scholarly journals Singularly Perturbed Oscillators with Exponential Nonlinearities

Author(s):  
S. Jelbart ◽  
K. U. Kristiansen ◽  
P. Szmolyan ◽  
M. Wechselberger

AbstractSingular exponential nonlinearities of the form $$e^{h(x)\epsilon ^{-1}}$$ e h ( x ) ϵ - 1 with $$\epsilon >0$$ ϵ > 0 small occur in many different applications. These terms have essential singularities for $$\epsilon =0$$ ϵ = 0 leading to very different behaviour depending on the sign of h. In this paper, we consider two prototypical singularly perturbed oscillators with such exponential nonlinearities. We apply a suitable normalization for both systems such that the $$\epsilon \rightarrow 0$$ ϵ → 0 limit is a piecewise smooth system. The convergence to this nonsmooth system is exponential due to the nonlinearities we study. By working on the two model systems we use a blow-up approach to demonstrate that this exponential convergence can be harmless in some cases while in other scenarios it can lead to further degeneracies. For our second model system, we deal with such degeneracies due to exponentially small terms by extending the space dimension, following the approach in Kristiansen (Nonlinearity 30(5): 2138–2184, 2017), and prove—for both systems—existence of (unique) limit cycles by perturbing away from singular cycles having desirable hyperbolicity properties.

2019 ◽  
Vol 29 (12) ◽  
pp. 1950160
Author(s):  
Zhihui Fan ◽  
Zhengdong Du

In this paper, we discuss the bifurcation of periodic orbits in planar piecewise smooth systems with discontinuities on finitely many smooth curves intersecting at the origin. We assume that the unperturbed system has either a limit cycle or a periodic annulus such that the limit cycle or each periodic orbit in the periodic annulus crosses every switching curve transversally multiple times. When the unperturbed system has a limit cycle, we give the conditions for its stability and persistence. When the unperturbed system has a periodic annulus, we obtain the expression of the first order Melnikov function and establish sufficient conditions under which limit cycles can bifurcate from the annulus. As an example, we construct a concrete nonlinear planar piecewise smooth system with three zones with 11 limit cycles bifurcated from the periodic annulus.


2021 ◽  
Vol 31 (15) ◽  
Author(s):  
Zhongjian Wang ◽  
Dingheng Pi

In this paper, we study bifurcations of the regularized systems of planar piecewise smooth systems, which have a visible fold-regular point and a sliding or grazing heteroclinic loop. Our results show that if the planar piecewise smooth system with a sliding heteroclinic loop undergoes sliding heteroclinic bifurcation, then the regularized system can bifurcate with a stable limit cycle passing through the regularized region and at most two limit cycles outside the regularized region. The regularized system can have at most three periodic orbits. When the upper subsystem is a Hamiltonian system, the regularized system can bifurcate with a semi-stable periodic orbit. Finally, we discuss two cases when the heteroclinic loop of a piecewise smooth system remains unbroken under a small perturbation. Our results show that the regularized system can bifurcate at most two limit cycles from an inner unstable grazing heteroclinic loop.


2021 ◽  
Vol 31 (15) ◽  
Author(s):  
Fang Wu ◽  
Lihong Huang ◽  
Jiafu Wang

The objective of this paper is to study the sliding bifurcation in a planar piecewise smooth system with an elliptic switching curve. Some new phenomena are observed, such as a crossing limit cycle containing four intersections with the switching curve, sliding cycles having four sliding segments, and sliding cycles consisting of the entire switching curve. Firstly, we investigate the bifurcation of sliding cycle from a sliding heteroclinic connection to two cusps and show the appearance of one sliding cycle with two folds. To plot the bifurcation diagram, a planar piecewise linear system with two zones separated by an ellipse are considered. Moreover, we study in more detail the unfolding of a sliding cycle connecting four cusps by exhibiting its complete bifurcation diagram. More precisely, we explore the necessary and sufficient conditions for the existence of limit cycles and derive the concrete bifurcation curves. Additionally, a simple piecewise smooth system with nonlinear subsystems is studied, which shows the possibility of the existence of two nested limit cycles. Finally, numerical simulations are given to confirm the theoretical analysis.


2021 ◽  
Vol 31 (10) ◽  
pp. 2150159
Author(s):  
Ai Ke ◽  
Maoan Han

We study bifurcations of limit cycles arising after perturbations of a special piecewise smooth system, which has a center and a homoclinic loop. By using the Picard–Fuchs equation, we give an upper bound of the maximum number of limit cycles bifurcated from the period annulus between the center and the homoclinic loop. Furthermore, by applying the method of first-order Melnikov function we obtain a lower bound of the maximum number of limit cycles bifurcated from the center.


