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that interest them. Pomeau and Ibanez combine their numerical calculations with the results of mathematical analysis, based on the use of
Poincaré sections. Stretching, folding, sensitivity to initial conditions are naturally brought in this context in connection with the Lorenz attractor. If the analysis is ultimately very mathematical, Pomeau and Ibanez follow, in a sense, a physicist approach, experimenting with the Lorenz system numerically.
533:
20:
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959:
41:
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An approximate
Koopman mode of the Hénon map found with a basis of 50x50 Gaussians evenly spaced over the domain. The standard deviation of the Gaussians is 3/45 and a 100x100 grid of points was used to fit the mode. This mode has eigenvalue 0.998, and it is the closest to 1. Notably, the dark blue
1723:
The second path suggested by Pomeau and Ibanez is the idea of realizing dynamical systems even simpler than that of Lorenz, but having similar characteristics, and which would make it possible to prove more clearly "evidences" brought to light by numerical calculations. Since the reasoning is based
1711:
who performs a series of numerical calculations with J.L. Ibanez. The analysis produces a kind of complement to the work of Ruelle (and
Lanford) presented in 1975. It is the Lorenz attractor, that is to say, the one corresponding to the original differential equations, and its geometric structure
1715:
Two openings are brought specifically by these experiences. They make it possible to highlight a singular behavior of the Lorenz system: there is a transition, characterized by a critical value of the parameters of the system, for which the system switches from a strange attractor position to a
1778:
1724:
on
Poincaré's section, he proposes to produce an application of the plane in itself, rather than a differential equation, imitating the behavior of Lorenz and its strange attractor. He builds one in an ad hoc manner which allows him to better base his reasoning.
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This point is unstable. Points close to this fixed point and along the slope 1.924 will approach the fixed point and points along the slope -0.156 will move away from the fixed point. These slopes arise from the linearizations of the
230:
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964:
1727:
In
January 1976, Pomeau presented his work during a seminar given at the Côte d'Azur Observatory, attended by Michel Hénon. Michel Hénon uses Pomeau’s suggestion to obtain a simple system with a strange attractor.
427:
2061:{\displaystyle \mathbf {s} (n+1)={\begin{bmatrix}s_{1}(n+1)\\s_{2}(n+1)\\s_{3}(n+1)\end{bmatrix}}={\begin{bmatrix}-\alpha s_{1}^{2}(n)+s_{3}(n)+1\\-\beta s_{1}(n)\\\beta s_{1}(n)+s_{2}(n)\end{bmatrix}}}
1332:
536:
Variation of 'b' showing the
Bifurcation diagram. The boomerang shape is further drawn in bold at the top. Initial coordinates for each cross-section is (0, -0.2). Achieved using Python and Matplotlib.
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2639:. Reprinted in: Chaos and Fractals, A Computer Graphical Journey: Ten Year Compilation of Advanced Research (Ed. C. A. Pickover). Amsterdam, Netherlands: Elsevier, pp. 69–71, 1998
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it is -2.3 to 1.0. All planar cross-sections that in each image of the video are empty indicates that for those cross-sections, the points diverged to infinity and were not plotted.
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Cvitanović et al. have shown how the structure of the Hénon strange attractor can be understood in terms of unstable periodic orbits within the attractor.
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configuration in a limit cycle. The importance will be revealed by Pomeau himself (and a collaborator, Paul
Manneville) through the "scenario" of
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3083:
378:
1744:
of this operator cannot be expressed in any nice form. Instead one must compute them numerically. These modes can give insight into the
2542:
Predrag
Cvitanović; Gemunu Gunaratne; Itamar Procaccia (1988). "Topological and metric properties of Hénon-type strange attractors".
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is produced. A Bifurcation diagram that is folded like a taco. Hence its boomerang shape when viewed in 2D from the top.
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The Hénon map may be decomposed into the composition of three functions acting on the domain one after the other.
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In practice the starting point (X,X) will follow a 4-point loop in two dimensions passing through all quadrants.
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300:. For the classical map, an initial point of the plane will either approach a set of points known as the Hénon
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1007:(i.e. one cube of space) at a time representing three axes, then moving along the fourth axis as time passes.
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The Hénon map maps two points into themselves: these are the invariant points. For the classical values of
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of 1.21 ± 0.01 or 1.25 ± 0.02 (depending on the dimension of the embedding space) and a
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281:. An overview of the type of behavior of the map at different parameter values may be obtained from its
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of the fixed point. The unstable manifold of the fixed point in the attractor is contained in the
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Many other generalizations have been proposed in the literature. One can generate, for example,
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is a natural linear operator on the space of scalar fields. For general nonlinear systems, the
1000:
999:, we obtain two additional dimensions for plotting. The Hénon map therefore, can be plotted in
568:
Classical Hénon map (15 iterations). Sub-iterations calculated using three steps decomposition.
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The Hénon map may also be deconstructed into a one-dimensional map, defined similarly to the
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3123:
3023:
2868:
2551:
2514:
2328:
Cong Zhang; Haipeng Li; Yueheng Lan (2022). "Phase space partition with
Koopman analysis".
