Classification of Dynamical Systems Near a Cosymmetric Equilibrium

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/h3876-8857-0078-b Classification of Dynamical Systems Near a Cosymmetric Equilibrium Kurakin, L. G. , Kurdoglyan, A. V. Vladikavkaz Mathematical Journal 2025. Vol. 27. Issue 4. Abstract: A local classification is developed in a neighborhood of a cosymmetric equilibrium for differential equations with invertible cosymmetry and a vector parameter, under the assumption that the kernel of the linearization matrix at the cosymmetric equilibrium is two-dimensional and that the entire stability spectrum, except for the double zero eigenvalue, is stable. Equations with such properties are of codimension one among even-dimensional systems with a cosymmetric equilibrium. In all cases, such a system admits a straightenable family of non-cosymmetric equilibria near the cosymmetric one. The classification is based on the following properties: the type of the cosymmetric equilibrium (node, focus, saddle); the relative position of the cosymmetric equilibrium and the family (including the case where the cosymmetric equilibrium belongs to the family); the number of boundary equilibria of the family separating its stable and unstable regions (\(\leqslant 3\)); the number of intersections of each separatrix of the cosymmetric saddle equilibrium with the family (\(\leqslant 3\)). Each property is determined by polynomial conditions, and the classification therefore reduces to identifying sets of conditions with a non-empty intersection. The defining polynomial conditions and corresponding phase portraits are presented for each identified class. The existence of each nonempty class is established by a scalable example for non-obvious cases, while the emptiness of the remaining classes is established separately. This work continues the studies of L. G. Kurakin and V. I. Yudovich [1, 2], where analogous results were obtained in the neighborhood of a non-cosymmetric equilibrium. Keywords: differential equation, equilibrium, cosymmetry, classification Language: English Download the full text Download JATS XML For citation: Kurakin, L. G. and Kurdoglyan, A. V. Classification of Dynamical Systems Near a Cosymmetric Equilibrium, Vladikavkaz Math. J., 2025, vol. 27, no. 4, pp. 86-102. DOI 10.46698/h3876-8857-0078-b + References 1. Kurakin, L. G. and Yudovich, V. I. Bifurcations Accompanying Monotonic Instability of an Equilibrium of a Cosymmetric Dynamical System, Chaos, 2000, vol. 10, no. 2, pp. 311-330. DOI: 10.1063/1.166497. 2. Kurakin, L. G. and Yudovich, V. I. Bifurcations under Monotone Loss of Stability of Equilibrium in a Cosymmetric Dynamical System, Doklady Akademii Nauk, 2000, vol. 61, no. 3, pp. 433-437 (in Russian). 3. Yudovich, V. I. Cosymmetry, Degeneration of Solutions of Operator Equations, and Onset of a Filtration Convection, Mathematical Notes, 1991, vol. 49, no. 5, pp. 540-545. DOI: 10.1007/BF01142654. 4. Lyubimov, D. V. Convective Motions in a Porous Medium Heated from Below, Journal of Applied Mechanics and Technical Physics, 1975, vol. 16, no. 2, pp. 257-261. DOI: 10.1007/BF00858924. 5. Kurakin, L. G. Critical Cases of Stability. Converse Implicit Function Theorem For Dynamical Systems With Cosymmetry, Journal of Applied Mechanics and Technical Physics, 1975, vol. 63, pp. 503-508. DOI: 10.1007/BF02311253. 6. Kurakin, L. G. and Kurdoglyan, A. V. On the Isolation/Nonisolation of a Cosymmetric Equilibrium and Bifurcations in its Neighborhood, Regular and Chaotic Dynamics, 2021, vol. 26, no. 3, pp. 258-270. DOI: 10.1134/S1560354721030047. 7. Vainberg, M. M. and Trenogin, V. A. Teoriya vetvleniya reshenij nelinejnyh uravnenij [Branching Theory of Solutions of Nonlinear Equations], Moscow, Nauka, 1969, 528 p. (in Russian). 8. Marsden, J. E. and McCracken, M. The Hopf Bifurcation and its Applications, vol. 19, New York, Springer Science & Business Media, 2012, 407 p. 9. Lyapunov, A. M. Obshchaya zadacha ob ustojchivosti dvizheniya [The General Problem of the Stability of Motion], Gostekhizdat, Moscow, 1950, 471 p. (in Russian). ← Contents of issue


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