2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Owen D. Jarman ◽  
Olivier Biner ◽  
John J. Wright ◽  
Judy Hirst

AbstractMitochondrial complex I (NADH:ubiquinone oxidoreductase) is a crucial metabolic enzyme that couples the free energy released from NADH oxidation and ubiquinone reduction to the translocation of four protons across the inner mitochondrial membrane, creating the proton motive force for ATP synthesis. The mechanism by which the energy is captured, and the mechanism and pathways of proton pumping, remain elusive despite recent advances in structural knowledge. Progress has been limited by a lack of model systems able to combine functional and structural analyses with targeted mutagenic interrogation throughout the entire complex. Here, we develop and present the α-proteobacterium Paracoccus denitrificans as a suitable bacterial model system for mitochondrial complex I. First, we develop a robust purification protocol to isolate highly active complex I by introducing a His6-tag on the Nqo5 subunit. Then, we optimize the reconstitution of the enzyme into liposomes, demonstrating its proton pumping activity. Finally, we develop a strain of P. denitrificans that is amenable to complex I mutagenesis and create a catalytically inactive variant of the enzyme. Our model provides new opportunities to disentangle the mechanism of complex I by combining mutagenesis in every subunit with established interrogative biophysical measurements on both the soluble and membrane bound enzymes.


2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.


mSphere ◽  
2017 ◽  
Vol 2 (4) ◽  
Author(s):  
Dianxiong Zou ◽  
Todd M. Hennessey

ABSTRACT Although many single-cell eukaryotes have served as classical model systems for chemosensory studies for decades, the major emphasis has been on chemoattraction and no chemorepellent receptor gene has been identified in any unicellular eukaryote. This is the first description of a gene that codes for a chemorepellent receptor in any protozoan. Integration of both depolarizing chemorepellent pathways and hyperpolarizing chemoattractant pathways is as important to chemoresponses of motile unicells as excitatory and inhibitory neurotransmitter pathways are to neurons. Therefore, both chemoattractant and chemorepellent pathways should be represented in a useful unicellular model system. Tetrahymena cells provide such a model system because simple behavioral bioassays, gene knockouts, biochemical analysis, and other approaches can be used with these eukaryotic model cells. This work can contribute to the basic understanding of unicellular sensory responses and provide insights into the evolution of chemoreceptors and possible chemorepellent approaches for preventing infections by some pathogenic protozoa. A conditioned supernatant from Tetrahymena thermophila contains a powerful chemorepellent for wild-type cells, and a gene called G37 is required for this response. This is the first genomic identification of a chemorepellent receptor in any eukaryotic unicellular organism. This conditioned supernatant factor (CSF) is small (<1 kDa), and its repellent effect is resistant to boiling, protease treatment, and nuclease digestion. External BAPTA eliminated the CSF response, suggesting that Ca2+ entry is required for the classical avoiding reactions (AR) used for chemorepulsion. A macronuclear G37 gene knockout (G37-KO) mutant is both nonresponsive to the CSF and overresponsive to other repellents such as quinine, lysozyme, GTP, and high potassium concentrations. All of these mutant phenotypes were reversed by overexpression of the wild-type G37 gene in a G37 overexpression mutant. Overexpression of G37 in the wild type caused increased responsiveness to the CSF and underresponsiveness to high K+ concentrations. Behavioral adaptation (by prolonged exposure to the CSF) caused decreases in responsiveness to all of the stimuli used in the wild type and the overexpression mutant but not in the G37-KO mutant. We propose that the constant presence of the CSF causes a decreased basal excitability of the wild type due to chemosensory adaptation through G37 and that all of the G37-KO phenotypes are due to an inability to detect the CSF. Therefore, the G37 protein may be the CSF receptor. The physiological role of these G37-mediated responses may be to both moderate basal excitability and detect the CSF as an indicator of high cell density growth. IMPORTANCE Although many single-cell eukaryotes have served as classical model systems for chemosensory studies for decades, the major emphasis has been on chemoattraction and no chemorepellent receptor gene has been identified in any unicellular eukaryote. This is the first description of a gene that codes for a chemorepellent receptor in any protozoan. Integration of both depolarizing chemorepellent pathways and hyperpolarizing chemoattractant pathways is as important to chemoresponses of motile unicells as excitatory and inhibitory neurotransmitter pathways are to neurons. Therefore, both chemoattractant and chemorepellent pathways should be represented in a useful unicellular model system. Tetrahymena cells provide such a model system because simple behavioral bioassays, gene knockouts, biochemical analysis, and other approaches can be used with these eukaryotic model cells. This work can contribute to the basic understanding of unicellular sensory responses and provide insights into the evolution of chemoreceptors and possible chemorepellent approaches for preventing infections by some pathogenic protozoa.


Sign in / Sign up

Export Citation Format

Share Document