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8:
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P. Grassberger; I. Procaccia (1983). "Measuring the strangeness of strange attractors".
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Michel Hénon and Yves Pomeau (1976). "Two strange attractors with a simple structure".
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A 3-D generalization for the Hénon map was proposed by Hitz and Zele. It is given by
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225:{\displaystyle {\begin{cases}x_{n+1}=1-ax_{n}^{2}+y_{n}\\y_{n+1}=bx_{n}.\end{cases}}}
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In the video example to the right, the three axes for each image in the video are
487:{\displaystyle y={\frac {3\left({\sqrt {609}}-7\right)}{280}}\approx 0.189406343.}
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The Hénon map does not have a strange attractor for all values of the parameters
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D.A. Russell; J.D. Hanson; E. Ott (1980). "Dimension of strange attractors".
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Attractor
Dimension Estimates for Dynamical Systems: Theory and Computation
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2159:
81:
80:. It is one of the most studied examples of dynamical systems that exhibit
2571:
2448:
Borges, Vinícius S.; Silva, Magno T. M.; Eisencraft, Marcio (2024-04-01).
336:. Higher density (darker) indicates increased probability of the variable
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of Hénon map in javascript (experiences.math.cnrs.fr) by Marc Monticelli.
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261:. For the classical values the Hénon map is chaotic. For other values of
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Cell-to-cell mapping: a method of global analysis for nonlinear systems
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If multiple Hénon maps are plotted, for each map varying the value of
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422:{\displaystyle x={\frac {{\sqrt {609}}-7}{28}}\approx 0.631354477,}
320:
of 1.261 ± 0.003 for the attractor of the classical map.
2682:
by C. Pellicer-Lostao and R. Lopez-Ruiz after work by Ed Pegg Jr,
1707:
In 1976 France, the Lorenz attractor is analyzed by the physicist
1034:
If one solves the one-dimensional Hénon map for the special case:
529:= 1.25 the Hénon map has a stable periodic orbit as an attractor.
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305:
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of chaotic maps like the Hénon map. In the mode provided, the
2250:
2607:
2454:
Communications in Nonlinear Science and Numerical Simulation
564:
2616:
2315:"Deux exemples français : Yves Pomeau et Michel Hénon"
1327:{\displaystyle X={b-1\pm {\sqrt {b^{2}-2b+1+4a}} \over 2a}}
372:
of the Hénon map, one of these points is on the attractor:
218:
19:
2215:
2399:
Borges, Vinícius S.; Eisencraft, Marcio (December 2022).
352:-- these can arise depending upon initial conditions for
2206:. Vol. 64. Springer Science & Business Media, 2013
1912:
1810:
1029:
543:
2579:
Carles Simó (1979). "On the Hénon-Pomeau attractor".
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111:
2503:"A two-dimensional mapping with a strange attractor"
525:
fixed at 0.3 the bifurcation diagram shows that for
1764:
region is the stable manifold of strange attractor.
304:, or diverge to infinity. The Hénon attractor is a
2146:
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2450:"Chaotic properties of an FIR filtered Hénon map"
3522:
2398:
1337:In the special case b=1, this is simplified to
856:
2209:
2674:Another interactive iteration of the Henon Map
1003:. We can visualize such a plot by viewing one
751:{\displaystyle (x_{2},y_{2})=(bx_{1},y_{1})\,}
658:{\displaystyle (x_{1},y_{1})=(x,1-ax^{2}+y)\,}
2717:
2643:Kuznetsov, Nikolay; Reitmann, Volker (2020).
983:Although the Hénon map can be plotted on the
953:
943:{\displaystyle x_{n+1}=1-ax_{n}^{2}+bx_{n-1}}
845:{\displaystyle (x_{3},y_{3})=(y_{2},x_{2})\,}
2366:Hitzl, Donald L.; Zele, Frank (March 1985).
340:acquiring that value for the given value of
2578:
2368:"An exploration of the Hénon quadratic map"
2244:
1375:{\displaystyle X={\pm {\sqrt {a}} \over a}}
2724:
2710:
2465:
2416:
2365:
841:
747:
654:
2500:
1758:
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563:
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348:regions of chaos and periodicity around
327:
312:in another. Numerical estimates yield a
102:in the plane and maps it to a new point
39:
18:
1101:{\displaystyle X=x_{n-1}=x_{n}=x_{n+1}}
3523:
2617:M. Michelitsch; O. E. Rössler (1989).
2610:Turbulence and Navier Stokes Equations
2507:Communications in Mathematical Physics
2131:generate chaotic signals with largest
1419:the formula is further simplified to
2705:
1111:One arrives at the simple quadradic:
552:, then stacking all maps together, a
332:Orbit diagram for the Hénon map with
2361:
2359:
2169:in the feedback loop of the system.
2864:Measure-preserving dynamical system
2746:
1030:Special cases and low-period orbits
544:Relationship to bifurcation diagram
235:The map depends on two parameters,
13:
2684:The Wolfram Demonstrations Project
1768:
1385:If, in addition, a is in the form
1230:{\displaystyle 0=-aX^{2}+(b-1)X+1}
14:
3542:
3432:Oleksandr Mykolayovych Sharkovsky
2654:
2356:
2962:
2954:
2731:
2123:it can be shown that almost all
1783:
1731:
559:
308:, smooth in one direction and a
2441:
2202:Section 13.3.2; Hsu, Chieh Su.
