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### Research Of Dynamical Systems Based on p-adic Analysis

Phyo Wai Linn, Uvarova L.A.

FSBEU HPE MSTU (STANKIN), Russia, 127055, Moscow, Vadkovsky Lane, 3A, +79999047908, E-mail: phyowailinnmipt@gmail.com +74999729459, E-mail: uvar11@yandex.ru

At present time, research in the field of chaotic systems is great interest. In particular, this is due to the need to find chaotic attractors, many of which have practical applications [1].

At the same time, it seems relevant to use p-adic analysis to study nonlinear dynamical systems. As showed result, which obtained in [2–4], this approach is presented as quite effective. In the present work, this approach is used to simulate the processes of phase transitions of the “liquid-gas” type. Molecular structures of phases are modeled by a node – communication system. In particular, it can be a Cayley tree with a root at the phase boundary. To analyze the p-adic model, use the Hamiltonian model:

$H=H_{v}+H_{g}$ (1)

$H_{v} \left ( \sigma \right ) =J_{v} \sum _{(x,y)\in L_{v}}\delta _{\sigma (x_{v})\sigma (y_{v}),} H_{g}(\sigma )=J_{g} \sum _{(x,y)\in L_{g}}\delta _{\sigma (x_{g})\sigma (y_{g})},$ (2)

Index $v$ refers to the liquid phase, index $g$ refers to the gas phase, $J_{v}$ $J_{g}$ are the coupling constants, $\delta _{ij}$ - the Kronecker delta, $L_{v},L_{g}$ characterize the geometry of the sets.

It is shown that the Gibbs energy can change abruptly from a limited value to infinity, which indicates the possibility of a phase transition and, accordingly, the breaking of bonds.

This work is supported by the Russian Scientific Foundation (grant No. 18-11-00247).

References

1. Shao Fu Wang, Da-Zhuan Xu. The dynamic analysis of a chaotic system .– Beijing, China: Advances in Mechanical Engineering, 2017. Vol. 9(3) 1–6.

2. Wang Z, Zhou L, Chan Z. Local bifurcation analysis and topological horseshoe 4D of a hyper-chaotic system. Nonlinear Dynamics 2016; 83: 2055-2066.

3. Farrukh Mukhamedov, Otabek Khakimov. Phase transition and chaos: P -adic Potts model on a Cayley tree. – Madrid, Italy: Chaos, Solitons and Fractals 87 (2016) 190–196.

4. Rozikov UA , Khakimov R . Periodic Gibbs measures for the Potts model on the Cayley tree. – Moscow, Russia: Theor Math Phys 2013 b;175:699–709

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