Investigating the relation between chaos and the three body problem
IInvestigating the relation between chaos and the three body problem
T.S.Sachin Venkatesh [email protected]
Delhi Technological University, Delhi 110042, India
Vishak Vikranth
Bengaluru 560076, India
We review the properties of fractals, the Mandelbrot set and how deterministic chaos ties to thepicture. A detailed study on three body systems, one of the major applications of chaos theorywas undertaken. Systems belonging to different families produced till date were studied and theirproperties were analysed. We then segregated them into three classes according to their properties.We suggest that such reviews be carried out in regular intervals of time as there are an infinitenumber of solutions for three body systems and some of them may prove to be useful in variousdomains apart from hierarchical systems.Key words - Celestial mechanics, Three-body problem, Gravitational interaction, Chaos, Orbits,Astronomical simulations
I. INTRODUCTION
Exploring the connections between different theories ofmathematics leads one through successive topics whichdiverge very little from one another, but looking at thethe path as a whole, the initial point of probing and thefinal point have very little in common. So a quick intro-duction of the topics we covered is listed below
A. Fractals
Fractals are infinitely complex patterns that are self-similar across different scales. They are created by re-peating a simple process over and over in an ongoingfeedback loop. A key characteristic of a fractal is its frac-tal dimension. Unlike the Euclidean dimension, fractaldimension is generally expressed by a non-integer and isan indicator of the complexity or roughness of a given fig-ure. Some common examples include Sierpi´nski trianglehaving a Hausdorff dimension of 1.585, Koch’s snowflakewith a Hausdorff dimension of 1.262 and the coastline ofBritain, whose fractal dimension is 1.21.
B. The Mandelbrot set and the logistic map
One of the most interesting fractals, the Mandelbrotset is the set of complex numbers c for which the func-tion x n +1 = x n + c does not diverge to infinity when iter-ated from z=0. The path of all such orbits when plottedgive us an image which is intuitive and rather easy tounderstand. All the points inside the main cardioid havea single fixed point, all the points inside the main bulbhave 2 limit points and subsequently all other bulbs rep-resent the set of numbers which have different numberof limit points. The logistic map is a polynomial map-ping of degree 2, often cited as an archetypal example ofhow complex, chaotic behaviour can arise from very sim- FIG. 1: The real line on the mandelbrot set lines upwith the bifurcations in the logistic map ple non-linear dynamical equations. By simple algebraicmanipulation, the logistic map x n +1 = rx n (1 − x n ), canbe recoded into the form x n +1 = x n + c . This leads us tofurther explore a relation which produces a rather inter-esting relation when we change our Point of View. Fromthe branching structure of the logistic bifurcation dia-gram we can read the cycle number of the correspondingfeatures of the Mandelbrot set. C. Deterministic Chaos
Deterministic chaos is the study of how systems thatfollow simple, straightforward, deterministic laws can ex-hibit very complicated and seemingly random long term a r X i v : . [ n li n . C D ] S e p behavior. One of the foundations of chaos theory is Ed-ward Lorenz’s discovery of systems where the dynamicsare sensitive to minute changes in the initial conditionsthus are unpredictable. Due to this sensitivity the be-haviour of systems appear to be random although modelis deterministic, which means that it is well defined andit does not contain any random parameters. In the be-ginning, this theory was used only in the field of meteo-rology but soon it was realized that it can also define theother chaotic systems in varied science disciplines whichwere based on the predictability of future behavior of thesystems. II. THEORYA. The relation between Logistic Map and Chaos
The logistic equation is of the form x n +1 = rx n (1 − x n ). By changing the values of r ,the following behaviour is observed : When r is in [0 , r is in [1 , r − r , regardless ofthe value for x ;though the rate of convergence of thefunction decreases as the value of r increases. When r is in [3 , √ r . When r is in [1 + √ , ∼ (3 . r increases beyond ∼ (3 . δ ≈ . . At r ≈ . x value yield drastically different resultsover time.For values of r beyond ∼ . r that show non-chaotic behavior; these are sometimescalled islands of stability. For example for r=1 + √ r oscillation among 6 limit points, then12 etc. B. Noise versus Deterministic Chaos
Noise is the random variation of values whereas chaoshappens when the initial state variables of the systemdiffer ever so slightly which lead to drastically differentoutcomes making it impossible to predict the initial valuefrom the output.We can also differentiate between noise and determin-istic chaos using fractals. The Hausdorff dimension of theattractor of a random model is usually infinite. But theattractor of a model of deterministic chaos always givesus a non integral, finite value. Hence, if the Hausdorff dimension of a model is finite it is good indicator thatit is a deterministic model that can be represented by asystem of non linear differential equations . C. Lorenz Attractor
