<p>Transcritical and saddle-node bifurcation behaviors of nonlinear ion-acoustic wave (IAW) characteristics are examined in a five-component plasma system in the Venusian upper ionosphere at an altitude of 1000-2000 km. The plasma model comprises <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mo>+</mo> </msup> </math></EquationSource> <EquationSource Format="TEX">$H^{+}$</EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <msup> <mi>O</mi> <mo>+</mo> </msup> </math></EquationSource> <EquationSource Format="TEX">$O^{+}$</EquationSource> </InlineEquation> ions, solar wind protons (sp), along with Maxwellian distributed planetary electrons (<InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mi>e</mi> </math></EquationSource> <EquationSource Format="TEX">$e$</EquationSource> </InlineEquation>) and solar wind electrons (se). The governing model equations are transformed into an ordinary differential equation (ODE) using a non-perturbative approach. A planar dynamical system is then derived by applying the theory of phase plane analysis. Various possible phase portraits are constructed to explore the associated nonlinear wave phenomena. The effects of plasma parameters such as <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> <EquationSource Format="TEX">$\rho $</EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> <EquationSource Format="TEX">$\omega $</EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="MATHML"><math> <mi>ζ</mi> </math></EquationSource> <EquationSource Format="TEX">$\zeta $</EquationSource> </InlineEquation> (unperturbed number density ratios), <InlineEquation ID="IEq7"> <EquationSource Format="MATHML"><math> <msub> <mi>σ</mi> <mrow> <mi>s</mi> <mi>e</mi> </mrow> </msub> </math></EquationSource> <EquationSource Format="TEX">$\sigma _{se}$</EquationSource> </InlineEquation> (the temperature ratio), and <InlineEquation ID="IEq8"> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> <EquationSource Format="TEX">$\lambda $</EquationSource> </InlineEquation> (the travelling wave speed) on nonlinear wave features are systematically investigated. Specifically, the influences of these parameters on the features of ion-acoustic solitary waves (IASWs) and nonlinear periodic ion-acoustic waves (NPIAWs) are analyzed. It is observed that a decrease in <InlineEquation ID="IEq9"> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> <EquationSource Format="TEX">$\rho $</EquationSource> </InlineEquation> enhances amplitude with minimal change in width of the IASW. It is also observed that increasing <InlineEquation ID="IEq10"> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> <EquationSource Format="TEX">$\omega $</EquationSource> </InlineEquation>, <InlineEquation ID="IEq11"> <EquationSource Format="MATHML"><math> <mi>ζ</mi> </math></EquationSource> <EquationSource Format="TEX">$\zeta $</EquationSource> </InlineEquation>, and <InlineEquation ID="IEq12"> <EquationSource Format="MATHML"><math> <msub> <mi>σ</mi> <mrow> <mi>s</mi> <mi>e</mi> </mrow> </msub> </math></EquationSource> <EquationSource Format="TEX">$\sigma _{se}$</EquationSource> </InlineEquation> results in a decrease in amplitude while having a negligible effect on width of the IASW. Additionally, it is observed that a decrease in <InlineEquation ID="IEq13"> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> <EquationSource Format="TEX">$\lambda $</EquationSource> </InlineEquation> leads to a reduction in amplitude along with a noticeable broadening of the IASW. It is also observed that increasing <InlineEquation ID="IEq14"> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> <EquationSource Format="TEX">$\rho $</EquationSource> </InlineEquation>, <InlineEquation ID="IEq15"> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> <EquationSource Format="TEX">$\omega $</EquationSource> </InlineEquation>, <InlineEquation ID="IEq16"> <EquationSource Format="MATHML"><math> <mi>ζ</mi> </math></EquationSource> <EquationSource Format="TEX">$\zeta $</EquationSource> </InlineEquation>, and <InlineEquation ID="IEq17"> <EquationSource Format="MATHML"><math> <msub> <mi>σ</mi> <mrow> <mi>s</mi> <mi>e</mi> </mrow> </msub> </math></EquationSource> <EquationSource Format="TEX">$\sigma _{se}$</EquationSource> </InlineEquation> reduces amplitude and slightly narrows the width of the NPIAW. It is further observed that an increase in <InlineEquation ID="IEq18"> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> <EquationSource Format="TEX">$\lambda $</EquationSource> </InlineEquation> significantly increases both amplitude and width of the NPIAW. Vector field diagrams are generated for various values of the control parameter <InlineEquation ID="IEq19"> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> <EquationSource Format="TEX">$\lambda $</EquationSource> </InlineEquation>, and the corresponding bifurcation curve is plotted. Based on the variation in <InlineEquation ID="IEq20"> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> <EquationSource Format="TEX">$\lambda $</EquationSource> </InlineEquation>, a combination of transcritical and saddle-node bifurcations is observed in the Venusian upper ionosphere. These findings contribute to a deeper understanding of the bifurcation features of IAWs in the Venusian ionospheric environment.</p>

Transcritical and saddle-node bifurcations of nonlinear ion-acoustic waves in the upper ionosphere of Venus

期刊论文
发表日期：

2025年12月17日

查看全文

Kusum Chettri,
Jharna Tamang,
Prasanta Chatterjee,
Asit Saha

摘要

Transcritical and saddle-node bifurcation behaviors of nonlinear ion-acoustic wave (IAW) characteristics are examined in a five-component plasma system in the Venusian upper ionosphere at an altitude of 1000-2000 km. The plasma model comprises $H^{+}$ $H^{+}$ , $O^{+}$ $O^{+}$ ions, solar wind protons (sp), along with Maxwellian distributed planetary electrons ( $e$ $e$ ) and solar wind electrons (se). The governing model equations are transformed into an ordinary differential equation (ODE) using a non-perturbative approach. A planar dynamical system is then derived by applying the theory of phase plane analysis. Various possible phase portraits are constructed to explore the associated nonlinear wave phenomena. The effects of plasma parameters such as $ρ$ $\rho $ , $ω$ $\omega $ , $ζ$ $\zeta $ (unperturbed number density ratios), $σ_{s e}$ $\sigma _{se}$ (the temperature ratio), and $λ$ $\lambda $ (the travelling wave speed) on nonlinear wave features are systematically investigated. Specifically, the influences of these parameters on the features of ion-acoustic solitary waves (IASWs) and nonlinear periodic ion-acoustic waves (NPIAWs) are analyzed. It is observed that a decrease in $ρ$ $\rho $ enhances amplitude with minimal change in width of the IASW. It is also observed that increasing $ω$ $\omega $ , $ζ$ $\zeta $ , and $σ_{s e}$ $\sigma _{se}$ results in a decrease in amplitude while having a negligible effect on width of the IASW. Additionally, it is observed that a decrease in $λ$ $\lambda $ leads to a reduction in amplitude along with a noticeable broadening of the IASW. It is also observed that increasing $ρ$ $\rho $ , $ω$ $\omega $ , $ζ$ $\zeta $ , and $σ_{s e}$ $\sigma _{se}$ reduces amplitude and slightly narrows the width of the NPIAW. It is further observed that an increase in $λ$ $\lambda $ significantly increases both amplitude and width of the NPIAW. Vector field diagrams are generated for various values of the control parameter $λ$ $\lambda $ , and the corresponding bifurcation curve is plotted. Based on the variation in $λ$ $\lambda $ , a combination of transcritical and saddle-node bifurcations is observed in the Venusian upper ionosphere. These findings contribute to a deeper understanding of the bifurcation features of IAWs in the Venusian ionospheric environment.