Behaviour near a hyperbolic equilibrium

1995 ◽  
pp. 242-258
Author(s):  
Odo Diekmann ◽  
Sjoerd M. Verduyn Lunel ◽  
Stephan A. van Gils ◽  
Hanns-Otto Walther
Keyword(s):  
Hyperbolic Equilibrium

scholarly journals Nonlinear Compartment Models with Time-Dependent Parameters

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1657
Author(s):  
Jochen Merker ◽  
Benjamin Kunsch ◽  
Gregor Schuldt
Keyword(s):  
Stable Equilibrium ◽  
Compartment Model ◽  
Basins Of Attraction ◽  
Dynamical Behaviour ◽  
Time Dependent ◽  
Compartment Models ◽  
Autonomous Case ◽  
Hyperbolic Equilibrium ◽  
Interior Points ◽  
Two Parameter

A nonlinear compartment model generates a semi-process on a simplex and may have an arbitrarily complex dynamical behaviour in the interior of the simplex. Nonetheless, in applications nonlinear compartment models often have a unique asymptotically stable equilibrium attracting all interior points. Further, the convergence to this equilibrium is often wave-like and related to slow dynamics near a second hyperbolic equilibrium on the boundary. We discuss a generic two-parameter bifurcation of this equilibrium at a corner of the simplex, which leads to such dynamics, and explain the wave-like convergence as an artifact of a non-smooth nearby system in C0-topology, where the second equilibrium on the boundary attracts an open interior set of the simplex. As such nearby idealized systems have two disjoint basins of attraction, they are able to show rate-induced tipping in the non-autonomous case of time-dependent parameters, and induce phenomena in the original systems like, e.g., avoiding a wave by quickly varying parameters. Thus, this article reports a quite unexpected path, how rate-induced tipping can occur in nonlinear compartment models.

SINGULAR CYCLES AND Ck-ROBUST TRANSITIVE SET ON MANIFOLD WITH BOUNDARY

2011 ◽  
Vol 13 (02) ◽  
pp. 191-211 ◽  
Author(s):  
D. CARRASCO-OLIVERA ◽  
C. A. MORALES ◽  
B. SAN MARTÍN
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Manifold With Boundary ◽  
Transitive Set ◽  
Singular Cycle ◽  
Hyperbolic Equilibrium

Let M be a 3-manifold with boundary ∂M. Let X be a C∞, vector field on M, tangent to ∂M, exhibiting a singular cycle associated to a hyperbolic equilibrium σ∈∂M with real eigenvalues λss < λs < 0 < λu satisfying λs - λss - 2λu > 0. We prove under generic conditions and k large enough the existence of a Ck robust transitive set of X, that is, any Ck vector field Ck close to X exhibits a transitive set containing the cycle. In particular, C∞ vector fields exhibiting Ck robust transitive sets, for k large enough, which are not singular-hyperbolic do exist on any compact 3-manifold with boundary.

Oxygen binding of erythrocruorin and coelomic cell haemoglobin from the terebellid polychaete Neoamphitrite figulus related to some environmental factors

1982 ◽  
Vol 62 (2) ◽  
pp. 373-384 ◽  
Author(s):  
Rufus M. G. Wells ◽  
Lynda M. Warren
Keyword(s):  
Haemoglobin Concentration ◽  
Log P ◽  
Oxygen Binding ◽  
Transfer System ◽  
Equilibrium Curve ◽  
High Affinity ◽  
Coelomic Cells ◽  
Hyperbolic Equilibrium ◽  
Oxygen Equilibrium Curve

Measurements of pH, oxygen content, O2-combining capacity, and haemoglobin concentration were made for the vascular blood of the burrowing polychaete Neoamphitrite figulus in order to assess the role of its two respiratory pigments in respiration. The oxygen equilibrium curve of the erythrocruorin (extracellular haemoglobin) in the vessels was sigmoidal, having an n50 value of 1·5 and a low affinity for oxygen as determined by the P50 which was 26 mmHg at pH 7·31 and 18 °C. O2-binding by the erythrocruorin is sensitive to changes in pH (Δ log P50/Δ log pH = –0·24 to –0·29). The coelomic cell haemoglobin has a hyperbolic equilibrium curve (n50 = 1·0) and a high affinity for oxygen (P50 = 4·5 mmHg) independent of pH, suggesting an oxygen transfer system from the erythrocruorin to the coelomic cells.

STABILITY BOUNDARY CHARACTERIZATION OF NONLINEAR AUTONOMOUS DYNAMICAL SYSTEMS IN THE PRESENCE OF A SUPERCRITICAL HOPF EQUILIBRIUM POINT

2013 ◽  
Vol 23 (12) ◽  
pp. 1350196 ◽  
Author(s):  
JOSAPHAT R. R. GOUVEIA ◽  
FABÍOLO MORAES AMARAL ◽  
LUÍS F. C. ALBERTO
Keyword(s):  
Dynamical Systems ◽  
Equilibrium Point ◽  
Equilibrium Points ◽  
Stability Boundary ◽  
Complete Characterization ◽  
Center Manifolds ◽  
The Stability ◽  
Hyperbolic Equilibrium ◽  
Autonomous Dynamical Systems

A complete characterization of the boundary of the stability region (or area of attraction) of nonlinear autonomous dynamical systems is developed admitting the existence of a particular type of nonhyperbolic equilibrium point on the stability boundary, the supercritical Hopf equilibrium point. Under a condition of transversality, it is shown that the stability boundary is comprised of all stable manifolds of the hyperbolic equilibrium points lying on the stability boundary union with the center-stable and\or center manifolds of the type-k, k ≥ 1, supercritical Hopf equilibrium points on the stability boundary.