An attractor is a point, set or even a fractal (strange at-tractor) around which it the function seems to converge.Edward Lorenz, an American physicist and meteorologistwho pioneered chaos theory is famous for demonstratingchaotic motion using ”Lorenz water wheel”: a water-wheel with holes at the bottom and a constant streampouring from above. The emptying and refilling of thewheel produces unpredictable changes in its direction ofrotation and angular velocity.Lorenz’s paper in 1963,tries to model weather and usedthese non linear differential equations :d x d t = σ ( y − x )d y d t = x ( ρ − z ) − y d z d t = xy − βz (1)According to Lorenz’s account, while working on theweather model he was running simulations and wanted torepeat a previous simulation. He started the simulationfrom a random iteration and since it was a computerprogram it should have given him the exact same results.But the results were in stark disagreement. He initiallyshrugged this off as an error, but then realized he hadnot entered the initial conditions exactly. The computerwas taking values precise upto six decimal places but theprinter only displayed three. He re-entered the roundedoff values and this minuscule error caused drastic changesin the outcome.FIG. 2: Lorenz attractor Lorenz considered the case where σ = 10, β = 8/3, ρ =28 with ( x , y , z ) = (0 , , . D. Three Body Problem
The three body problem is a system of ordinary differ-ential equations modelling three bodies of masses undermutual gravitation in two or three dimensions. The mo-tion of such a system was first pondered upon by Newtonand was reconsidered by many mathematicians and sci-entists. The famous three-body problem has had a greatinfluence on physics, mathematics and non-linear dynam-ics. It has paved way to a new field in modern science,chaotic dynamics.Until 1975, there were not many models of three bodysystems and the ones that existed were too restrictiveand specific about the parameters of the model. In 1890,Poincar´e proved the non existence of the uniform first in-tegral of a three-body problem in general, and also high-lighted the sensitive dependence to initial conditions ofits trajectories . Three body systems without mass hi-erarchy are never thought to be stable for very long .They can certainly exist for some period of time, butthey aren’t found to be long term stable. In these sys-tems, each body orbits the center of mass of the system.Mostly, two of the bodies form a close binary system,and the third body orbits this binary at a distance muchlarger than that of the orbit of the close binary. Thisarrangement is called hierarchical. The reason for thisbehaviour is that if the inner and outer orbits are com-parable in size, the system may become dynamically un-stable, leading to a body being ejected from the system.And if the system in question consists of disproportion-ate mass bodies, the Hill sphere mechanism comes intoplay.A lot of work has been carried out and various mod-els have been explored such as Poincar´e’s planar circu-lar restricted three body problem (PCR3BP) , Sitnikovmodels in the early 1900s and the recent discovery ofthirteen families of stable planar equal mass three bodysystems. The recent discoveries have formulated a newmethod to check for duplicate orbits and sort different in-stances, they use an abstract space called a ‘shape spacesphere’ which describes the shape of the orbits in termsof the relative distances between the objects. Three spotsaround the sphere’s equator mark where two of the par-ticles would collide, and a line drawn over the ball, whichmust avoid those spots, maps how near the objects getto each other. One of their original solutions nicknamed‘yarn’ is shown. III. MODELING AND OBSERVATIONS
We simulated different models of the three body sys-tem ranging from the more popular ‘Montgomery 8’ or-bit to custom made orbits like ‘perturbed circular orbits’.Most of these models are equal mass systems, but thereare some special orbits that were simulated for unequalmass systems, which were observed to remain stable overextensive number of iterations. The simulated models FIG. 3: Shape sphere showing the relative positionsof the orbits in a 3-body problem. FIG. 4: The same orbits in real euclidean spacerelative to centre of mass of the system. can broadly be classified into three classes of stability;namely, Long-term stable, Quasi-stable and Chaotic sys-tems. A. Long term stable
The families of such three body systems are extremelyrare to compute in a general setup due to the increasedcomplexity in computation owing to the increase in thenumber of parameters. Some of the first solutions to thethree body system were given by Euler and Lagrange forspecial cases like ‘Circular Restricted 3 Body Problem’and ‘Planar Restricted 3 Body Problem’. Until 2013, spe-cific solutions could be sorted into just three families: theLagrange-Euler family, the Broucke-H´enon family, andthe figure-eight family. Since then there have been morethan a thousand periodic solutions for the planar threebody problem and 13 families of solutions to the moreFIG. 5: Montgomery 8 - the system comprises of 3equal masses with a net angular momentum of zero. FIG. 6: Newton’s cradle-esque system of equalmasses that meet at only two points.FIG. 7: Double ring system. The blue body actslike a field regulator.general, non planar three body problem.The Montgomery 8 orbit uses a variational method toexhibit a simple periodic orbit for the Newtonian problemof three equal masses in the plane .Other stable orbits include Newton’s cradle-esque or-bit (FIG. 6) which looks as if two binary orbits have co-alesced together to form a system with a period of two.There are two separate levels of orbital systems inter-twined into one. The bodies try to slingshot each otherbut owing to their gravitational pull, they aren’t able toescape the orbit, thus when they try to slingshot, theyform a larger orbit but when they are unable to exit thesystem, they are dominated by the gravitational poten-tial and form a shorter orbit.A more rational system that has a higher probability of being detected in the universe is the system shown be-low (FIG. 7). This is a solution for an equal mass systemwhere two bodies move in the same orbit while the thirdbody acts as the field regulator, making sure the othertwo bodies stay in the same orbit. The probability of theexistence of such system is relatively high due to the sim-plicity of the system and the lack of complex orbits whichare very sensitive to even the slightest of perturbations. B. Quasi-stable
