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• W92074458 abstract "Dynamical systems with two degrees of freedom can be reduced to the study of an area-preserving mapping. We consider here, as a model problem, the mapping given by the quadratic equations: <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x 1 equals x cosine alpha minus left-parenthesis y minus x squared right-parenthesis sine alpha> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>x</mml:mi> <mml:mi>cos</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>y</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>sin</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>α<!-- α --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>{x_1} = xcos alpha - left ( {y - {x^2}} right )sin alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=y 1 equals x sine alpha plus left-parenthesis y minus x squared right-parenthesis cosine alpha> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>y</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>x</mml:mi> <mml:mi>sin</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mo>+</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>y</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mspace linebreak=newline /> <mml:mi>cos</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>α<!-- α --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>{y_1} = xsin alpha + left ( {y - {x^2}} right ) cos alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which is shown to be in a sense the simplest nontrivial mapping. Some analytical properties are given, and numerical results are exhibited in Figs. 2 to 14." @default.
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• W92074458 title "Numerical study of quadratic area-preserving mappings" @default.
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• W92074458 doi "https://doi.org/10.1090/qam/253513" @default.
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