On Invariant Analysis, Doubly Periodic Solutions, Series Solutions, Dynamical Behavior of the Transmission line Circuit of a Travelling Wave Parametric Amplifier
Gupta R.K. Singla K. Shaikhova G.
August 2025Birkhauser
Qualitative Theory of Dynamical Systems
2025#24Issue 4
This work presents an extensive study on the travelling wave parametric amplifier-superconducting nonlinear asymmetric inductive element (TWPA-SNAIL) transmission line circuit equation. The TWPA-SNAIL circuit combines nonlinear dynamics, parametric amplification, and superconducting phenomena to enable low-noise amplification of microwave signals. The circuit equation is relevant as it captures the complex nonlinear and dispersive physics of TWPAs and superconducting transmission lines, indicating its significance in quantum computing and sensitive signal amplification. The analysis begins with the identification of inherent symmetries presented in the governing equation. The symmetry reduction process is implemented to reduce the number of independent variables in the governing PDE. After that, the power series approach and the F-expansion method are applied to derive invariant solutions, including both series representations and double-periodic forms expressed in terms of Jacobi and Weierstrass elliptic functions, that provide valuable physics insight into the dynamics of TWPA circuits with nonlinear asymmetric inductive elements. These solutions involve both positive and negative powers of the elliptic functions that are very rare in the literature. This work also investigates the complex dynamics of the governing equation resulting from bifurcation and chaotic analysis. Phase diagrams are constructed with the help of Maple software by systematically changing the values of the crucial parameters, and thus, critical transitions of system behavior are identified. Using 2D, 3D phase space trajectories, and time series plots, the chaotic and quasiperiodic nature of the system is further analyzed. The novelty of the presented work is reflected in the obtained symmetries, the derivation of diverse analytic solutions, and the investigation of its dynamical behavior.
Bifurcation , Chaotic behavior , Jacobi elliptic functions , Series solutions , SNAIL , TWPA , Weierstrass functions
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Department of Mathematics, Central University of Haryana, Mahendergarh, India
L.N. Gumilyov Eurasian National University, Astana, Kazakhstan
Department of Mathematics
L.N. Gumilyov Eurasian National University
10 лет помогаем публиковать статьи Международный издатель
Книга Публикация научной статьи Волощук 2026 Book Publication of a scientific article 2026