On inclination-flip homoclinic orbit associated to a saddle-node singularity

1996 ◽  
Vol 27 (2) ◽  
pp. 145-160 ◽  
Author(s):  
C. A. Morales
Keyword(s):  
Homoclinic Orbit ◽  
Saddle Node ◽  
Inclination Flip

scholarly journals Resonant Homoclinic Flips Bifurcation in Principal Eigendirections

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Tiansi Zhang ◽  
Xiaoxin Huang ◽  
Deming Zhu
Keyword(s):  
Periodic Orbit ◽  
Coordinate System ◽  
Homoclinic Orbit ◽  
Principal Eigenvalue ◽  
Small Neighborhood ◽  
Homoclinic Bifurcation ◽  
Bifurcation Equation ◽  
Orbit Flip ◽  
Inclination Flip ◽  
Poincaré Return Map

A codimension-4 homoclinic bifurcation with one orbit flip and one inclination flip at principal eigenvalue direction resonance is considered. By introducing a local active coordinate system in some small neighborhood of homoclinic orbit, we get the Poincaré return map and the bifurcation equation. A detailed investigation produces the number and the existence of 1-homoclinic orbit, 1-periodic orbit, and double 1-periodic orbits. We also locate their bifurcation surfaces in certain regions.

scholarly journals A non-transverse homoclinic orbit to a saddle-node equilibrium

1996 ◽  
Vol 16 (3) ◽  
pp. 431-450 ◽  
Author(s):  
Alan R. Champneys ◽  
Jörg Härterich ◽  
Björn Sandstede
Keyword(s):  
Homoclinic Orbit ◽  
Homoclinic Orbits ◽  
Bifurcation Curve ◽  
Numerical Computations ◽  
Parameter Region ◽  
Codimension Two ◽  
Saddle Node ◽  
Unstable Manifolds ◽  
The Creation ◽  
Shift Dynamics

AbstractA homoclinic orbit is considered for which the center-stable and center-unstable manifolds of a saddle-node equilibrium have a quadratic tangency. This bifurcation is of codimension two and leads generically to the creation of a bifurcation curve defining two independent transverse homoclinic orbits to a saddle-node. This latter case was shown by Shilnikov to imply shift dynamics. It is proved here that in a large open parameter region of the codimension-two singularity, the dynamics are completely described by a perturbation of the Hénon-map giving strange attractors, Newhouse sinks and the creation of the shift dynamics. In addition, an example system admitting this bifurcation is constructed and numerical computations are performed on it.

Codimension-1 Sliding Bifurcations of a Filippov Pest Growth Model with Threshold Policy

2014 ◽  
Vol 24 (10) ◽  
pp. 1450122 ◽  
Author(s):  
Sanyi Tang ◽  
Guangyao Tang ◽  
Wenjie Qin
Keyword(s):  
Homoclinic Orbit ◽  
Sliding Mode ◽  
Control Strategies ◽  
Control Policy ◽  
Threshold Level ◽  
Total Density ◽  
Boundary Node ◽  
Threshold Policy ◽  
Saddle Node ◽  
Pest Outbreaks

A Filippov system is proposed to describe the stage structured nonsmooth pest growth with threshold policy control (TPC). The TPC measure is represented by the total density of both juveniles and adults being chosen as an index for decisions on when to implement chemical control strategies. The proposed Filippov system can have three pieces of sliding segments and three pseudo-equilibria, which result in rich sliding mode bifurcations and local sliding bifurcations including boundary node (boundary focus, or boundary saddle) and tangency bifurcations. As the threshold density varies the model exhibits the interesting global sliding bifurcations sequentially: touching → buckling → crossing → sliding homoclinic orbit to a pseudo-saddle → crossing → touching bifurcations. In particular, bifurcation of a homoclinic orbit to a pseudo-saddle with a figure of eight shape, to a pseudo-saddle-node or to a standard saddle-node have been observed for some parameter sets. This implies that control outcomes are sensitive to the threshold level, and hence it is crucial to choose the threshold level to initiate control strategy. One more sliding segment (or pseudo-equilibrium) is induced by the total density of a population guided switching policy, compared to only the juvenile density guided policy, implying that this control policy is more effective in terms of preventing multiple pest outbreaks or causing the density of pests to stabilize at a desired level such as an economic threshold.

NONRESONANT BIFURCATIONS OF HETEROCLINIC LOOPS WITH ONE INCLINATION FLIP

2011 ◽  
Vol 21 (01) ◽  
pp. 255-273 ◽  
Author(s):  
SHULIANG SHUI ◽  
JINGJING LI ◽  
XUYANG ZHANG
Keyword(s):  
Periodic Orbit ◽  
Homoclinic Orbit ◽  
Vector Fields ◽  
Heteroclinic Orbits ◽  
Dimensional Vector ◽  
Heteroclinic Loop ◽  
Stable Foliation ◽  
Local Coordinates ◽  
Inclination Flip

Heteroclinic bifurcations in four-dimensional vector fields are investigated by setting up local coordinates near a heteroclinic loop. This heteroclinic loop consists of two principal heteroclinic orbits, but there is one stable foliation that involves an inclination flip. The existence, nonexistence, coexistence and uniqueness of the 1-heteroclinic loop, 1-homoclinic orbit, and 1-periodic orbit are studied. Also, the nonexistence, existence of the 2-homoclinic and 2-periodic orbit are demonstrated.

