zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Sensitivity tools vs. Poincaré sections. (English) Zbl 1092.37531
Summary: We introduce a modification of the fast Lyapunov indicator (FLI) denominated $\text{OFLI}_{\text{TT}}^2$ indicator that may provide a global picture of the evolution of a dynamical system. Therefore, it gives an alternative or a complement to the pictures given by the classical Poincaré sections and, besides, it may be used for any dimension. We present several examples comparing with the Poincaré sections in two classical problems, the Hénon-Heiles and the extensible-pendulum problems. Besides, we show the application to Hamiltonians of three degrees of freedom as an isotropic harmonic oscillator in three dimensions perturbed by a cubic potential and non-Hamiltonian problems as a four-dimensional chaotic system. Finally, a numerical method especially designed for its computation is presented in the appendix.

37N05Dynamical systems in classical and celestial mechanics
37D45Strange attractors, chaotic dynamics
70K55Transition to stochasticity (chaotic behavior)
70H14Stability problems (mechanics of particles and systems)
Full Text: DOI
[1] Galias, Z.; Zgliczyński, P.: Computer assisted proof of chaos in the Lorenz equations. Physica D 115, 165-188 (1998) · Zbl 0941.37018
[2] Zgliczyński, P.: Computer assisted proof of chaos in the hénon map and in the Rössler equations. Nonlinearity 10, 243-252 (1997) · Zbl 0907.58048
[3] Tucker, W.: A rigorous ODE solver and Smale’s 14th problem. Found comput math 2, 53-117 (2002) · Zbl 1047.37012
[4] Morbidelli, A.: Modern celestical mechanics. (2002)
[5] Contopoulos, G.; Voglis, N.: Spectra of stretching numbers and helicity angles in dynamical systems. Celes mech dyn astr 64, 1-20 (1996) · Zbl 0880.58021
[6] Contopoulos, G.; Voglis, N.: A fast method for distinguishing between order and chaotic orbits. Astr astrophys 317, 73-82 (1997)
[7] Froeschlé, C.; Lega, E.: Twist angles: a method for distinguing islands, tori and weak chaotic orbits. Comparison with other methods of analysis. Astr astrophys 334, 355-362 (1998)
[8] Laskar, J.: The chaotic motion of the solar system: a numerical estimate of the size of the chaotic zones. Icarus 88, 266-291 (1990)
[9] Laskar, J.: Frequency analysis for multi-dimensional systems. Global dynamics and diffusion. Physica D 67, 257-281 (1993) · Zbl 0783.58027
[10] Cincotta, P. M.; Giordano, C. M.; Simó, C.: Phase space structure of multidimensional systems by means of the mean exponential growth factor of nearby orbits. Physica D 182, 151-178 (2003) · Zbl 1032.37042
[11] Froeschlé, C.; Lega, E.: On the structure of symplectic mappings. The fast Lyapunov indicator: a very sensitivity tool. Celes mech dyn astr 78, 167-195 (2000) · Zbl 0986.37075
[12] Fouchard, M.; Lega, E.; Froeschlé, C.; Froeschlé, C.: On the relationship between fast Lyapunov indicator and periodic orbits for continuous flows. Celes mech dyn astr 83, 205-222 (2002) · Zbl 1083.70011
[13] Skokos, Ch.: Alignment indices: a new, simple method for determining the ordered or chaotic nature of orbits. J phys A: math gen 34, 10029-10043 (2001) · Zbl 1004.37021
[14] Skokos, Ch.; Antonopoulos, Ch.; Bountis, T. C.; Vrahatis, M. N.: Detecting order and chaos in Hamiltonian systems by the SALI method. J phys A: math gen 37, 6269-6284 (2004)
[15] Froeschlé, C.; Guzzo, M.; Lega, E.: Graphical evolution of the Arnold’s web: from order to chaos. Science 289, 2108-2110 (2000)
[16] Guzzo, M.; Lega, E.; Froeschlé, C.: On the numerical detection of the effective stability of chaotic motions in quasi-integrable systems. Physica D 163, l-25 (2002) · Zbl 0986.37076
[17] Hénon, M.; Heiles, C.: The applicability of the third integral of motion: some numerical experiments. Astron J 50, 73-79 (1964)
[18] Carretero-González, R.; Núñez-Yépez, H. N.; Salas-Brito, A. L.: Regular and chaotic behaviour in extensible pendulum. Eur J phys 15, 139-148 (1994)
[19] Hovland, P.; Bischof, C.; Spiegelman, D.; Casella, M.: Efficient derivative codes through automatic differentiation and interface contraction: an application in biostatistics. SIAM J sci comput 18, 1056-1060 (1997) · Zbl 0891.65016
[20] Sandu, A.; Carmichael, G. R.; Potra, F. A.: Sensitivity analysis for atmospheric chemistry models via automatic differentiation. Atmos environ 31, 475-489 (1997)
[21] Barrio R. Performance of the Taylor series method for ODEs/DAEs. Appl Math Comput, in press · Zbl 1067.65063
[22] Barrio R, Blesa F, Lara M. VSVO formulation of Taylor methods for the numerical solution of ODEs. Preprint · Zbl 1085.65056
[23] Simó C. Dynamical systems, numerical experiments and super-computing. Preprints of the Barcelona UB-UPC Dynamical Systems Group, 2003
[24] Hairer, E.; Hairer, M.: Gnicodes--Matlab programs for geometric numerical integration. Frontiers in numerical analysis, 199-240 (2003) · Zbl 1028.65136
[25] Ferrer, S.; Hanßmann, H.; Palacián, J.; Yanguas, P.: On perturbed oscillators in 1-1-1 resonance: the case of axially symmetric cubic potentials. J geom phys 40, 320-369 (2002) · Zbl 1037.34031
[26] Hanßmann, H.; Van Der Meer, J. C.: On the Hamiltonian Hopf bifurcations in the 3D hénon-Heiles family. J dyn diff eqns 14, 675-695 (2002) · Zbl 1035.37038
[27] Chandré, C.; Wiggins, S.; Uzer, T.: Time-frequency analysis of chaotic systems. Physica D 181, 171-196 (2003) · Zbl 1027.70027
[28] Qi, G.; Du, S.; Chen, G.; Chen, Z.; Yuan, Z.: On a four-dimensional chaotic system. Chaos, solitons & fractals 23, 1671-1682 (2005) · Zbl 1071.37025
[29] Guckenheimer, J.; Meloon, B.: Computing periodic orbits and their bifurcations with automatic differentiation. SIAM J sci comput 22, 951-985 (2000) · Zbl 0976.65111
[30] Corliss, G. F.; Chang, Y. F.: Solving ordinary differential equations using Taylor series. ACM trans math software 8, 114-144 (1982) · Zbl 0503.65046
[31] Griewank, A.: Evaluating derivatives. (2000) · Zbl 0958.65028
[32] Simó, C.; Valls, C.: A formal approximation of the splitting of separatrices in the classical Arnold’s example of diffusion with two equal parameters. Nonlinearity 14, 1707-1760 (2001) · Zbl 1001.37051
[33] Barrio R. Sensitivity analysis of ODEs/DAEs using the Taylor series method. Preprint · Zbl 1108.65077
[34] Eberhard, P.; Bischof, C.: Automatic differentiation of numerical integration algorithms. Math comput 68, 717-731 (1999) · Zbl 1017.65062