1412:{\displaystyle {1 \over c^{n}}}
971:Hénon map in 4D. The range for
3197:Rabinovich–Fabrikant equations
2680:Orbit Diagram of the Hénon Map
2619:"A New Feature in Hénon's Map"
2581:Journal of Statistical Physics
2405:Chaos, Solitons & Fractals
2392:
2373:Physica D: Nonlinear Phenomena
2321:
2307:
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1638:{\displaystyle (-X,-X)=(-X,X)}
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1578:{\displaystyle (X,-X)=(-X,-X)}
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611:
585:
84:. The Hénon map takes a point
16:Discrete-time dynamical system
1:
2690:Matlab code for the Hénon Map
2494:
1459:{\displaystyle X=\pm c^{n/2}}
1158:{\displaystyle X=1-aX^{2}+bX}
857:One-dimensional decomposition
292:as a simplified model of the
2635:10.1016/0097-8493(89)90070-8
2386:10.1016/0167-2789(85)90092-2
2238:10.1016/0167-2789(83)90298-1
2090:{\displaystyle \alpha =1.07}
1692:{\displaystyle (-X,X)=(X,X)}
1518:{\displaystyle (X,X)=(X,-X)}
1026:axis that is moved through.
1022:. As time passes, it is the
762:3) a reflection in the line
575:1) an area-preserving bend:
323:
7:
2932:Poincaré recurrence theorem
2476:10.1016/j.cnsns.2024.107845
2427:10.1016/j.chaos.2022.112865
2273:10.1103/PhysRevLett.45.1175
2172:
521:. For example, by keeping
10:
3547:
2927:Poincaré–Bendixson theorem
2116:{\displaystyle \beta =0.3}
1702:
954:Four-dimensional extension
288:The map was introduced by
71:Hénon–Pomeau attractor/map
3480:
3297:
3279:Swinging Atwood's machine
3224:
3162:
3032:
3019:
2971:
2952:
2922:Krylov–Bogolyubov theorem
2902:
2799:
2739:
1736:In dynamical system, the
3187:Lotka–Volterra equations
3011:Synchronization of chaos
2814:axiom A dynamical system
2623:Computers & Graphics
2564:10.1103/PhysRevA.38.1503
2189:
669:2) a contraction in the
3172:Double scroll attractor
2937:Stable manifold theorem
2844:False nearest neighbors
2301:"L'attracteur de Hénon"
2253:Physical Review Letters
975:is -1.5 to 0.5 and for
3212:Van der Pol oscillator
3192:Mackey–Glass equations
2824:Box-counting dimension
2401:"A filtered Hénon map"
2148:
2117:
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1001:four-dimensional space
980:
944:
846:
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659:
569:
537:
488:
423:
361:
318:Box Counting dimension
226:
58:
37:
3362:Svetlana Jitomirskaya
3269:Multiscroll attractor
3114:Interval exchange map
3067:Dyadic transformation
3052:Complex quadratic map
2894:Topological conjugacy
2829:Correlation dimension
2804:Anosov diffeomorphism
2661:Interactive Henon map
2149:
2118:
2092:
2063:
1762:
1756:can be clearly seen.
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1103:
970:
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424:
331:
314:correlation dimension
227:
43:
22:
3372:Edward Norton Lorenz
2287:"Pomeau_Ibanez 1976"
2147:{\displaystyle 0.23}
2138:
2101:
2075:
1779:
1720:, proposed in 1979.
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872:
777:
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434:
379:
109:
44:Hénon attractor for
23:Hénon attractor for
3332:Mitchell Feigenbaum
3274:Population dynamics
3259:Hénon–Heiles system
3119:Irrational rotation
3072:Dynamical billiards
3057:Coupled map lattice
2917:Liouville's theorem
2849:Hausdorff dimension
2834:Conservative system
2819:Bifurcation diagram
2556:1988PhRvA..38.1503C
2519:1976CMaPh..50...69H
2265:1980PhRvL..45.1175R
2230:1983PhyD....9..189G
1935:
917:
554:Bifurcation diagram
277:, or converge to a
245:classical Hénon map
162:
69:, sometimes called
3510:Santa Fe Institute
3377:Aleksandr Lyapunov
3207:Three-body problem
3094:Gingerbreadman map
2981:Bifurcation theory
2859:Lyapunov stability
2612:. Springer: 29–68.
2593:10.1007/BF01009612
2527:10.1007/BF01608556
2144:
2125:initial conditions
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991:-axes, by varying
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863:Fibonacci Sequence
842:
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510:of the Hénon map.
484:
419:
362:
222:
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148:
59:
38:
3518:
3517:
3382:Benoît Mandelbrot
3347:Martin Gutzwiller
3337:Peter Grassberger
3220:
3219:
3202:Rössler attractor
2950:
2949:
2854:Invariant measure
2776:Lyapunov exponent
2649:. Cham: Springer.
