◽  
Saddle Node ◽  
The Stability ◽  
Autonomous Dynamical Systems

MULTIPLE ATTRACTORS AND BIFURCATIONS IN HARD OSCILLATORS DRIVEN BY CONSTANT INPUTS

2012 ◽  
Vol 22 (11) ◽  
pp. 1250267 ◽  
Author(s):  
VALENTINA LANZA ◽  
LINDA PONTA ◽  
MICHELE BONNIN ◽  
FERNANDO CORINTO
Keyword(s):  
Dynamical Systems ◽  
Limit Cycle ◽  
Equilibrium Points ◽  
External Input ◽  
Global Bifurcations ◽  
Multiple Attractors ◽  
Neural Excitability ◽  
Bifurcation Phenomena ◽  
Homoclinic Bifurcations ◽  
Input Strength

Hard oscillators are dynamical systems that show the coexistence of qualitatively different attractors, in the form of limit cycles and equilibrium points. In the presence of external inputs their dynamic behavior is significantly different from those of oscillators, called soft, with a limit cycle as unique attractor. This paper studies the dynamics of a simple hard oscillator under the influence of a constant external input. It is shown that, despite the apparent simplicity, when the input strength and the oscillator's natural frequency are varied the system exhibits many different bifurcation phenomena, including global bifurcations as saddle-node on limit cycle and homoclinic bifurcations. The model under investigation can play a role in neuroscience, as it exhibits two different mechanisms of class I neural excitability and one mechanism for class II. It also highlights a mechanism of transition between the two classes.

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