Zero Lyapunov exponent in the vicinity of the saddle-node bifurcation point in the presence of noise

2008 ◽  
Vol 78 (3) ◽  
Author(s):  
Alexander E. Hramov ◽  
Alexey A. Koronovskii ◽  
Maria K. Kurovskaya
Keyword(s):  
Lyapunov Exponent ◽  
Bifurcation Point ◽  
Saddle Node Bifurcation ◽  
Saddle Node

COHERENCE RESONANCE–INDUCED STOCHASTIC NEURAL FIRING AT A SADDLE-NODE BIFURCATION

2011 ◽  
Vol 25 (29) ◽  
pp. 3977-3986 ◽  
Author(s):  
HUAGUANG GU ◽  
HUIMIN ZHANG ◽  
CHUNLING WEI ◽  
MINGHAO YANG ◽  
ZHIQIANG LIU ◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Point ◽  
Firing Pattern ◽  
Neuronal System ◽  
Neural Firing ◽  
Coherence Resonance ◽  
Saddle Node Bifurcation ◽  
Hopf Bifurcation Point ◽  
Firing Patterns ◽  
Saddle Node

Coherence resonance at a saddle-node bifurcation point and the corresponding stochastic firing patterns are simulated in a theoretical neuronal model. The characteristics of noise-induced neural firing pattern, such as exponential decay in histogram of interspike interval (ISI) series, independence and stochasticity within ISI series are identified. Firing pattern similar to the simulated results was discovered in biological experiment on a neural pacemaker. The difference between this firing and integer multiple firing generated at a Hopf bifurcation point is also given. The results not only revealed the stochastic dynamics near a saddle-node bifurcation, but also gave practical approaches to identify the saddle-node bifurcation and to distinguish it from the Hopf bifurcation in neuronal system. In addition, many previously observed firing patterns can be attribute to stochastic firing pattern near such a saddle-node bifurcation.

Frequency-Amplitude Response of Subharmonic Resonance of One-Third Order of Electrostatically Actuated MEMS Circular Plates

2020 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Julio Beatriz ◽  
Miguel Martinez
Keyword(s):  
Steady State ◽  
Circular Plate ◽  
Bifurcation Point ◽  
Multiple Scales ◽  
Subharmonic Resonance ◽  
Third Order ◽  
Amplitude Response ◽  
Saddle Node Bifurcation ◽  
Electrostatically Actuated ◽  
Saddle Node

Abstract This work deals with the subharmonic resonance of one-third order of electrostatically actuated clamped MEMS circular plate resonators. The system consists of flexible MEMS circular plate parallel to a ground plate actuated only by AC voltage. Hard excitations due to large enough AC voltage of frequency near three-halves of the natural frequency of the MEMS plate resonator lead it into a subharmonic resonance. The partial differential equation describing the motion of the resonator is nondimensionalized and two reduced order models are developed. The first one consists of a one mode of vibration model which is solved using the Method of Multiple Scales (MMS). The frequency-amplitude response (bifurcation diagram) is predicted. Hard excitations were modeled by keeping the first term of the Taylor polynomial of the electrostatic force as a large term and the rest of them as small terms. The second model uses two modes of vibration, and it is solved through numerical integration. This produces time responses of the resonator. Both methods show a zero-amplitude steady-state stable branch for the entire range of resonant frequencies. Also, two branches, one unstable and one stable, with a saddle node bifurcation point are predicted for non-zero steady state amplitudes. One can notice that non-zero steady state amplitudes can be reached only from large enough initial amplitudes. Both methods are in agreement for amplitudes up to 0.7 of the gap. The effect of damping and voltage on the frequency response are reported. As the damping increases, the saddle-node bifurcation point and consequently the non-zero steady-state branches are shifted to larger amplitudes. As the voltage increases, the saddle-node bifurcation point and the non-zero branches are shifted to lower amplitudes and lower frequency.

scholarly journals Fast-slow Variable Dissection with Two Slow Variables Related to Calcium Concentrations: A Case Study to Bursting in a Neural Pacemaker Model

2021 ◽  
Author(s):  
Yuye Li ◽  
Huaguang Gu ◽  
Yanbing Jia ◽  
Kaihua Ma
Keyword(s):  
Bifurcation Point ◽  
Slow Variable ◽  
Quiescent State ◽  
Bifurcation Curve ◽  
Fast Subsystem ◽  
Saddle Node Bifurcation ◽  
Initiation Point ◽  
Saddle Node ◽  
Pathological Pain ◽  
Slow Variables

Abstract Neuronal bursting is an electrophysiological behavior participating in physiological or pathological functions and a complex nonlinear alternating between burst and quiescent state modulated by slow variables. Identification of dynamics of bursting modulated by two slow variables is still an open problem. In the present paper, a novel fast-slow variable dissection method with two slow variables is proposed to analyze the complex bursting in a 4-dimensional neuronal model to describe bursting associated with pathological pain. The lumenal (Clum) and intracellular (Cin) calcium concentrations are the slowest variables respectively in the quiescent state and burst duration. Questions encountered when the traditional method with one low variable is used. When Clum is taken as slow variable, the burst is successfully identified to terminate near the saddle-homoclinic bifurcation point of the fast subsystem and begin not from the saddle-node bifurcation. With Cin chosen as slow variable, Clum value of initiation point is far from the saddle-node bifurcation point, due to Clum not contained in the equation of membrane potential. To overcome this problem, both Cin and Clum are regarded as slow variables, the two-dimensional fast subsystem exhibits a saddle-node bifurcation point, which is extended to a saddle-node bifurcation curve by introducing Clum dimension. Then, the initial point of burst is successfully identified to be near the saddle-node bifurcation curve. The results present a feasible method for fast-slow variable dissection and deep understanding to the complex bursting behavior with two slow variables, which is helpful for the modulation to pathological pain.

Attractor-preserving control to avoid saddle-node bifurcation

2014 ◽  
Vol 2 ◽  
pp. 150-153
Author(s):  
Daisuke Ito ◽  
Tetsushi Ueta ◽  
Shigeki Tsuji ◽  
Kazuyuki Aihara
Keyword(s):  
Saddle Node Bifurcation ◽  
Saddle Node
Sign in / Sign up

Export Citation Format

Share Document