2544:Physical Review A
2501:M. Hénon (1976).
2342:10.1063/5.0079812
2133:Lyapunov exponent
1754:strange attractor
1746:symbolic dynamics
1407:
1370:
1364:
1322:
1311:
1242:quadratic formula
968:
508:strange attractor
504:unstable manifold
476:
459:
408:
396:
302:strange attractor
3538:
3490:Butterfly effect
3402:Itamar Procaccia
3352:Brosl Hasslacher
3249:Elastic pendulum
3177:Duffing equation
3124:Kaplan–Yorke map
3042:Arnold's cat map
3030:
3029:
3006:Stability theory
2991:Dynamical system
2986:Control of chaos
2966:
2958:
2942:Takens's theorem
2874:Poincaré section
2744:
2743:
2726:
2719:
2712:
2703:
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2613:
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2550:(3): 1503–1520.
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2224:(1–2): 189–208.
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1738:Koopman operator
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843:
837:
836:
824:
823:
805:
804:
792:
791:
757:
755:
754:
749:
743:
742:
730:
729:
708:
707:
695:
694:
664:
662:
661:
656:
644:
643:
610:
609:
597:
596:
493:
491:
490:
485:
477:
472:
471:
467:
460:
455:
444:
428:
426:
425:
420:
409:
404:
397:
392:
389:
294:Poincaré section
268:
264:
260:
253:
243:, which for the
242:
238:
231:
229:
228:
223:
221:
220:
211:
210:
195:
194:
175:
174:
161:
156:
135:
134:
101:
82:chaotic behavior
78:dynamical system
57:
50:
36:
29:
3546:
3545:
3541:
3540:
3539:
3537:
3536:
3535:
3521:
3520:
3519:
3514:
3482:
3476:
3422:Caroline Series
3317:Mary Cartwright
3299:
3293:
3244:Double pendulum
3226:
3216:
3165:
3158:
3084:Exponential map
3035:
3021:
3015:
2973:
2967:
2960:
2946:
2912:Ergodic theorem
2905:
2898:
2889:Stable manifold
2879:Recurrence plot
2795:
2749:
2735:
2730:
2665:Henon attractor
2657:
2497:
2492:
2491:
2446:
2442:
2397:
2393:
2364:
2357:
2326:
2322:
2313:
2312:
2308:
2299:
2298:
2294:
2285:
2284:
2280:
2249:
2245:
2214:
2210:
2201:
2197:
2192:
2184:Takens' theorem
2175:
2167:digital filters
2163:chaotic signals
2157:
2139:
2136:
2135:
2102:
2099:
2098:
2076:
2073:
2072:
2051:
2050:
2035:
2031:
2013:
2009:
2003:
2002:
1987:
1983:
1974:
1973:
1952:
1948:
1930:
1925:
1908:
1907:
1897:
1896:
1875:
1871:
1868:
1867:
1846:
1842:
1839:
1838:
1817:
1813:
1806:
1805:
1782:
1780:
1777:
1776:
1771:
1769:Generalizations
1750:stable manifold
1734:
1705:
1651:
1648:
1647:
1591:
1588:
1587:
1531:
1528:
1527:
1477:
1474:
1473:
1446:
1442:
1438:
1427:
1424:
1423:
1401:
1397:
1392:
1390:
1387:
1386:
1359:
1355:
1353:
1345:
1342:
1341:
1314:
1281:
1277:
1275:
1262:
1260:
1252:
1249:
1248:
1194:
1190:
1176:
1173:
1172:
1140:
1136:
1119:
1116:
1115:
1086:
1082:
1073:
1069:
1054:
1050:
1042:
1039:
1038:
1032:
958:
956:
928:
924:
912:
907:
879:
875:
873:
870:
869:
859:
832:
828:
819:
815:
800:
796:
787:
783:
778:
775:
774:
738:
734:
725:
721:
703:
699:
690:
686:
681:
678:
677:
639:
635:
605:
601:
592:
588:
583:
580:
579:
562:
546:
500:stable manifold
454:
453:
449:
445:
443:
435:
432:
431:
391:
390:
388:
380:
377:
376:
326:
269:the map may be
266:
262:
255:
248:
247:have values of
240:
236:
216:
215:
206:
202:
184:
180:
177:
176:
170:
166:
157:
152:
124:
120:
113:
112:
110:
107:
106:
98:
91:
85:
52:
