scholarly journals Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xianjun Wang ◽  
Huaguang Gu ◽  
Bo Lu
Keyword(s):  
Homoclinic Orbit ◽  
Threshold Curve

scholarly journals Homoclinic orbits and Lie rotated vector fields

2000 ◽  
Vol 24 (3) ◽  
pp. 187-192
Author(s):  
Jie Wang ◽  
Chen Chen
Keyword(s):  
Limit Cycles ◽  
Homoclinic Orbit ◽  
Vector Fields ◽  
Homoclinic Orbits ◽  
Singular Points ◽  
Definition Of

Based on the definition of Lie rotated vector fields in the plane, this paper gives the property of homoclinic orbit as parameter is changed and the singular points are fixed on Lie rotated vector fields. It gives the conditions of yielding limit cycles as well.

Stabilization of the Furuta pendulum around its homoclinic orbit

2002 ◽  
Vol 75 (6) ◽  
pp. 390-398 ◽  
Author(s):  
Isabelle Fantoni ◽  
Rogelio Lozano
Keyword(s):  
Homoclinic Orbit

Central Mechanism of Hearing in Insects

1961 ◽  
Vol 38 (3) ◽  
pp. 545-558 ◽  
Author(s):  
NOBUO SUGA ◽  
YASUJI KATSUKI
Keyword(s):  
Sound Source ◽  
Threshold Curve ◽  
Inhibitory Interaction ◽  
Response Range ◽  
Metathoracic Ganglion ◽  
Hair Sensilla ◽  
Prothoracic Ganglion ◽  
Tympanic Organ ◽  
Large Fibre ◽  
The Brain

1. The impulses from the tympanic organ are transmitted at the prothoracic ganglion to a central neuron, the auditory T large fibre, which lies in the cord between the brain and the metathoracic ganglion. The impulses in the T large fibre are conducted rostrally and caudally with the same discharge pattern. Information is sent up to the brain, and down to the metathoracic ganglion, after a delay of about 12 msec. 2. The impulses from the cercal hair sensilla are transmitted to two similar auditory C large fibres which lie in the cord between the metathoracic and last (6th) abdominal ganglia and are then sent up to the mesothoracic ganglia by other auditory large fibres. 3. Central inhibitory interaction between the impulses from the tympanic nerves of the two sides are shown by a marked increase of impulses in the T large fibre following section of one of the tympanic nerves. No inhibitory interaction is found between the impulses from the two cercal nerves. 4. The auditory T large fibre receives not only the excitatory effect from the ipsilateral tympanic nerve at the prothoracic ganglion, but also the inhibitory and weak excitatory effects from the contralateral one. 5. The response range of the T large fibre is narrower than the threshold curve of the tympanic nerve and corresponds with one type of response range in the tympanic neurons. The response ranges of the C large fibres correspond closely with the threshold curve of the cercal nerve. 6. A large difference in threshold between the two T large fibres is found in the response to sound incident from the side. The number of impulses in the T large fibre nearer to the sound source is greater than in that farther from the source. 7. The difference in the number of impulses between the two T large fibres is most marked in the response to sound of the frequency which is dominant in stridulation. This difference is due to the mutual inhibitory interaction of neurons which modifies the number of impulses without changing the threshold of the tympanic large fibre. 8. It is suggested that the central inhibitory interaction increases the information about a sound source and plays an important role in the mechanism of the directional sense. 9. The stridulation of the group activates the tympanic nerve and evokes synchronized discharge in the T large fibre, but scarcely at all in the primary C large fibre. The tympanic organ and its neural network seem well adapted to reception of stridulation. 10. It is concluded that though neither of the two sound receptive organs--the tympanic organ and the cercal hair sensilla--can perform frequency analysis, the insect may be able to do so by making use of both organs, since they have different frequency ranges and are served by different auditory large-fibre tracts.

scholarly journals Bifurcations and Stability of Nondegenerated Homoclinic Loops for Higher Dimensional Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Yinlai Jin ◽  
Feng Li ◽  
Han Xu ◽  
Jing Li ◽  
Liqun Zhang ◽  
...  
Keyword(s):  
Periodic Orbit ◽  
Homoclinic Orbit ◽  
Variational Equation ◽  
Small Neighborhood ◽  
Unperturbed System ◽  
Homoclinic Loop ◽  
Resonant Condition ◽  
Local Current ◽  
The Stability ◽  
Linear Variational Equation

By using the foundational solutions of the linear variational equation of the unperturbed system along the homoclinic orbit as the local current coordinates system of the system in the small neighborhood of the homoclinic orbit, we discuss the bifurcation problems of nondegenerated homoclinic loops. Under the nonresonant condition, existence, uniqueness, and incoexistence of 1-homoclinic loop and 1-periodic orbit, the inexistence ofk-homoclinic loop andk-periodic orbit is obtained. Under the resonant condition, we study the existence of 1-homoclinic loop, 1-periodic orbit, 2-fold 1-periodic orbit, and two 1-periodic orbits; the coexistence of 1-homoclinic loop and 1-periodic orbit. Moreover, we give the corresponding existence fields and bifurcation surfaces. At last, we study the stability of the homoclinic loop for the two cases of non-resonant and resonant, and we obtain the corresponding criterions.

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