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Display information for equation id:math.1354.116 on revision:1354
* Page found: Dynamische Systeme und deterministisches Chaos (eq math.1354.116)
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Hash: 4a1112a7942d15f449d44502efd9f95e
TeX (original user input):
\begin{align}
& \bar{\varpi }{{*}^{(1)}}:{{\varpi }_{1}}=\varpi ,{{\varpi }_{2}}=0,{{\varpi }_{3}}=0 \\
& 0=\det (A-\lambda 1)=\left| \begin{matrix}
-\lambda & 0 & 0 \\
0 & -\lambda & {{k}_{2}}{{\omega }_{{}}} \\
0 & -{{k}_{3}}\omega & -\lambda \\
\end{matrix} \right|=-\lambda \left( {{\lambda }^{2}}+{{k}_{2}}{{k}_{3}}{{\omega }^{2}} \right) \\
& \Rightarrow {{\lambda }_{1}}^{(1)}=0,{{\lambda }_{2/3}}^{(1)}=\pm i\omega \sqrt{{{k}_{2}}{{k}_{3}}} \\
\end{align}
TeX (checked):
{\begin{aligned}&{\bar {\varpi }}{{*}^{(1)}}:{{\varpi }_{1}}=\varpi ,{{\varpi }_{2}}=0,{{\varpi }_{3}}=0\\&0=\det(A-\lambda 1)=\left|{\begin{matrix}-\lambda &0&0\\0&-\lambda &{{k}_{2}}{{\omega }_{}}\\0&-{{k}_{3}}\omega &-\lambda \\\end{matrix}}\right|=-\lambda \left({{\lambda }^{2}}+{{k}_{2}}{{k}_{3}}{{\omega }^{2}}\right)\\&\Rightarrow {{\lambda }_{1}}^{(1)}=0,{{\lambda }_{2/3}}^{(1)}=\pm i\omega {\sqrt {{{k}_{2}}{{k}_{3}}}}\\\end{aligned}}
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<math class="mwe-math-element" xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>ϖ</mi><mo>¯</mo></mover></mrow></mrow><msup><mo>*</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mrow></msup><mi>:</mi><msub><mi>ϖ</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mo>=</mo><mi>ϖ</mi><mo>,</mo><msub><mi>ϖ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><mo>=</mo><mn>0</mn><mo>,</mo><msub><mi>ϖ</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mo>=</mo><mn>0</mn></mtd></mtr><mtr><mtd></mtd><mtd><mn>0</mn><mo>=</mo><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>A</mi><mo>−</mo><mi>λ</mi><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><mrow data-mjx-texclass="ORD"><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><mo>−</mo><mi>λ</mi></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>−</mo><mi>λ</mi></mtd><mtd><msub><mi>k</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><msub><mi>ω</mi><mrow data-mjx-texclass="ORD"></mrow></msub></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>−</mo><msub><mi>k</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><mi>ω</mi></mtd><mtd><mo>−</mo><mi>λ</mi></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow><mo data-mjx-texclass="CLOSE">|</mo></mrow><mo>=</mo><mo>−</mo><mi>λ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>λ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo>+</mo><msub><mi>k</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><msub><mi>k</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub><msup><mi>ω</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo><msup><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mrow></msup><mo>=</mo><mn>0</mn><mo>,</mo><msup><msub><mi>λ</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mo>/</mo><mn>3</mn></mrow></mrow></msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mrow></msup><mo>=</mo><mo>±</mo><mi>i</mi><mi>ω</mi><mrow data-mjx-texclass="ORD"><msqrt><mrow data-mjx-texclass="ORD"><msub><mi>k</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msub><msub><mi>k</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msub></mrow></msqrt></mrow></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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