Bifurcation analysis and solitary waves in a stochastic FitzHugh-Nagumo model

Jianping Zheng; Muhammad Abuzar; Luai Abdulla Aldoghan; Mohammed Ahmed Alomair; Jianping Zheng; Muhammad Abuzar; Luai Abdulla Aldoghan; Mohammed Ahmed Alomair

doi:10.3934/math.2026374

AIMS Mathematics

2026, Volume 11, Issue 4: 9066-9092. doi: 10.3934/math.2026374

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Bifurcation analysis and solitary waves in a stochastic FitzHugh-Nagumo model

1.
College of Information Engineering, Hefei University of Economics, Hefei 230011, PR China
2.
School of Mathematical Sciences, Guizhou Normal University, Guiyang, Yunyan 550003, PR China
3.
Department of Quantitative Methods, School of Business, King Faisal University, Al-Ahsa 31982, Saudi Arabia

Received: 18 January 2026 Revised: 24 March 2026 Accepted: 26 March 2026 Published: 02 April 2026
MSC : 35C08, 37K40

The stochastic FitzHugh-Nagumo (FHN) equation is a fundamental model for excitable media, widely used to describe neuronal signal transmission and population dynamics under random perturbations. However, analytical studies that simultaneously incorporate stochastic effects, nonlinear wave propagation, and bifurcation behavior are still limited. In this work, we investigated a stochastic FHN model with multiplicative Gaussian noise and developed a unified analytical framework by integrating the unified method with a novel auxiliary equation technique. Using a stochastic transformation and expectation operator, the governing stochastic partial differential equation was reduced to a deterministic traveling-wave ordinary differential equation. This framework enables the systematic derivation of multiple classes of exact solutions, including rational, trigonometric, and hyperbolic soliton structures such as kink, anti-kink, periodic, and singular waves. Compared with existing methods, the proposed approach provides a more general mechanism for constructing diverse solution families within a single analytical structure. Furthermore, the reduced equation was reformulated into a planar dynamical system via a Galilean transformation, allowing bifurcation and phase-portrait analysis. Graphical results clearly illustrate the influence of noise intensity on wave morphology and temporal dynamics. The findings provide new analytical insights into noise-modulated excitation phenomena in stochastic nonlinear systems.
- stochastic model,
- analytical methods,
- solitary waves,
- bifurcation analysis,
- soliton solutions
Citation: Jianping Zheng, Muhammad Abuzar, Luai Abdulla Aldoghan, Mohammed Ahmed Alomair. Bifurcation analysis and solitary waves in a stochastic FitzHugh-Nagumo model[J]. AIMS Mathematics, 2026, 11(4): 9066-9092. doi: 10.3934/math.2026374

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Abstract

The stochastic FitzHugh-Nagumo (FHN) equation is a fundamental model for excitable media, widely used to describe neuronal signal transmission and population dynamics under random perturbations. However, analytical studies that simultaneously incorporate stochastic effects, nonlinear wave propagation, and bifurcation behavior are still limited. In this work, we investigated a stochastic FHN model with multiplicative Gaussian noise and developed a unified analytical framework by integrating the unified method with a novel auxiliary equation technique. Using a stochastic transformation and expectation operator, the governing stochastic partial differential equation was reduced to a deterministic traveling-wave ordinary differential equation. This framework enables the systematic derivation of multiple classes of exact solutions, including rational, trigonometric, and hyperbolic soliton structures such as kink, anti-kink, periodic, and singular waves. Compared with existing methods, the proposed approach provides a more general mechanism for constructing diverse solution families within a single analytical structure. Furthermore, the reduced equation was reformulated into a planar dynamical system via a Galilean transformation, allowing bifurcation and phase-portrait analysis. Graphical results clearly illustrate the influence of noise intensity on wave morphology and temporal dynamics. The findings provide new analytical insights into noise-modulated excitation phenomena in stochastic nonlinear systems.

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