scholarly journals A theorem on S-shaped bifurcation curve for a positone problem with convex–concave nonlinearity and its applications to the perturbed Gelfand problem

2011 ◽  
Vol 251 (2) ◽  
pp. 223-237 ◽  
Author(s):  
Kuo-Chih Hung ◽  
Shin-Hwa Wang
Keyword(s):  
Bifurcation Curve ◽  
Gelfand Problem

scholarly journals Partial regularity of stable solutions to the fractional Geľfand-Liouville equation

2021 ◽  
Vol 10 (1) ◽  
pp. 1316-1327
Author(s):  
Ali Hyder ◽  
Wen Yang
Keyword(s):  
Liouville Equation ◽  
Weak Solutions ◽  
Partial Regularity ◽  
Singular Set ◽  
Stable Solutions ◽  
Gelfand Problem

Abstract We analyze stable weak solutions to the fractional Geľfand problem ( − Δ ) s u = e u i n Ω ⊂ R n . $$\begin{array}{} \displaystyle (-{\it\Delta})^su = e^u\quad\mathrm{in}\quad {\it\Omega}\subset\mathbb{R}^n. \end{array}$$ We prove that the dimension of the singular set is at most n − 10s.

Detection of Synergistic Combinatorial Perturbations by a Bifurcation-Based Approach

2021 ◽  
Vol 31 (12) ◽  
pp. 2150175
Author(s):  
Min Luo ◽  
Dasong Huang ◽  
Jianfeng Jiao ◽  
Ruiqi Wang
Keyword(s):  
Gene Expression ◽  
Rational Design ◽  
Epithelial To Mesenchymal Transition ◽  
Regulation Of Gene Expression ◽  
Drug Combinations ◽  
Bifurcation Curve ◽  
Combinatorial Regulation ◽  
Mesenchymal Transition ◽  
Combinatorial Control ◽  
Two Parameters

Drug combination has become an attractive strategy against complex diseases, despite the challenges in handling a large number of possible combinations among candidate drugs. How to detect effective drug combinations and determine the dosage of each drug in the combination is still a challenging task. When regarding a drug as a perturbation, we propose a bifurcation-based approach to detect synergistic combinatorial perturbations. In the approach, parameters of a dynamical system are divided into two groups according to their responses to perturbations. By combining two parameters chosen from two groups, three types of combinations can be obtained. Synergism for different perturbation combinations can be detected by relative positions of the bifurcation curve and the isobole. The bifurcation-based approach can be used not only to detect combinatorial perturbations but also to determine their perturbation quantities. To demonstrate the effectiveness of the approach, we apply it to the epithelial-to-mesenchymal transition (EMT) network. The approach has implications for the rational design of drug combinations and other combinatorial control, e.g. combinatorial regulation of gene expression.

GWN-INDUCED BURSTING, SPIKING, AND RANDOM SUBTHRESHOLD IMPULSING OSCILLATION BEFORE HOPF BIFURCATIONS IN THE CHAY MODEL

2004 ◽  
Vol 14 (12) ◽  
pp. 4143-4159 ◽  
Author(s):  
ZHUOQIN YANG ◽  
QISHAO LU ◽  
HUAGUANG GU ◽  
WEI REN
Keyword(s):  
Hysteresis Loop ◽  
Gaussian White Noise ◽  
Global Bifurcation ◽  
Intrinsic Noise ◽  
Stable Node ◽  
Bifurcation Curve ◽  
Fast System ◽  
Integer Multiple ◽  
Rest State ◽  
Stable Focus

Gaussian white noise (GWN), as an intrinsic noise source, can give rise to various firing activities at the rest state before a supercritical or subcritical Hopf bifurcation (supH or subH) in the Chay system without or with external current input, when VK, VC, λn and I are considered as changeable control parameters. These firing activities are closely related to the global bifurcation mechanism of the whole system and the fast/slow dynamical subsystems, and can be tackled by means of bifurcation analysis. GWN can induce some typical bursting phenomena in the stochastic Chay system. Firstly, integer multiple "fold/homoclinic" or "circle/homoclinic" bursting due to GWN, with only one spike per burst, can arise from rest states before both subH and supH (with respect to the parameter VK), and their respective trajectories have the same shape and property. However, less spikes appear and their peaks are lower before supH, comparing with those before subH. Secondly, a "fold/fold" point–point hysteresis loop bursting due to GWN is generated before supH (with respect to the parameter VC) on the upper branch of a "Z"-shaped bifurcation curve between two fold bifurcations of the fast system. Thirdly, at a rest state before subH (with respect to the additional current I) situated on the lower branch of a "S"-shaped bifurcation curve between two fold bifurcations of the fast system, a GWN-induced firing pattern appears and is classified as "Hopf/homoclinic" bursting via "fold/homoclinic" point–point hysteresis loop. GWN-induced firing activities other than bursting can also be observed in the stochastic Chay system. For example, sometimes GWN-induced continuous spiking without any particular shape may arise at a rest state before supH (with respect to the parameter VK) for certain values of parameters. Moreover, under the situation that a stable node and a stable focus coexist before subH (with respect to the parameter I) and the attractive region of the stable node is larger than that of the stable focus, GWN only provoke random subthreshold impulsing oscillation near the stable node.