45:
31:
24:
17:
12:
11:
5:
3544:
3534:
3533:
3516:
3515:
3513:
3512:
3507:
3505:Predictability
3502:
3497:
3492:
3486:
3484:
3478:
3477:
3475:
3474:
3472:Lai-Sang Young
3469:
3467:James A. Yorke
3464:
3462:Amie Wilkinson
3459:
3454:
3449:
3444:
3439:
3434:
3429:
3424:
3419:
3414:
3409:
3404:
3399:
3397:Henri Poincaré
3394:
3389:
3384:
3379:
3374:
3369:
3364:
3359:
3354:
3349:
3344:
3339:
3334:
3329:
3324:
3319:
3314:
3309:
3303:
3301:
3295:
3294:
3292:
3291:
3286:
3281:
3276:
3271:
3266:
3264:Kicked rotator
3261:
3256:
3251:
3246:
3241:
3236:
3234:Chua's circuit
3230:
3228:
3222:
3221:
3218:
3217:
3215:
3214:
3209:
3204:
3199:
3194:
3189:
3184:
3179:
3174:
3168:
3166:
3163:
3160:
3159:
3157:
3156:
3154:Zaslavskii map
3151:
3149:Tinkerbell map
3146:
3141:
3136:
3131:
3126:
3121:
3116:
3111:
3106:
3101:
3096:
3091:
3086:
3081:
3080:
3079:
3069:
3064:
3059:
3054:
3049:
3044:
3038:
3036:
3033:
3027:
3017:
3016:
3014:
3013:
3008:
3003:
2998:
2996:Ergodic theory
2993:
2988:
2983:
2977:
2975:
2969:
2968:
2953:
2951:
2948:
2947:
2945:
2944:
2939:
2934:
2929:
2924:
2919:
2914:
2908:
2906:
2903:
2900:
2899:
2897:
2896:
2891:
2886:
2881:
2876:
2871:
2866:
2861:
2856:
2851:
2846:
2841:
2836:
2831:
2826:
2821:
2816:
2811:
2806:
2800:
2797:
2796:
2794:
2793:
2788:
2786:Periodic point
2783:
2778:
2773:
2768:
2763:
2758:
2752:
2750:
2747:
2741:
2737:
2736:
2729:
2728:
2721:
2714:
2706:
2700:
2699:
2693:
2687:
2677:
2671:
2656:
2655:External links
2653:
2652:
2651:
2640:
2629:(2): 263–265.
2614:
2605:
2587:(4): 465–494.
2576:
2539:
2496:
2493:
2490:
2489:
2440:
2391:
2380:(3): 305–326.
2355:
2320:
2306:
2292:
2278:
2243:
2208:
2194:
2193:
2191:
2188:
2187:
2186:
2181:
2174:
2171:
2143:
2112:
2109:
2106:
2086:
2083:
2080:
2055:
2049:
2046:
2043:
2038:
2034:
2030:
2027:
2024:
2021:
2016:
2012:
2008:
2005:
2004:
2001:
1998:
1995:
1990:
1986:
1982:
1979:
1976:
1975:
1972:
1969:
1966:
1963:
1960:
1955:
1951:
1947:
1944:
1941:
1938:
1933:
1928:
1924:
1920:
1917:
1914:
1913:
1911:
1906:
1901:
1895:
1892:
1889:
1886:
1883:
1878:
1874:
1870:
1869:
1866:
1863:
1860:
1857:
1854:
1849:
1845:
1841:
1840:
1837:
1834:
1831:
1828:
1825:
1820:
1816:
1812:
1811:
1809:
1804:
1801:
1798:
1795:
1792:
1789:
1785:
1770:
1767:
1742:eigenfunctions
1733:
1730:
1704:
1701:
1700:
1699:
1688:
1685:
1682:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1658:
1655:
1645:
1634:
1631:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1585:
1574:
1571:
1568:
1565:
1562:
1559:
1556:
1553:
1550:
1547:
1544:
1541:
1538:
1535:
1525:
1514:
1511:
1508:
1505:
1502:
1499:
1496:
1493:
1490:
1487:
1484:
1481:
1467:
1466:
1453:
1449:
1445:
1441:
1437:
1434:
1431:
1404:
1400:
1396:
1383:
1382:
1369:
1363:
1358:
1352:
1349:
1335:
1334:
1320:
1317:
1310:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1284:
1280:
1274:
1271:
1268:
1265:
1259:
1256:
1238:
1237:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1197:
1193:
1189:
1186:
1183:
1180:
1166:
1165:
1154:
1151:
1148:
1143:
1139:
1135:
1132:
1129:
1126:
1123:
1109:
1108:
1095:
1092:
1089:
1085:
1081:
1076:
1072:
1068:
1063:
1060:
1057:
1053:
1049:
1046:
1031:
1028:
955:
952:
951:
950:
937:
934:
931:
927:
923:
920:
915:
910:
906:
902:
899:
896:
893:
