Level Sets Lemmas and Unicity of Critical Point of Invariants, Tools for Local Stability and Topological Properties of Dynamical Systems
DOI:
https://doi.org/10.5644/SJM.08.2.08Keywords:
Difference equations, dynamical systems, local stability, level setsAbstract
We prove that the level curves of some differentiable functions of two variables with unique critical point are diffeomorphic to the circle ${\mathbb T}$, and show how this result can be used in the study of local stability of dynamical systems in dimension 2 with invariant function, without using the Hessian. We extend the results to the level sets of an invariant function of dynamical systems, with a synthesis exposition of examples of improvements of previously studied order $q$ difference equations with invariant. In fine we present some differential tools for the study of the topological nature of invariant level sets in dimension at least three.
2000 Mathematics Subject Classification. 37E75, 39A20, 39A30
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References
E. Barbeau, B. Gelbord and S. Tanny, Periodicities of solutions of the generalized Lyness recursion, J. Difference Equ. Appl., 1 (1995) 291-306.
G. Bastien and M. Rogalski, Global behaviour of the solutions of Lyness' difference equations, J. Difference Equ. Appl., 10 (2004), 977-1003.
G. Bastien and M. Rogalski, On some algebraic difference equations $u_{n+2}u_n=psi(u_{n+1})$ in ${mathbb R^+_*}$, related to families of conics or cubics: generalization of the Lyness' sequences, J. Math. Anal. Appl., 300 (2004), 303-333.
G. Bastien and M. Rogalski, emph{On the algebraic difference equation $u_{n+2}u_n=psi(u_{n+1})$ in ${mathbb R}_*^+$ related to a family of elliptic quartics in the plane, Adv. Difference Equ., 3 (2005), 227-261.
G. Bastien and M. Rogalski, On the algebraic difference equations $u_{n+2}+u_n=psi(u_{n+1})$ in ${mathbb R}$, related to a
family of elliptic quartics in the plane, J. Math. Anal. Appl., 326 (2007), 822-844.
G. Bastien and M. Rogalski, Results and conjectures about order $q$ Lyness' difference equation $u_{n+q},u_n=a+u_{n+q-1}+{dots}+u_{n+1}$ in ${mathbb R}_*^+$, with a particular study of the case $q=3$, Adv. Difference Equ., 2009, article ID 134749.
G. Bastien and M. Rogalski, On the topological nature of the stable sets associated to the second invariant of the order $q$ standard Lyness' equation, Sarajevo J. Math., 7 (19) (2011), 1-8.
S. Cairns, An elementary proof of the Jordan-Schönflies Theorem, Proc. Amer. Math. Soc., 2 (6) (1951), 860-867.
A. Cima, A. Gasull and V. Manosa, On two and three periodic Lyness difference equations, J. Difference Equ. Appl., 18 (5) (2012), 849-864.
A. Cima, A. Gasull and V. Manosa, Dynamics of the third order Lyness' difference equation, J. Difference Equ. Appl., 13 (10) (2007), 855-884.
J. Esch and T. D. Rogers, The screensaver map: Dynamics on elliptic curves arising from polygonal folding, Discrete Comput. Geom., 25 (2001), 477-502.
A. Gasull, V. Manosa and X. Xarles, Rational periodic sequences for the Lyness recurrence, Discrete Cont. Dyn. Syst. Ser. A, 32 (2) (2012), 587-604.
E. J. Janowski, M. R. S. Kulenović and Z. Nurkanović, Stability of the $k$-th order Lyess' equation with a period-$k$ coefficient, Int. J. Bifurcations Chaos, 17 (1) (2007), 143-152.
M. R. S. Kulenović, Invariants and related Liapunov functions for difference equations, Appl. Math. Lett., 13 (2000), 1-8.
J. Lafontaine, Introduction aux variétées différentiables, Presses Universitaires de Grenoble, Grenoble (1996).
W. Rudin, Real and Complex Analysis, second edition, McGraw-Hill, 1974.
E. C. Zeeman, Geometric unfolding of a difference equation, unpublished paper: preprint, Hertford College, Oxford, 1996.