HOMOCLINIC MANIFOLDS, CENTER MANIFOLDS AND EXACT SOLUTIONS OF FOUR-DIMENSIONAL TRAVELING WAVE SYSTEMS FOR TWO CLASSES OF NONLINEAR WAVE EQUATIONS

2011 ◽  
Vol 21 (02) ◽  
pp. 527-543 ◽  
Author(s):  
JIBIN LI ◽  
YI ZHANG
Keyword(s):  
Exact Solutions ◽  
Traveling Wave ◽  
Wave Equations ◽  
Periodic Wave ◽  
Nonlinear Wave Equations ◽  
Center Manifolds ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Gap Soliton ◽  
Hyperbolic Equilibrium

For the Lax KdV5 equation and the KdV–Sawada–Kotera–Ramani equation, their corresponding four-dimensional traveling wave systems are studied by using Congrove's method. Exact explicit gap soliton, embedded soliton, periodic and quasi-periodic wave solutions are obtained. The existence of homoclinic manifolds to three kinds of equilibria including a hyperbolic equilibrium, a center-saddle and an equilibrium with zero pair of eigenvalues is revealed. The bifurcation conditions of equilibria are given.

Seismic Response Analysis on a Chaotic System

2013 ◽  
Vol 639-640 ◽  
pp. 911-916
Author(s):  
Cui Xiang Liang
Keyword(s):  
Seismic Response ◽  
Chaotic System ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Response Analysis ◽  
Bifurcation Diagrams ◽  
Seismic Response Analysis ◽  
Nonlinear Seismic Response ◽  
Hyperbolic Equilibrium ◽  
Hyperbolic Equilibrium Point

This paper is concerned with the dynamical behavior of a chaotic system which is a model for seismic response of structures. The local bifurcation of the non-hyperbolic equilibrium point of the chaotic system is investigated by using center manifold method. The transcritical bifurcation is analyzed in detail. Based on numerical simulations, spectrums of maximal Lyapunov exponent and the bifurcation diagrams are presented for the dynamic analysis. The method proposed can be used as a reference of nonlinear seismic response analysis.

scholarly journals THE EQUILIBRIUM BETWEEN CYTOCHROME OXIDASE AND CARBON MONOXIDE

1957 ◽  
Vol 40 (4) ◽  
pp. 593-608 ◽  
Author(s):  
George Wald ◽  
David W. Allen
Keyword(s):  
Carbon Monoxide ◽  
Cytochrome Oxidase ◽  
Hill Equation ◽  
Bohr Effect ◽  
Degree Of Saturation ◽  
Percentage Saturation ◽  
Equilibrium Function ◽  
Hyperbolic Equilibrium ◽  
High Degree ◽  
The Hill

An evolution argument which attempted to trace the development of hemoglobins from such respiratory pigments as cytochrome oxidase presupposed that the latter possesses, in addition to its high affinity for oxygen, an approximately hyperbolic equilibrium function, and little if any Bohr effect (decline in affinity for oxygen with rise in acidity). Since cytochrome oxidase, unlike hemoglobin, is irreversibly oxidized by oxygen, the present experiments examine its combination with carbon monoxide, with which, like hemoglobin, it yields a true equilibrium. In all known hemoglobins the form of the equilibrium function and the vigor of the Bohr effect are similar with carbon monoxide and with oxygen, so that observations involving the former gas are relevant to the relations of the latter. The equilibrium function of cytochrome oxidase with carbon monoxide—percentage saturation vs. partial pressure of CO—is slightly inflected (in the Hill equation n = 1.26; for a hyperbola, n = 1). No Bohr effect is present in the range of pH 7–8. The pressure of carbon monoxide at which half-saturation occurs (p50) is about 0.17 mm. at 10–13°C. The affinity for carbon monoxide is therefore higher than commonly supposed. These properties are consistent with the evolution argument. They are important also for the physiological functioning of cytochrome oxidase, the nearly hyperbolic equilibrium function facilitating a high degree of saturation, and the lack of Bohr effect making this enzyme impervious to hyperacidity. The slight inflection of the equilibrium function shows that the Fe-porphyrin units of cytochrome oxidase interact to a degree, hence that the enzyme must contain more than one such unit per molecule. It is suggested that in cytochrome oxidase two Fe-porphyrin groups may unite with one oxygen in the manner Fe++-O2-Fe++; and that the evolution of hemoglobins proceeded over a first stage in which the hemes were separated so that each combines with only one molecule of oxygen, so tending to remain reduced; to a further stage in which the separated hemes interact through the protein to facilitate one another in combining with oxygen.

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