888:
885:
882:
878:
858:
855:
854:
853:
840:
835:
831:
827:
822:
818:
814:
811:
808:
803:
799:
795:
790:
786:
782:
760:
759:
746:
741:
737:
733:
728:
724:
720:
717:
714:
711:
706:
702:
698:
693:
689:
685:
667:
666:
653:
650:
647:
642:
638:
634:
631:
628:
625:
622:
619:
616:
613:
608:
604:
600:
595:
591:
587:
561:
558:
545:
542:
495:
494:
483:
480:
475:
470:
466:
463:
458:
452:
448:
442:
439:
429:
418:
415:
412:
407:
403:
400:
395:
387:
384:
344:. Notice the
325:
322:
279:periodic orbit
233:
232:
219:
214:
209:
205:
201:
198:
193:
190:
187:
183:
179:
178:
173:
169:
165:
160:
155:
151:
147:
144:
141:
138:
133:
130:
127:
123:
119:
118:
116:
96:
89:
15:
9:
6:
4:
3:
2:
3543:
3532:
3529:
3528:
3526:
3511:
3508:
3506:
3503:
3501:
3500:Edge of chaos
3498:
3496:
3493:
3491:
3488:
3487:
3485:
3479:
3473:
3470:
3468:
3465:
3463:
3460:
3458:
3457:Marcelo Viana
3455:
3453:
3450:
3448:
3447:Audrey Terras
3445:
3443:
3442:Floris Takens
3440:
3438:
3435:
3433:
3430:
3428:
3425:
3423:
3420:
3418:
3415:
3413:
3410:
3408:
3405:
3403:
3400:
3398:
3395:
3393:
3390:
3388:
3385:
3383:
3380:
3378:
3375:
3373:
3370:
3368:
3365:
3363:
3360:
3358:
3355:
3353:
3350:
3348:
3345:
3343:
3342:Celso Grebogi
3340:
3338:
3335:
3333:
3330:
3328:
3325:
3323:
3322:Chen Guanrong
3320:
3318:
3315:
3313:
3310:
3308:
3307:Michael Berry
3305:
3304:
3302:
3296:
3290:
3287:
3285:
3282:
3280:
3277:
3275:
3272:
3270:
3267:
3265:
3262:
3260:
3257:
3255:
3252:
3250:
3247:
3245:
3242:
3240:
3237:
3235:
3232:
3231:
3229:
3223:
3213:
3210:
3208:
3205:
3203:
3200:
3198:
3195:
3193:
3190:
3188:
3185:
3183:
3182:Lorenz system
3180:
3178:
3175:
3173:
3170:
3169:
3167:
3161:
3155:
3152:
3150:
3147:
3145:
3142:
3140:
3137:
3135:
3132:
3130:
3129:Langton's ant
3127:
3125:
3122:
3120:
3117:
3115:
3112:
3110:
3107:
3105:
3104:Horseshoe map
3102:
3100:
3097:
3095:
3092:
3090:
3087:
3085:
3082:
3078:
3075:
3074:
3073:
3070:
3068:
3065:
3063:
3060:
3058:
3055:
3053:
3050:
3048:
3045:
3043:
3040:
3039:
3037:
3031:
3028:
3025:
3018:
3012:
3009:
3007:
3004:
3002:
3001:Quantum chaos
2999:
2997:
2994:
2992:
2989:
2987:
2984:
2982:
2979:
2978:
2976:
2970:
2965:
2961:
2957:
2943:
2940:
2938:
2935:
2933:
2930:
2928:
2925:
2923:
2920:
2918:
2915:
2913:
2910:
2909:
2907:
2901:
2895:
2892:
2890:
2887:
2885:
2882:
2880:
2877:
2875:
2872:
2870:
2867:
2865:
2862:
2860:
2857:
2855:
2852:
2850:
2847:
2845:
2842:
2840:
2837:
2835:
2832:
2830:
2827:
2825:
2822:
2820:
2817:
2815:
2812:
2810:
2809:Arnold tongue
2807:
2805:
2802:
2801:
2798:
2792:
2789:
2787:
2784:
2782:
2779:
2777:
2774:
2772:
2769:
2767:
2764:
2762:
2759:
2757:
2754:
2753:
2751:
2745:
2742:
2738:
2734:
2727:
2722:
2720:
2715:
2713:
2708:
2707:
2704:
2697:
2694:
2691:
2688:
2685:
2681:
2678:
2675:
2672:
2670:
2666:
2662:
2659:
2658:
2648:
2647:
2641:
2636:
2632:
2628:
2624:
2620:
2615:
2611:
2606:
2602:
2598:
2594:
2590:
2586:
2582:
2577:
2573:
2569:
2565:
2561:
2557:
2553:
2549:
2545:
2540:
2536:
2532:
2528:
2524:
2520:
2516:
2512:
2508:
2504:
2499:
2498:
2485:
2481:
2477:
2473:
2468:
2463:
2459:
2455:
2451:
2444:
2436:
2432:
2428:
2424:
2419:
2414:
2410:
2406:
2402:
2395:
2387:
2383:
2379:
2375:
2374:
2369:
2362:
2360:
2351:
2347:
2343:
2339:
2336:(6): 063132.