A STRANGE ATTRACTOR WITH LARGE ENTROPY IN THE UNFOLDING OF A LOW RESONANT DEGENERATE HOMOCLINIC ORBIT

2006 ◽  
Vol 16 (12) ◽  
pp. 3509-3522 ◽  
Author(s):  
M. MARTENS ◽  
V. NAUDOT ◽  
J. YANG
Keyword(s):  
Vector Field ◽  
Lebesgue Measure ◽  
Parameter Space ◽  
Homoclinic Orbit ◽  
Strange Attractor ◽  
Linear Part ◽  
Unperturbed System ◽  
Positive Lebesgue Measure ◽  
Open Set ◽  
Inclination Flip

The unfolding of a vector field exhibiting a degenerate homoclinic orbit of inclination-flip type is studied. The linear part of the unperturbed system possesses a resonance but the coefficient of the corresponding monomial vanishes. We show that for an open set in the parameter space, the system possesses a suspended cubic Hénon-like map. As a consequence, strange attractors with entropy close to log 3 persist in a positive Lebesgue measure set.

scholarly journals The Twisting Bifurcations of Double Homoclinic Loops with Resonant Eigenvalues

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Xiaodong Li ◽  
Weipeng Zhang ◽  
Fengjie Geng ◽  
Jicai Huang
Keyword(s):  
Periodic Orbit ◽  
Homoclinic Orbit ◽  
Dimensional System ◽  
Bifurcation Diagrams ◽  
Saddle Node Bifurcation ◽  
Saddle Node ◽  
Higher Dimensional

The twisting bifurcations of double homoclinic loops with resonant eigenvalues are investigated in four-dimensional systems. The coexistence or noncoexistence of large 1-homoclinic orbit and large 1-periodic orbit near double homoclinic loops is given. The existence or nonexistence of saddle-node bifurcation surfaces is obtained. Finally, the complete bifurcation diagrams and bifurcation curves are also given under different cases. Moreover, the methods adopted in this paper can be extended to a higher dimensional system.

scholarly journals Multiple and Complex Codimension-2 Bifurcations underlying Post-inhibitory Rebound Spike and Excitability Transition

2021 ◽  
Author(s):  
Xianjun Wang ◽  
Huaguang Gu ◽  
Yuye Li ◽  
Bo Lu
Keyword(s):  
Hopf Bifurcation ◽  
Homoclinic Orbit ◽  
Neuronal Model ◽  
Type I ◽  
Type Ii ◽  
Neural Firing ◽  
Saddle Node Bifurcation ◽  
Saddle Node ◽  
Snic Bifurcation ◽  
Degenerate Bifurcation

Abstract Neuron exhibits nonlinear dynamics such as excitability transition and post-inhibitory rebound (PIR) spike related to bifurcations, which are associated with information processing, locomotor modulation, or brain disease. PIR spike is evoked by inhibitory stimulation instead of excitatory stimulation, which presents a challenge to the threshold concept. In the present paper, 7 codimension-2 or degenerate bifurcations related to 10 codimension-1 bifurcations are acquired in a neuronal model, which presents the bifurcations underlying the excitability transition and PIR spike. Type I excitability corresponds to saddle-node bifurcation on an invariant cycle (SNIC) bifurcation, and type II excitability to saddle-node (SN) bifurcation or sub-critical Hopf (SubH) bifurcation or sup-critical Hopf (SupH) bifurcation. The excitability transition from type I to II corresponds to the codimension-2 bifurcation, Saddle-Node Homoclinic orbit (SNHO) bifurcation, via which SNIC bifurcation terminates and meanwhile big homoclinic orbit (BHom) bifurcation and SN bifurcation emerge. A degenerate bifurcation via which BHom bifurcation terminates and fold limit cycle (LPC) bifurcation emerges is responsible for spiking transition from type I to II, and the roles of other codimension-2 bifurcations (Cusp, Bogdanov-Takens, and Bautin) are discussed. In addition, different from the widely accepted viewpoint that PIR spike is mainly evoked near Hopf bifurcation rather than SNIC bifurcation, PIR spike is identified to be induced near SNIC or BHom or LPC bifurcations, and threshold curves resemble that of Hopf bifurcation. The complex bifurcations present comprehensive and deep understandings of excitability transition and PIR spike, which are helpful for the modulation to neural firing activities and physiological functions.

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