We provide sufficient conditions for the existence of periodic solutions and their stability of the 4-order differential equations <Equation ID="Equ25"> <EquationSource Format="TEX">$ \ddddot{u} +(a_{1}u+a_{0}) \dddot{u}+(b_{1}u+1+b_{0})\ddot{u}+(c_{1}u+a_{0})\dot{u}+c_{2}u^{2}+b_{0}u=\varepsilon ^{2}F(t,u,\dot{u}, \ddot{u}, \dddot{u},\varepsilon ), $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mover accent="true"> <mi>u</mi> <mo>⃜</mo> </mover> <mo>+</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mi>u</mi> <mo>+</mo> <msub> <mi>a</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mover accent="true"> <mi>u</mi> <mo>⃛</mo> </mover> <mo>+</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mn>1</mn> </msub> <mi>u</mi> <mo>+</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>b</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mover accent="true"> <mi>u</mi> <mo>¨</mo> </mover> <mo>+</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> <mi>u</mi> <mo>+</mo> <msub> <mi>a</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mover accent="true"> <mi>u</mi> <mo>˙</mo> </mover> <mo>+</mo> <msub> <mi>c</mi> <mn>2</mn> </msub> <msup> <mi>u</mi> <mn>2</mn> </msup> <mo>+</mo> <msub> <mi>b</mi> <mn>0</mn> </msub> <mi>u</mi> <mo>=</mo> <msup> <mi>ε</mi> <mn>2</mn> </msup> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mover accent="true"> <mi>u</mi> <mo>˙</mo> </mover> <mo>,</mo> <mover accent="true"> <mi>u</mi> <mo>¨</mo> </mover> <mo>,</mo> <mover accent="true"> <mi>u</mi> <mo>⃛</mo> </mover> <mo>,</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$a_{0},a_{1,}b_{0,}b_{1},c_{1},c_{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow> <mn>1</mn> <mo>,</mo> </mrow> </msub> <msub> <mi>b</mi> <mrow> <mn>0</mn> <mo>,</mo> </mrow> </msub> <msub> <mi>b</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>c</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> are real parameters, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation> is a small parameter and F is a <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$C^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$2\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation>-periodic function in the variable t. Moreover, we provide some applications.

Periodic orbits and their stability for a class of non-autonomous 4-order differential equations

期刊论文
发表日期：

2026年6月16日

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Amina Feddaoui,
Jaume Llibre,
Amar Makhlouf

摘要

We provide sufficient conditions for the existence of periodic solutions and their stability of the 4-order differential equations $ \ddddot{u} +(a_{1}u+a_{0}) \dddot{u}+(b_{1}u+1+b_{0})\ddot{u}+(c_{1}u+a_{0})\dot{u}+c_{2}u^{2}+b_{0}u=\varepsilon ^{2}F(t,u,\dot{u}, \ddot{u}, \dddot{u},\varepsilon ), $ $\overset{⃜}{u} + (a_{1} u + a_{0}) \overset{⃛}{u} + (b_{1} u + 1 + b_{0}) \ddot{u} + (c_{1} u + a_{0}) \dot{u} + c_{2} u^{2} + b_{0} u = ε^{2} F (t, u, \dot{u}, \ddot{u}, \overset{⃛}{u}, ε),$ where $a_{0},a_{1,}b_{0,}b_{1},c_{1},c_{2}$ $a_{0}, a_{1,} b_{0,} b_{1}, c_{1}, c_{2}$ are real parameters, $\varepsilon $ $ε$ is a small parameter and F is a $C^2$ $C^{2}$ $2\pi $ $2 π$ -periodic function in the variable t. Moreover, we provide some applications.