2335:
2331:
2324:
2316:
2310:
2302:
2296:
2288:
2282:
2274:
2270:
2266:
2262:
2258:
2254:
2247:
2239:
2235:
2231:
2227:
2223:
2219:
2212:
2205:
2199:
2195:
2185:
2182:
2180:
2179:Horseshoe map
2177:
2176:
2170:
2168:
2164:
2161:
2155:
2141:
2134:
2130:
2126:
2110:
2107:
2104:
2084:
2081:
2078:
2069:
2053:
2044:
2036:
2032:
2028:
2022:
2014:
2010:
2006:
1996:
1988:
1984:
1980:
1977:
1970:
1967:
1961:
1953:
1949:
1945:
1939:
1931:
1926:
1922:
1918:
1915:
1909:
1904:
1899:
1890:
1887:
1884:
1876:
1872:
1861:
1858:
1855:
1847:
1843:
1832:
1829:
1826:
1818:
1814:
1807:
1802:
1796:
1793:
1790:
1774:
1761:
1757:
1755:
1751:
1747:
1743:
1739:
1732:Koopman modes
1729:
1725:
1721:
1719:
1718:Intermittency
1713:
1710:
1683:
1680:
1677:
1671:
1665:
1662:
1659:
1656:
1646:
1629:
1626:
1623:
1620:
1614:
1608:
1605:
1602:
1599:
1596:
1586:
1569:
1566:
1563:
1560:
1557:
1551:
1545:
1542:
1539:
1536:
1526:
1509:
1506:
1503:
1500:
1494:
1488:
1485:
1482:
1472:
1471:
1470:
1451:
1447:
1443:
1439:
1435:
1432:
1429:
1422:
1421:
1420:
1402:
1398:
1394:
1367:
1361:
1356:
1350:
1347:
1340:
1339:
1338:
1318:
1315:
1308:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1282:
1278:
1272:
1269:
1266:
1263:
1257:
1254:
1247:
1246:
1245:
1243:
1224:
1221:
1218:
1212:
1209:
1206:
1200:
1195:
1191:
1187:
1184:
1181:
1178:
1171:
1170:
1169:
1152:
1149:
1146:
1141:
1137:
1133:
1130:
1127:
1124:
1121:
1114:
1113:
1112:
1093:
1090:
1087:
1083:
1079:
1074:
1070:
1066:
1061:
1058:
1055:
1051:
1047:
1044:
1037:
1036:
1035:
1027:
1025:
1021:
1017:
1013:
1008:
1006:
1002:
998:
994:
990:
986:
978:
974:
935:
932:
929:
925:
921:
918:
913:
908:
904:
900:
897:
894:
891:
886:
883:
880:
876:
868:
867:
866:
864:
833:
829:
825:
820:
816:
809:
801:
797:
793:
788:
784:
773:
772:
771:
769:
766: =
765:
739:
735:
731:
726:
722:
718:
712:
704:
700:
696:
691:
687:
676:
675:
674:
672:
648:
645:
640:
636:
632:
629:
626:
623:
620:
614:
606:
602:
598:
593:
589:
578:
577:
576:
573:
566:
560:Decomposition
557:
555:
551:
541:
534:
530:
528:
524:
520:
516:
511:
509:
505:
501:
481:
478:
473:
468:
464:
461:
456:
450:
446:
440:
437:
430:
416:
413:
410:
405:
401:
398:
393:
385:
382:
375:
374:
373:
371:
367:
359:
355:
351:
347:
343:
339:
335:
330:
321:
319:
315:
311:
307:
303:
299:
295:
291:
286:
284:
283:orbit diagram
280:
276:
272:
258:
251:
246:
212:
207:
203:
199:
196:
191:
188:
185:
181:
171:
167:
163:
158:
153:
149:
145:
142:
139:
136:
131:
128:
125:
121:
114:
105:
104:
103:
99:
92:
83:
79:
76:
75:discrete-time
72:
68:
64:
55:
48:
42:
34:
27:
21:
3531:Chaotic maps
3452:Mary Tsingou
3417:David Ruelle
3412:Otto Rössler
3357:Michel Hénon
3327:Leon O. Chua
3284:Tilt-A-Whirl
3254:FPUT problem
3139:Standard map
3134:Logistic map
3098:
2959:
2733:Chaos theory
2669:Chaotic Maps
2645:
2626:
2622:
2609:
2584:
2580:
2547:
2543:
2513:(1): 69–77.
2510:
2506:
2457:
2453:
2443:
2408:
2404:
2394:
2377:
2371:
2333:
2329:
2323:
2309:
2295:
2281:
2259:(14): 1175.
2256:
2252:
2246:
2221:
2217:
2211:
2203:
2198:
2160:band-limited
2156:
2070:
1775:
1772:
1735:
1726:
1722:
1714:
1706:
1468:
1384:
1336:
1239:
1167:
1110:
1033:
1023:
1019:
1015:
1011:
1009:
996:
992:
988:
984:
982:
976:
972:
860:
767:
763:
761:
670:
668:
574:
571:
549:
547:
539:
526:
522:
518:
514:
512:
496:
482:0.189406343.
369:
365:
363:
357:
353:
349:
345:
341:
337:
333:
298:Lorenz model
290:Michel Hénon
287:
275:intermittent
256:
249:
244:
234:
94:
87:
70:
66:
60:
53:
46:
32:
25:
3437:Nina Snaith
3427:Yakov Sinai
3312:Rufus Bowen
3062:Duffing map
3047:Baker's map
2972:Theoretical
2884:SRB measure
2791:Phase space
2761:Bifurcation
2129:unit sphere
2127:inside the
1709:Yves Pomeau
673:direction:
414:0.631354477
63:mathematics
3495:Complexity
3392:Edward Ott
3239:Convection
3164:Continuous
2839:Ergodicity
2696:Simulation
2692:by M.Suzen
2676:by A. Luhn
2495:References
2467:2401.10281
2460:: 107845.