Three body systems corresponding to this family aremuch harder to find since these systems remain stable forsome period of time which may range anywhere from aFIG. 8: Overlapping rings with a common centroid.The equal mass bodies are placed at the vertices ofan equilateral triangle. The highlighted curvesrepresent the quasi-stable orbits of the systems,while the dashed lines represent the path of thebodies after destabilizing. FIG. 9: Unequal mass system destabilised due toperturbations. The highlighted curves represent thequasi-stable orbits of the systems, while the dashedlines represent the path of the bodies afterdestabilizing.few moments to a few hundreds of years but over longerperiods of time, they seem to diverge from their orbitsand the three body system breaks down to a two bodysystem with the third body being flung away. Such sys-tems are perfect examples to demonstrate sensitive de-pendence to initial conditions (SDIC), even the slightestchanges in the initial conditions of positions, velocity,mass can alter their course and send them travelling in acompletely different direction or configuration.One such system is where the equal mass bodies areinitially placed at the vertices of an equilateral triangleand the centre of mass of the system lies at it’s cen-troid. The bodies revolve around the centre of mass in asymmetric orbit but after some period of time they areflung away from their seemingly stable orbit and chaoticmotion takes over the system. Changing the initial con-ditions such that the bodies now lie at the vertices ofa slightly bigger equilateral triangle produces the sametype of orbit but the chaotic regime of this configurationis completely different from that of the initial configura-tion.An unequal mass system that demonstrates similar be-haviour is shown alongside (FIG. 9). The blue bodyhas double the mass of other bodies. When the bod-ies approach the centre of mass of the system (which isslightly shifted towards blue), the other bodies go aroundin loops. This perturbation destabilises the system.
C. Chaotic systems
The systems corresponding to this classification are ex-tremely chaotic in nature. Even the slightest change inthe initial conditions ensures that the system traversesa path in the phase space that can never be replicated.Such systems are extremely susceptible to perturbations.The model shown below (FIG. 10) demonstrates one ofthe defining features of chaotic three body systems, thereduction of the complexity of the system. Every systemin the universe tries to attain stability and these systemsare no different. They obtain stability by reducing froma three body system to a two body system by flingingaway the third body.Another system with completely different initial con-ditions, unequal masses and a planar restriction is shownbelow (FIG. 11). The outcome remains the same, in thesense that it’s impossible to reproduce this system with-out taking into consideration every digit of the mantissainto subsequent calculations.
IV. CONCLUSION
Re-coding the Mandelbrot set to form the logistic mapprovided us with an explanation for why the bifurcationslined up with the real line and probed us to further studyit’s fractal nature. But a more interesting thing we foundabout the logistic map was it’s chaotic behaviour at cer-tain values of r and the periodically distributed islandsFIG. 10: Chaotic behaviour of an equal masssystem. The highlighting feature displayed in bothsystems is how the three body system reduces to atwo body system to attain stability. The third bodyis always flung away with the help of the gravityassist of other two bodies. FIG. 11: Chaotic behaviour of an unequal masssystem. The highlighting feature displayed in bothsystems is how the three body system reduces to atwo body system to attain stability. The third bodyis always flung away with the help of the gravityassist of other two bodies.of stability, this lead us to probe deeper into chaos the-ory and the problems it posed. Lorenz demonstrated theonset of chaos and sensitive dependence on initial condi-tions in his experiment - the Lorenz water wheel. One ofthe widely studied application of chaos is the three bodysystem and the N body system as a whole.The three body problem lacked an analytical solutionfor centuries and Poincar´e proved why it won’t ever besolved analytically. In 1912, Karl Sundman found a seriessolution for the problem but it converges so slowly thatit is infeasible to produce practically relevant results fromit. Since then the three body problem has been solved forseveral restrictive cases and the search is still going on formore viable and general systems. Classifying three bodysystems into long term stable, quasi-stable and chaoticsystems helps to not only bring to light the difficulty infinding solutions for such systems but also it’s complex- ity and the infinite solutions it has in certain regions ofinitial conditions. These can be equated to the islands ofstability, only the problem here is that these islands areextremely scattered and thus give us a very small windowof exploring them. V. ACKNOWLEDGEMENTS
We would like to thank our mentor Andres LopezMoreno for guiding and supporting us throughout theproject. We would also like to thank ISEC for providingus this opportunity to be able to research and review top-ics on our own under expert guidance. I would also liketo thank my colleague Shobhit Ranjan for proofreadingthe article. G.-J. Lay,
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