Some modified Michaelis–Menten equations having stable closed trajectories

1988 ◽  
Vol 109 (3-4) ◽  
pp. 341-359 ◽  
Author(s):  
Russell A. Smith
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Global Convergence ◽  
Critical Point ◽  
Closed Trajectory ◽  
Biochemical Reactions ◽  
3 Dimensional ◽  
Convergence Of Solutions ◽  
Autonomous Ordinary Differential Equation

A 3-dimensional autonomous ordinary differential equation is studied which models certain cellular biochemical reactions. Extended Poincaré-Bendixson theory is used to obtain algebraic conditions on the parameters which are sufficient for the existence of at least one stable closed trajectory. Similar conditions are also obtained for the absence of chaos and for the global convergence of solutions to a critical point.

ON THE STRUCTURE OF THE SOLUTIONS OF A TWO-PARAMETER FAMILY OF THREE-DIMENSIONAL ORDINARY DIFFERENTIAL EQUATIONS

2003 ◽  
Vol 13 (05) ◽  
pp. 1287-1298 ◽  
Author(s):  
SERKAN T. IMPRAM ◽  
RUSSELL JOHNSON ◽  
RAFFAELLA PAVANI
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Wide Range ◽  
Two Parameters ◽  
Two Parameter ◽  
Autonomous Ordinary Differential Equation

We analyze the global structure of the solutions of a three-dimensional, autonomous ordinary differential equation which depends on two parameters. We use graphical, heuristic, and rigorous arguments to show that as the parameters vary, a wide range of dynamical behavior is displayed.

scholarly journals On the third order boundary value problems with asymmetric nonlinearity

2011 ◽  
Vol 16 (2) ◽  
pp. 231-241 ◽  
Author(s):  
Sergey Smirnov
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Third Order ◽  
Number Of Solutions ◽  
The Third ◽  
Asymmetric Nonlinearity ◽  
Autonomous Ordinary Differential Equation

The author considers two point third order boundary value problem with asymmetric nonlinearity. The structure and oscillatory properties of solutions of the third order nonlinear autonomous ordinary differential equation are discussed. Results on the estimation of the number of solutions to boundary value problem are provided. An illustrative example is given.

scholarly journals Existence of Nontrivial Solutions for Sixth-Order Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1852
Author(s):  
Gabriele Bonanno ◽  
Pasquale Candito ◽  
Donal O’Regan
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Critical Point ◽  
Nontrivial Solution ◽  
Critical Point Theory ◽  
Point Theory ◽  
Sixth Order ◽  
Nontrivial Solutions ◽  
Order Ordinary Differential Equation

We show the existence of at least one nontrivial solution for a nonlinear sixth-order ordinary differential equation is investigated. Our approach is based on critical point theory.

scholarly journals On Computer-Assisted Proving The Existence Of Periodic And Bounded Orbits

2015 ◽  
Vol 29 (1) ◽  
pp. 7-17
Author(s):  
Roman Srzednicki
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Periodic Solutions ◽  
Poincaré Map ◽  
Conley Index ◽  
Computer Assisted ◽  
Small Step ◽  
Index Pair ◽  
Time Periodic ◽  
Autonomous Ordinary Differential Equation

AbstractWe announce a new result on determining the Conley index of the Poincaré map for a time-periodic non-autonomous ordinary differential equation. The index is computed using some singular cycles related to an index pair of a small-step discretization of the equation. We indicate how the result can be applied to computer-assisted proofs of the existence of bounded and periodic solutions. We provide also some comments on computer-assisted proving in dynamics.

scholarly journals Semi-discrete optimal transport methods for the semi-geostrophic equations

2022 ◽  
Vol 61 (1) ◽  
Author(s):  
David P. Bourne ◽  
Charlie P. Egan ◽  
Beatrice Pelloni ◽  
Mark Wilkinson
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Weak Solutions ◽  
Compact Support ◽  
Optimal Transport ◽  
Constructive Proof ◽  
3 Dimensional ◽  
Eulerian Coordinates ◽  
Numerical Solver ◽  
Explicit Relation

AbstractWe give a new and constructive proof of the existence of global-in-time weak solutions of the 3-dimensional incompressible semi-geostrophic equations (SG) in geostrophic coordinates, for arbitrary initial measures with compact support. This new proof, based on semi-discrete optimal transport techniques, works by characterising discrete solutions of SG in geostrophic coordinates in terms of trajectories satisfying an ordinary differential equation. It is advantageous in its simplicity and its explicit relation to Eulerian coordinates through the use of Laguerre tessellations. Using our method, we obtain improved time-regularity for a large class of discrete initial measures, and we compute explicitly two discrete solutions. The method naturally gives rise to an efficient numerical method, which we illustrate by presenting simulations of a 2-dimensional semi-geostrophic flow in geostrophic coordinates generated using a numerical solver for the semi-discrete optimal transport problem coupled with an ordinary differential equation solver.

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