2418:2211.16964
2411:: 112865.
1005:hyperplane
310:Cantor set
3407:Mary Rees
3367:Bryna Kra
3300:theorists
3109:Ikeda map
3099:Hénon map
3089:Gauss map
2771:Limit set
2756:Attractor
2601:122545201
2484:1007-5704
2435:254095983
2105:β
2079:α
2007:β
1981:β
1978:−
1919:α
1916:−
1657:−
1621:−
1606:−
1597:−
1567:−
1558:−
1543:−
1507:−
1436:±
1357:±
1288:−
1273:±
1267:−
1210:−
1185:−
1131:−
1059:−
933:−
898:−
630:−
479:≈
462:−
411:≈
399:−
346:satellite
324:Attractor
143:−
67:Hénon map
3525:Category
3483:articles
3225:Physical
3144:Tent map
3034:Discrete
2974:branches
2904:Theorems
2740:Concepts
2535:12772992
2350:35778118
2173:See also
1244:yields:
3481:Related
3289:Weather
3227:systems
3020:Chaotic
2766:Fractal
2572:9900529
2552:Bibcode
2515:Bibcode
2261:Bibcode
2226:Bibcode
2218:Physica
1752:of the
1703:History
350:a=1.075
306:fractal
296:of the
271:chaotic
73:, is a
3387:Hee Oh
3022:maps (
2869:Mixing
2599:
2570:
2533:
2482:
2433:
2348:
2165:using
1018:, and
987:- and
65:, the
3298:Chaos
3077:outer
2781:Orbit
2597:S2CID
2531:S2CID
2462:arXiv
2431:S2CID
2413:arXiv
2330:Chaos
2190:Notes
334:b=0.3
259:= 0.3
252:= 1.4
56:= 0.3
49:= 1.4
35:= 0.3
28:= 1.4
3024:list
2748:Core
2663:and
2568:PMID
2480:ISSN
2346:PMID
2142:0.23
2097:and
2085:1.07
2071:For
1240:The
995:and
517:and
502:and
368:and
356:and
265:and
254:and
239:and
51:and
30:and
2667:in
2631:doi
2589:doi
2560:doi
2523:doi
2472:doi
2458:131
2423:doi
2409:165
2382:doi
2338:doi
2269:doi
2234:doi
2111:0.3
1168:Or
474:280
457:609
394:609
61:In
3527::
2627:13
2625:.
2621:.
2595:.
2585:21
2583:.
2566:.
2558:.
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2546:.
2529:.
2521:.
2511:50
2509:.
2505:.
2478:.
2470:.
2456:.
2452:.
2429:.
2421:.
2407:.
2403:.
2378:14
2376:.
2370:.
2358:^
2344:.
2334:32
2332:.
2267:.
2257:45
2255:.
2232:.
2222:9D
2220:.
2154:.
2068:.
1014:,
865:.
770::
406:28
285:.
273:,
93:,
3026:)
2725:e
2718:t
2711:v
2686:.
2637:.
2633::
2603:.
2591::
2574:.
2562::
2554::
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2525::
2517::
2486:.
2474::
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2425::
2415::
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2303:.
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2271::
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2240:.
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2108:=
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2048:)
2045:n
2042:(
2037:2
2033:s
2029:+
2026:)
2023:n
2020:(
2015:1
2011:s
2000:)
1997:n
1994:(
1989:1
1985:s
1971:1
1968:+
1965:)
1962:n
1959:(
1954:3
1950:s
1946:+
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1940:n
1937:(
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1853:(
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1827:n
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1784:s
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1684:X
1681:,
1678:X
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1666:X
1663:,
1660:X
1654:(
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1630:X
1627:,
1624:X
1618:(
1615:=
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1609:X
1603:,
1600:X
1594:(
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1570:X
1564:,
1561:X
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1540:,
1537:X
1534:(
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1510:X
1504:,
1501:X
1498:(
1495:=
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1489:X
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1480:(
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1448:/
1444:n
1440:c
1433:=
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1403:n
1399:c
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1362:a
1351:=
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1319:a
1316:2
1309:a
1306:4
1303:+
1300:1
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1294:b
1291:2
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1279:b
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1255:X
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1222:+
1219:X
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1213:1
1207:b
1204:(
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1196:2
1192:X
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1182:=
1179:0
1153:X
1150:b
1147:+
1142:2
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1134:a
1128:1
1125:=
1122:X
1094:1
1091:+
1088:n
1084:x
1080:=
1075:n
1071:x
1067:=
1062:1
1056:n
1052:x
1048:=
1045:X
1024:a
1020:b
1016:y
1012:x
997:b
993:a
989:y
985:x
977:a
973:b
936:1
930:n
926:x
922:b
919:+
914:2
909:n
905:x
901:a
895:1
892:=
887:1
884:+
881:n
877:x
852:.
839:)
834:2
830:x
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821:2
817:y
813:(
810:=
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802:3
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208:n
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200:b
197:=
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168:y
164:+
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150:x
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140:1
137:=
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115:{
100:)
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88:x
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