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Rigorous verification of saddle–node bifurcations in ODEs

Indagationes Mathematicae, 2016
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Jean-Philippe Lessard
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SHILNIKOV’S SADDLE-NODE BIFURCATION

International Journal of Bifurcation and Chaos, 1996
In 1969, Shilnikov described a bifurcation for families of ordinary differential equations involving n≥2 trajectories bi-asymptotic to a non-hyperbolic stationary point. At nearby parameter values the system has trajectories in correspondence with the full shift on n symbols.
Glendinning, Paul, Sparrow, Colin
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A NONAUTONOMOUS SADDLE-NODE BIFURCATION PATTERN

Stochastics and Dynamics, 2004
In this paper we study certain differential equations depending on a small parameter ε which exhibit a bifurcation of saddle-node type as ε passes through zero. We use a classical averaging technique together with methods and results from the modern theory of nonautonomous differential equations.
FABBRI, ROBERTA, JOHNSON, RUSSELL ALLAN, F. MANTELLINI   +2 more
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Control of the saddle-node and transcritical bifurcations

IFAC Postprint Volumes IPPV / International Federation of Automatic Control, 2004
Abstract In this paper, the control of the saddle-node and transcritical bifurcations in nonlinear systems is treated. A new approach is presented to find sufficient conditions in terms of the original vector fields. The analysis of the system dynamics is reduced to dimension one through the center manifold theorem.
Fernando Verduzco
exaly   +2 more sources

Predicting non-stationary and stochastic activation of saddle-node bifurcation in non-smooth dynamical systems

Nonlinear Dynamics, 2018
Saddle-node bifurcation can cause dynamical systems undergo large and sudden transitions in their response, which is very sensitive to stochastic and non-stationary influences that are unavoidable in practical applications.
Jinki Kim, K W Wang
exaly   +2 more sources

Saddle-Node Bifurcations on Fractal Basin Boundaries

Physical Review Letters, 1995
We demonstrate and analyze a bifurcation producing a type of fractal basin boundary which has the strange property that any point that is on the boundary of that basin is also simultaneously on the boundary of at least two other basins. We give rigorous general criteria guaranteeing this phenomenon, present illustrative numerical examples, and discuss ...
, Nusse, , Ott, , Yorke
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Critical saddle-node horseshoes: bifurcations and entropy

Nonlinearity, 2003
The paper deals with a one-parameter family \(\mathcal{O}(D)\) of \(C^\infty\)-diffeomorphisms \((f_\mu)_{\mu \in [-1,1]}\) of a compact disk \(D\) of a two-dimensional manifold \(M.\) It is assumed that for \(\mu < 0\) the limit set of \(f_\mu\) in \(D\) is the union of a sink \(S_\mu\) and a hyperbolic saddle set \(\Lambda_\mu\) conjugate to a ...
Díaz, Lorenzo J., Rios, Isabel L.
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NONAUTONOMOUS SADDLE-NODE BIFURCATION IN A CANONICAL ELECTROSTATIC MEMS

International Journal of Bifurcation and Chaos, 2013
We study the existence and stability of periodic solutions of a canonical mass-spring model of electrostatically actuated Micro-Electro-Mechanical System (MEMS) by means of classical topological techniques like a priori bounds, Leray–Schauder degree and topological index. A saddle-node bifurcation is revealed, in analogy with the autonomous case.
Alexander Gutiérrez, Pedro J. Torres
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The Saddle-Node Separatrix-Loop Bifurcation

SIAM Journal on Mathematical Analysis, 1987
In the paper vector fields \(x'=f(x)\), \(x\in {\mathbb{R}}^ 2\), having at some point an equilibrium of saddle-node type with a separatrix loop, are studied. Such vector fields fill a codimension two submanifold \(\Sigma\) of an appropriate Banach space. Author gives analytic conditions that determine whether a two-parameter perturbation of \(x'=f(x)\)
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Saddle-node bifurcations in the spectrum of HOCl

The Journal of Chemical Physics, 2000
A detailed analysis of the bound-state spectrum of HOCl (hypoclorous acid) in the ground electronic state is presented. Exact quantum mechanical calculations (filter diagonalization) are performed employing an ab initio potential energy surface, which has been constructed using the multireference configuration-interaction method and a quintuple-zeta ...
J. Weiß, J. Hauschildt, S. Yu. Grebenshchikov, R. Düren, R. Schinke, J. Koput, S. Stamatiadis, S. C. Farantos   +7 more
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