TY - JOUR

T1 - Differential Equations with a Small Parameter and Multipeak Oscillations

AU - Chumakov, G. A.

AU - Chumakova, N. A.

N1 - The work was carried out with financial support from the Ministry of Science and Higher Education of the Russian Federation within the framework of state assignments of Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences, project FWNF-2022-0005, and Boreskov Institute of Catalysis of the Siberian Branch of the Russian Academy of Sciences, project no. FWUR-2024-0037. No additional grants to carry out or direct this particular research were obtained.

PY - 2024/3

Y1 - 2024/3

N2 - In this paper, we study a nonlinear dynamical system of autonomous ordinary differentialequations with a small parameter such that two variables and are fast and another one is slow. If we take the limit as, then this becomes a “degeneratesystem” included in the one-parameter family of two-dimensional subsystems offast motions with the parameter in some interval. It is assumed that in each subsystem there existsa structurally stable limit cycle. In addition, in the completedynamical system there is some structurally stable periodic orbit that tends to a limit cycle for some as tends to zero. We can define the first return map, or the Poincarémap, on a local cross section in the hyperplane orthogonal to at some point. We prove that the Poincaré map has an invariantmanifold for the fixed point corresponding to the periodic orbit on a guaranteed interval over the variable, and the interval length is separated from zero as tends to zero. The proved theorem allows one to formulate some sufficientconditions for the existence and/or absence of multipeak oscillations in the complete dynamicalsystem. As an example of application of the obtained results, we consider some kinetic model ofthe catalytic reaction of hydrogen oxidation on nickel.

AB - In this paper, we study a nonlinear dynamical system of autonomous ordinary differentialequations with a small parameter such that two variables and are fast and another one is slow. If we take the limit as, then this becomes a “degeneratesystem” included in the one-parameter family of two-dimensional subsystems offast motions with the parameter in some interval. It is assumed that in each subsystem there existsa structurally stable limit cycle. In addition, in the completedynamical system there is some structurally stable periodic orbit that tends to a limit cycle for some as tends to zero. We can define the first return map, or the Poincarémap, on a local cross section in the hyperplane orthogonal to at some point. We prove that the Poincaré map has an invariantmanifold for the fixed point corresponding to the periodic orbit on a guaranteed interval over the variable, and the interval length is separated from zero as tends to zero. The proved theorem allows one to formulate some sufficientconditions for the existence and/or absence of multipeak oscillations in the complete dynamicalsystem. As an example of application of the obtained results, we consider some kinetic model ofthe catalytic reaction of hydrogen oxidation on nickel.

KW - Poincaré map

KW - invariant manifold

KW - kinetic model

KW - limit cycle

KW - multipeak self-oscillations

KW - ordinary differential equation

KW - small parameter

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85191370942&origin=inward&txGid=0002e8c5455d66395b714abdfaa63311

UR - https://elibrary.ru/item.asp?id=67311794

UR - https://www.mendeley.com/catalogue/329cabe4-942e-3605-8604-a144563dce11/

U2 - 10.1134/S1990478924010034

DO - 10.1134/S1990478924010034

M3 - Article

VL - 18

SP - 18

EP - 35

JO - Journal of Applied and Industrial Mathematics

JF - Journal of Applied and Industrial Mathematics

SN - 1990-4789

IS - 1

M1 - 3

ER -