On the algebraic difference equations <it>u</it><sub><it>n</it>+2</sub><it>u</it><sub><it>n</it></sub> = <it>ψ</it>(<it>u</it><sub><it>n</it>+1</sub>) in <inline-formula><graphic file="1687-1847-2005-948567-i1.gif"/></inline-formula>, related to a family of elliptic quartics in the plane
<p/> <p>We continue the study of algebraic difference equations of the type <it>u</it><sub><it>n</it>+2</sub><it>u</it><sub><it>n</it></sub> = <it>ψ</it>(<it>u</it><sub><it&g...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2005-01-01
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| Series: | Advances in Difference Equations |
| Online Access: | http://www.advancesindifferenceequations.com/content/2005/948567 |
| Summary: | <p/> <p>We continue the study of algebraic difference equations of the type <it>u</it><sub><it>n</it>+2</sub><it>u</it><sub><it>n</it></sub> = <it>ψ</it>(<it>u</it><sub><it>n</it>+1</sub>), which started in a previous paper. Here we study the case where the algebraic curves related to the equations are quartics <it>Q</it>(<it>K</it>) of the plane. We prove, as in "on some algebraic difference equations <it>u</it><sub><it>n</it>+2</sub><it>u</it><sub><it>n</it></sub> = <it>ψ</it>(<it>u</it><sub><it>n</it>+1</sub>) in <inline-formula><graphic file="1687-1847-2005-948567-i2.gif"/></inline-formula>, related to families of conics or cubics: generalization of the Lyness' sequences" (2004), that the solutions <it>M</it><sub><it>n</it></sub> = (<it>u</it><sub><it>n</it>+1</sub>, <it>u</it><sub><it>n</it></sub>) are persistent and bounded, move on the positive component <it>Q</it><sup>0</sup>(<it>K</it>) of the quartic <it>Q</it>(<it>K</it>) which passes through <it>M</it><sub>0</sub>, and diverge if <it>M</it><sub>0</sub> is not the equilibrium, which is locally stable. In fact, we study the dynamical system <it>F</it>(<it>x</it>, <it>y</it>) = ((<it>a</it> + <it>bx</it> + <it>cx</it><sup>2</sup>)/<it>y</it>(<it>c</it> + <it>dx</it> + <it>x</it><sup>2</sup>), <it>x</it>), (<it>a</it>, <it>b</it>, <it>c</it>, <it>d</it>) ∈ ℝ<sup>+4</sup>, <it>a</it> + <it>b</it> > 0, <it>b</it> + <it>c</it> + <it>d</it> > 0, in <inline-formula><graphic file="1687-1847-2005-948567-i3.gif"/></inline-formula>, and show that its restriction to <it>Q</it><sup>0</sup> (<it>K</it>) is conjugated to a rotation on the circle. We give the possible periods of solutions, and study their global behavior, such as the density of initial periodic points, the density of trajectories in some curves, and a form of sensitivity to initial conditions. We prove a dichotomy between a form of pointwise chaotic behavior and the existence of a common minimal period to all nonconstant orbits of <it>F</it>.</p> |
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| ISSN: | 1687-1839 1687-1847 |