Control Barrier Function-Based Quadratic Programs Introduce Undesirable Asymptotically Stable Equilibria

Of concern is the global dynamics of a two-species Holling-II amensalism system with nonlinear growth rate. The existence and stability of trivial equilibrium, semi-trivial equilibria, interior equilibria and infinite singularity are studied. Under different parameters, there exist two stable equilibria which means that this model is not always globally asymptotically stable. Together with the existence of all possible equilibria and their stability, saddle connection and close orbits, we derive some conditions for transcritical bifurcation and saddle-node bifurcation. Furthermore, the global dynamics of the model is performed. Next, we incorporate Allee effect on the first species and offer a new analysis of equilibria and bifurcation discussion of the model. Finally, several numerical examples are performed to verify our theoretical results.

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Control Barrier Function Based Quadratic Programs for Safety Critical Systems

IEEE Transactions on Automatic Control ◽

10.1109/tac.2016.2638961 ◽

2017 ◽

Vol 62 (8) ◽

pp. 3861-3876 ◽

Cited By ~ 165

Author(s):

Aaron D. Ames ◽

Xiangru Xu ◽

Jessy W. Grizzle ◽

Paulo Tabuada

Keyword(s):

Barrier Function ◽

Critical Systems ◽

Quadratic Programs ◽

Safety Critical ◽

Safety Critical Systems

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Asymptotically stable equilibria for monotone semiflows

Discrete and Continuous Dynamical Systems ◽

10.3934/dcds.2006.14.385 ◽

2006 ◽

Vol 14 (3) ◽

pp. 385-398 ◽

Cited By ~ 8

Author(s):

M. W. Hirsch ◽

◽

Hal L. Smith ◽

Keyword(s):

Asymptotically Stable ◽

Stable Equilibria

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BIFURCATIONS IN A PREDATOR-PREY MODEL WITH DIFFUSION AND MEMORY

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127406015751 ◽

2006 ◽

Vol 16 (06) ◽

pp. 1855-1863 ◽

Cited By ~ 8

Author(s):

SHABAN ALY

Keyword(s):

Hopf Bifurcation ◽

Allee Effect ◽

Stationary Solutions ◽

Asymptotically Stable ◽

Predator Prey ◽

Predator Prey Model ◽

The Past ◽

Prey System ◽

Stable Equilibria ◽

Per Capita

In this paper we formulate a delayed predator-prey system in two patches in which the per capita migration rate of each species is influenced only by its own density, i.e. there is no response to the density of the other and the growth rate of the predator depends on the prey that was available in the past. If the equilibrium point lies in the Allée effect zone and when the diffusion is present only, we show that at a critical value of the bifurcation parameter the system undergoes a Turing bifurcation, patterns emerge, the spatially homogeneous equilibrium loses its stability and two new spatially non-constant stable equilibria emerge which are asymptotically stable. When the delay is present only, the increase of delay destabilizes the system and causes the occurrence of periodic oscillations, Andronov–Hopf bifurcation. For the full general model (with both diffusion and delay) if the bifurcation parameters are increased through critical values of diffusion and delay the two new spatially nonconstant stationary solutions lose their stability by Hopf bifurcation.

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Stability Analysis of a Class of Noise Perturbed Neural Networks

International Journal of Neural Systems ◽

10.1142/s0129065797000306 ◽

1997 ◽

Vol 08 (03) ◽

pp. 295-300 ◽

Cited By ~ 1

Author(s):

Anke Meyer-Bäse

Keyword(s):

Neural Networks ◽

Stability Analysis ◽

Sliding Mode ◽

Neural System ◽

Order System ◽

Analysis Tool ◽

Asymptotically Stable ◽

Parameter Perturbations ◽

Stable Equilibria ◽

External Stimuli

We establish robustness stability results for a specific type of artificial neural networks for associative memories under parameter perturbations and determine conditions that ensure the existence of asymptotically stable equilibria of the perturbed neural system that are near the asymptotically stable equilibria of the original unperturbed neural network. The proposed stability analysis tool is the sliding mode control and it facilitates the analysis by considering only a reduced-order system instead of the original one and time-dependent external stimuli.

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The persistence of synchronization under environmental noise

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2005.1484 ◽

2005 ◽

Vol 461 (2059) ◽

pp. 2257-2267 ◽

Cited By ~ 23

Author(s):

Tomás Caraballo ◽

Peter E Kloeden

Keyword(s):

Additive Noise ◽

Environmental Noise ◽

Dissipative Systems ◽

Asymptotically Stable ◽

Stable Equilibria

It is shown that the synchronization of dissipative systems persists when they are disturbed by additive noise, no matter how large the intensity of the noise, provided asymptotically stable stationary-stochastic solutions are used instead of asymptotically stable equilibria.

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