scholarly journals Complex Behaviors of Epidemic Model with Nonlinear Rewiring Rate

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Ding Fang ◽  
Yongxin Zhang ◽  
Wendi Wang
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Epidemic Model ◽  
Multiple Equilibria ◽  
Homoclinic Bifurcation ◽  
Propagation Model ◽  
Bifurcation Curve ◽  
Saddle Node Bifurcation ◽  
Saddle Node

An SIS propagation model with the nonlinear rewiring rate on an adaptive network is considered. It is found by bifurcation analysis that the model has the complex behaviors which include the transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, and Bogdanov–Takens bifurcation. Especially, a bifurcation curve with “S” shape emerges due to the nonlinear rewiring rate, which leads to multiple equilibria and twice saddle-node bifurcations. Numerical simulations show that the model admits a homoclinic bifurcation and a saddle-node bifurcation of the limit cycle.

A note on the persistence of an s-shaped bifurcation curve

1987 ◽  
Vol 24 (3) ◽  
pp. 175-179 ◽  
Author(s):  
Ratnasingham Shivaji
Keyword(s):  
Bifurcation Curve

scholarly journals Numerical Exploration of Kaldorian Macrodynamics: Enhanced Stability and Predominance of Period Doubling and Chaos with Flexible Exchange Rates

2008 ◽  
Vol 2008 ◽  
pp. 1-23 ◽  
Author(s):  
Toichiro Asada ◽  
Christos Douskos ◽  
Panagiotis Markellos
Keyword(s):  
Exchange Rates ◽  
Stability Region ◽  
Extreme Values ◽  
Period Doubling ◽  
Model Parameters ◽  
Flip Bifurcation ◽  
Bifurcation Curve ◽  
The Stability ◽  
Flexible Exchange Rates ◽  
Enhanced Stability

We explore a discrete Kaldorian macrodynamic model of an open economy with flexible exchange rates, focusing on the effects of variation of the model parameters, the speed of adjustment of the goods marketα, and the degree of capital mobilityβ. We determine by a numerical grid search method the stability region in parameter space and find that flexible rates cause enhanced stability of equilibrium with respect to variations of the parameters. We identify the Hopf-Neimark bifurcation curve and the flip bifurcation curve, and find that the period doubling cascades which leads to chaos is the dominant behavior of the system outside the stability region, persisting to large values ofβ. Cyclical behavior of noticeable presence is detected for some extreme values of a state parameter. Bifurcation and Lyapunov exponent diagrams are computed illustrating the complex dynamics involved. Examples of attractors and trajectories are presented. The effect of the speed of adaptation of the expected rate is also briefly discussed. Finally, we explore a special model variation incorporating the “wealth effect” which is found to behave similarly to the basic model, contrary to the model of fixed exchange rates in which incorporation of this effect causes an entirely different behavior.

scholarly journals Accurate Approximation of Homoclinic Solutions in Gray–Scott Kinetic Model

2015 ◽  
Vol 25 (09) ◽  
pp. 1550125 ◽  
Author(s):  
Yu. A. Kuznetsov ◽  
H. G. E. Meijer ◽  
B. Al-Hdaibat ◽  
W. Govaerts
Keyword(s):  
Kinetic Model ◽  
Normal Form ◽  
Homoclinic Orbits ◽  
Analytic Solutions ◽  
Homoclinic Bifurcation ◽  
Second Order ◽  
Numerical Continuation ◽  
Accurate Approximation ◽  
Bifurcation Curve ◽  
Systems Of Differential Equations

The second-order predictor for the homoclinic orbit is applied to the Gray–Scott model. The problem is used to illustrate the approximation of the homoclinic orbits near a generic Bogdanov–Takens bifurcation in n-dimensional systems of differential equations. In the process, we show that it is necessary to take (usually ignored) cubic terms in the Bogdanov–Takens normal form into account to derive a correct second-order prediction for the homoclinic bifurcation curve. The analytic solutions are compared with those obtained by numerical continuation.

scholarly journals Asymmetric gravity–capillary solitary waves on deep water

2014 ◽  
Vol 759 ◽  
Author(s):  
Z. Wang ◽  
J.-M. Vanden-Broeck ◽  
P. A. Milewski
Keyword(s):  
Wave Propagation ◽  
Solitary Waves ◽  
Travelling Wave ◽  
Propagation Direction ◽  
Travelling Wave Solutions ◽  
Two Dimensional ◽  
Bifurcation Curve ◽  
Wave Propagation Direction ◽  
Wave Solutions ◽  
Deep Fluid

AbstractWe present new families of gravity–capillary solitary waves propagating on the surface of a two-dimensional deep fluid. These spatially localised travelling-wave solutions are non-symmetric in the wave propagation direction. Our computation reveals that these waves appear from a spontaneous symmetry-breaking bifurcation, and connect two branches of multi-packet symmetric solitary waves. The speed–energy bifurcation curve of asymmetric solitary waves features a zigzag behaviour with one or more turning points.

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