Bifurcation and chaos in N-type and S-type muscular blood vessel models

Wenxin Zhang; Lijun Pei; Wenxin Zhang; Lijun Pei

doi:10.3934/era.2025057

Electronic Research Archive

2025, Volume 33, Issue 3: 1285-1305. doi: 10.3934/era.2025057

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Bifurcation and chaos in N-type and S-type muscular blood vessel models

Wenxin Zhang ,
Lijun Pei ^,

School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China

Received: 06 November 2024 Revised: 31 January 2025 Accepted: 07 February 2025 Published: 04 March 2025

In this paper, we have constructed two muscular blood vessel systems influenced by external disturbances by using the non-autonomous delayed differential equation (DDE). As an important physiological structure of the human body, muscle blood vessels participate in many activities such as blood flow. Many diseases are associated with abnormal dynamics of muscle blood vessels. From a mathematical point of view, vasospasm is caused by a chaotic state of blood vessels. Vasospasm is the manifestation of this disease in the vascular system, which can cause blood vessel blockage and even harm human health when it is serious. We conducted the dynamical analysis of the S-type and N-type muscular blood vessel systems by utilizing bifurcation diagrams, time histories, and pseudo-phase portraits, and investigated the effects of different parameters on these systems. Specifically, when parameters change, rich dynamical phenomena occur, such as equilibria, periodic solutions, quasi-periodic solutions, Hopf bifurcation, and chaos, as well as the route of period-doubling bifurcation to chaos. Meanwhile, we analyzed approximate analytical solutions by using the method of multiple scales (MMS) and determined the stability of steady-state solutions through the Routh-Hurwitz criterion. The results indicate that the MMS can deduce better analytical results for some non-autonomous DDEs.
- muscular blood vessel systems,
- chaos,
- MMS,
- non-autonomous DDE,
- Hopf bifurcation
Citation: Wenxin Zhang, Lijun Pei. Bifurcation and chaos in N-type and S-type muscular blood vessel models[J]. Electronic Research Archive, 2025, 33(3): 1285-1305. doi: 10.3934/era.2025057

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Abstract

In this paper, we have constructed two muscular blood vessel systems influenced by external disturbances by using the non-autonomous delayed differential equation (DDE). As an important physiological structure of the human body, muscle blood vessels participate in many activities such as blood flow. Many diseases are associated with abnormal dynamics of muscle blood vessels. From a mathematical point of view, vasospasm is caused by a chaotic state of blood vessels. Vasospasm is the manifestation of this disease in the vascular system, which can cause blood vessel blockage and even harm human health when it is serious. We conducted the dynamical analysis of the S-type and N-type muscular blood vessel systems by utilizing bifurcation diagrams, time histories, and pseudo-phase portraits, and investigated the effects of different parameters on these systems. Specifically, when parameters change, rich dynamical phenomena occur, such as equilibria, periodic solutions, quasi-periodic solutions, Hopf bifurcation, and chaos, as well as the route of period-doubling bifurcation to chaos. Meanwhile, we analyzed approximate analytical solutions by using the method of multiple scales (MMS) and determined the stability of steady-state solutions through the Routh-Hurwitz criterion. The results indicate that the MMS can deduce better analytical results for some non-autonomous DDEs.

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