Generation of limit cycles from a separatrix forming a loop and from the separatrix of an equilibrium state of saddle-node type

1963 ◽  
pp. 189-231 ◽  
Author(s):  
A. A. and Leontovič Andronov
Keyword(s):  
Equilibrium State ◽  
Limit Cycles ◽  
Saddle Node ◽  
Node Type

scholarly journals The Geometry of Quadratic Polynomial Differential Systems with a Finite and an Infinite Saddle-Node (A, B)

2014 ◽  
Vol 24 (04) ◽  
pp. 1450044 ◽  
Author(s):  
Joan C. Artés ◽  
Alex C. Rezende ◽  
Regilene D. S. Oliveira
Keyword(s):  
Normal Form ◽  
Bifurcation Diagram ◽  
Limit Cycles ◽  
Applied Mathematics ◽  
Algebraic Surfaces ◽  
Phase Portraits ◽  
Differential Systems ◽  
Bifurcation Set ◽  
Saddle Node ◽  
The Family

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers have been written on these systems, a complete understanding of this family is still missing. Classical problems, and in particular, Hilbert's 16th problem [Hilbert, 1900, 1902], are still open for this family. In this paper, we study the bifurcation diagram of the family QsnSN which is the set of all quadratic systems which have at least one finite semi-elemental saddle-node and one infinite semi-elemental saddle-node formed by the collision of two infinite singular points. We study this family with respect to a specific normal form which puts the finite saddle-node at the origin and fixes its eigenvectors on the axes. Our aim is to make a global study of the family [Formula: see text] which is the closure of the set of representatives of QsnSN in the parameter space of that specific normal form. This family can be divided into three different subfamilies according to the position of the infinite saddle-node, namely: (A) with the infinite saddle-node in the horizontal axis, (B) with the infinite saddle-node in the vertical axis and (C) with the infinite saddle-node in the bisector of the first and third quadrants. These three subfamilies modulo the action of the affine group and times homotheties are four-dimensional. Here, we provide the complete study of the geometry with respect to a normal form of the first two families, (A) and (B). The bifurcation diagram for the subfamily (A) yields 38 phase portraits for systems in [Formula: see text] (29 in QsnSN(A)) out of which only three have limit cycles and 13 possess graphics. The bifurcation diagram for the subfamily (B) yields 25 phase portraits for systems in [Formula: see text] (16 in QsnSN(B)) out of which 11 possess graphics. None of the 25 portraits has limit cycles. Case (C) will yield many more phase portraits and will be written separately in a forthcoming new paper. Algebraic invariants are used to construct the bifurcation set. The phase portraits are represented on the Poincaré disk. The bifurcation set of [Formula: see text] is the union of algebraic surfaces and one surface whose presence was detected numerically. All points in this surface correspond to connections of separatrices. The bifurcation set of [Formula: see text] is formed only by algebraic surfaces.

Canards of Folded Saddle-Node Type I

2015 ◽  
Vol 47 (4) ◽  
pp. 3235-3283 ◽  
Author(s):  
Theodore Vo ◽  
Martin Wechselberger
Keyword(s):  
Type I ◽  
Saddle Node ◽  
Node Type

scholarly journals Saddle-node of limit cycles in planar piecewise linear systems and applications

2019 ◽  
Vol 39 (9) ◽  
pp. 5275-5299
Author(s):  
Victoriano Carmona ◽  
◽  
Soledad Fernández-García ◽  
Antonio E. Teruel ◽  
◽  
...  
Keyword(s):  
Linear Systems ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Piecewise Linear Systems ◽  
Saddle Node

scholarly journals Periodic Orbits as a Probe to Reveal Exotic States in Vibrationally Excited Molecules: The Saddle-Node States

1993 ◽  
Vol 13 (2) ◽  
pp. 87-99 ◽  
Author(s):  
Stavros C. Farantos
Keyword(s):  
Periodic Orbits ◽  
Vibrationally Excited ◽  
Simultaneous Generation ◽  
Potential Barriers ◽  
Saddle Node ◽  
Families Of Periodic Orbits ◽  
Molecular Potentials ◽  
Localized Quantum States ◽  
Theoretical Results ◽  
Node Type

We present theoretical results which show the existence of isomerizing localized quantum states above potential barriers for the excited 1B2 states of SO2 and O3. These states are assigned by periodic orbits which emerge from saddle-node bifurcations, the characteristic of which is the simultaneous generation of two families of periodic orbits one stable and one unstable. Similar isomerizing states and bifurcations have been found for other molecules, and this leads to the conclusion, that the appearance of saddle-node type states may be a generic phenomenon for molecular potentials with barriers.

scholarly journals Algebraic Limit Cycles in Piecewise Linear Differential Systems

2018 ◽  
Vol 28 (03) ◽  
pp. 1850039 ◽  
Author(s):  
Claudio A. Buzzi ◽  
Armengol Gasull ◽  
Joan Torregrosa
Keyword(s):  
Limit Cycles ◽  
Piecewise Linear ◽  
Saddle Node Bifurcation ◽  
Differential Systems ◽  
Saddle Node ◽  
Linear Differential Systems ◽  
Linear Differential ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

This paper is devoted to study the algebraic limit cycles of planar piecewise linear differential systems. In particular, we present examples exhibiting two explicit hyperbolic algebraic limit cycles, as well as some one-parameter families with a saddle-node bifurcation of algebraic limit cycles. We also show that all degrees for algebraic limit cycles are allowed.

SEMISTABLE LIMIT CYCLES THAT BIFURCATE FROM CENTERS

2003 ◽  
Vol 13 (11) ◽  
pp. 3489-3498 ◽  
Author(s):  
HECTOR GIACOMINI ◽  
MIREILLE VIANO ◽  
JAUME LLIBRE
Keyword(s):  
Parameter Space ◽  
Limit Cycles ◽  
Analytic Functions ◽  
Arbitrary Order ◽  
Perturbation Parameter ◽  
Analytic Method ◽  
Analytic Center ◽  
Saddle Node Bifurcation ◽  
Saddle Node ◽  
Bifurcation Surface

Suppose that the differential system [Formula: see text] has a center at the origin for ε=0, where P0, Q0, aij and bij are analytic functions in their variables, such that aij(0)=bij(0)=0. We present an analytic method to compute the semistable limit cycles which bifurcate from the periodic orbits of the analytic center, up to an arbitrary order in the perturbation parameter ε. We also provide an algorithm for the computation of the saddle–node bifurcation hypersurface of limit cycles in the parameter space {aij,bij}1≤i,j≤m. As an example, we apply the method to compute, first, the anal ytic expression of the unique semistable limit cycle of the Liénard system [Formula: see text] and second, an approximation of the saddle-node bifurcation surface of limit cycles in the parameter space (a1, a3, a5). Both computations are valid for ε sufficiently small.

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