Stability of the systems of equations

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Problem: Investigate the stability of the system in at zeroresting point. \[ \left\{\begin{array}{c} \frac{d y}{d t}=-\sin y-x e^{-x} \\ \frac{d x}{d t}=\sin (-x)+\sin (-y) \end{array} .\right. \]

8.4.1.1 Stability of the systems of equations

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Problem: Investigate the stability of the system at zeroresting point. \[ \left\{\begin{array}{c} \frac{d y}{d t}=\tan ^{-1} \frac{A x}{\sqrt{1-(A x)^{2}}}+B \sin \left(\frac{\pi}{2}-y\right)-(B-y) e^{-D y} \\ \frac{d x}{d t}=C \cdot \cosh y+D \cdot \sinh (A x)-C \end{array}\right. \] \begin{tabular}{|l|l|l|l|} \hline\( A \) & \( B \) & \( C \) & \( D \) \\ \hline- & + & - & + \\ \hline \end{tabular}

8.4.1.2 Stability of the systems of equations

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Problem: Investigate the stability of the system at the zeroresting point. \[ \left\{\begin{array}{c} \frac{d y}{d t}=\tan ^{-1} \frac{A x}{\sqrt{1-(A x)^{2}}}+B \sin \left(\frac{\pi}{2}-y\right)-(B-y) e^{-D y} \\ \frac{d x}{d t}=C \cdot \cosh y+D \cdot \sinh (A x)-C \end{array}\right. \] \begin{tabular}{|l|l|l|l|} \hline- & - & + & + \\ \hline \end{tabular}

8.4.1.4 Stability of the systems of equations

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Problem: Find the resting points (equilibrium) of the system and investigate the stability of the system at these points. \[ \left\{\begin{array}{c} \frac{d y}{d t}=A \sin y+C x e^{-\frac{B}{D} x} \\ \frac{d x}{d t}=B \sinh (A x)+D \tan (C y) \end{array}\right. \] \begin{tabular}{|c|c|c|c|} \hline\( A \) & \( B \) & \( C \) & \( D \) \\ \hline+ & + & - & - \\ \hline \end{tabular}

8.4.1.3 Stability of the systems of equations

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Problem: Find the resting points (equilibrium) of the system and investigate the stability of the system at these points: \[ \left\{\begin{array}{l} \frac{d y}{d t}=\operatorname{sh}(A x)+\frac{\pi}{2} \operatorname{sh}(B y) \\ \frac{d x}{d t}=\cos \left(\frac{\pi}{2} x\right)+c e^{y}-D \sec y \end{array}\right. \] \begin{tabular}{|l|l|l|l|} \( A \) & \( B \) & \( C \) & \( D \) \\ \hline- & - & - & - \end{tabular}

8.4.1.5 Stability of the systems of equations

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Problem: Find the resting point (equilibrium) of the system and investigate the stability of the system at this points: \[ \left\{\begin{array}{l} \frac{d y}{d t}=C \cdot \arcsin \frac{B x}{\sqrt{1+(B x)^{2}}}+A \tan (D y) \\ \frac{d x}{d t}=A \operatorname{th}(B y)+D x e^{-\frac{C}{D} x} \end{array}\right. \] \begin{tabular}{|l|l|l|l|} \hline\( A \) & \( B \) & \( C \) & \( D \) \\ \hline+ & + & - & - \\ \hline \end{tabular}

8.4.1.6 Stability of the systems of equations

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Problem: Find the resting points (equilibrium) of the system and investigate the stability of the system at these points: \[ \left\{\begin{array}{l} \frac{d y}{d t}=C \cdot \arcsin \frac{B x}{\sqrt{1+(B x)^{2}}}+A \tan (D y) \\ \frac{d x}{d t}=A \cdot \operatorname{th}(B y)+D \cdot x e^{-\frac{C}{D} x} \end{array}\right. \] \begin{tabular}{|l|l|l|l|} \hline\( A \) & \( B \) & \( C \) & \( D \) \\ \hline- & + & - & + \\ \hline \end{tabular}

8.4.1.7 Stability of the systems of equations

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Problem: Determine at what values of the parameter \( \alpha \) the resting point of the system is stable: \[ x^{\prime}=-3 x+\alpha y, \quad y^{\prime}=-\alpha x+y . \]

8.4.1.8 Stability of the systems of equations

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Problem: Investigate the stability of the solution of the following system of equations: \[ x^{\prime}=x+y, \quad y^{\prime}=x-y, \quad x(0)=y(0)=0 . \]

8.4.1.9 Stability of the systems of equations

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Problem: Investigate the stability of the solution of the following system of equations: \[ x^{\prime}=-2 x-3 y, y^{\prime}=x+y, x(0)=y(0)=0 . \]

8.4.1.10 Stability of the systems of equations

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Problem: Determine the character of the resting point of the following system: \[ x^{\prime}=x+2 y, y^{\prime}=-3 x+y \text {. } \]

8.4.1.11 Stability of the systems of equations

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Problem: Determine the character of the resting point of the following system: \[ x^{\prime}=-2 x+\frac{1}{3} y, \quad y^{\prime}=-2 x+\frac{1}{2} y . \]

8.4.1.12 Stability of the systems of equations

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Problem: Determine the character of the resting point of the following system: \[ x^{\prime}=-y, \quad y^{\prime}=x-2 y \text {. } \]

8.4.1.13 Stability of the systems of equations

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