HÉNON HEILES HAMILTONIAN CHAOS IN 2 D

Transcription

1 ABSTRACT HÉNON HEILES HAMILTONIAN CHAOS IN D MODELING CHAOS & COMPLEXITY 008 YOUVAL DAR PHYSICS DAR@PHYSICS.UCDAVIS.EDU Chaos in two degrees of freedom, demonstrated b using the Hénon Heiles Hamiltonian INTRODUCTION A star moving about a galactic center can be considered a point mass. The Hamiltonian governing this motion will have three degrees of freedom (si coordinates in phase space) and will look like H = p / m + V( ρ, z), p. In 964, while investigating the motion of stars around the galactic center, Michael Hénon and Carl Heiles wanted to simplif the problem, the reduced the degrees of freedom b arguing invariant energ l z, angular momentum (due to the clindrical smmetr) supported b the observed motion that was confined to two dimensions. The chose to use a Hamiltonian that is used for motion in a near elliptic equilibrium. In this paper I will eamine Hénon and Heiles Hamiltonian dnamic properties, to some etent. In particular, I will look at the chaotic nature of the motion described b the equation of motion that derived from this Hamiltonian. E PROJECT S OBJECTIVE To gain an understanding of the dnamics of this specific Hamiltonian, to locate the onset of chaos and to observe what general lesson we can learn from it. To do that using computational tools learned in the course and using Pthon. BASIC ANALYSIS The Hénon Heiles Hamiltonian: H p r p θ kr λ r sin( θ ) = (.) m mr In Cartesian coordinates p p ( ) = λ( ) (.) H k m m And converting, normalizing and making dimensionless constants

2 H ( = p + p + + ) + (.) This equation can be further generalized in the form H = ( p + p + A + B ) + D C (.4) Deriving the equation of motion using H q = p H p = q (.5) The equations of motion are = p = p p = p = (.6) For the fied points, we want = = p = p = 0 At the same time, so p p = 0 = 0 So = = 0, = 0, or 0, =, is a function of E (.7) We also have the condition of invariant energ which is reduced to ( ) E = + + (.8) So our fied points should compl with both (.7) and (.8)

3 The Jacobian: J = (.9) The Jacobian determinant: Det( J ) = Tr( J ) = 0 (.0) The Eigen values are: λ λ,,4 =± =± + (.) The sum of the Eigen values is zero, which is consistent with energ conservation. For chaos in torus we epect the Lapunov Characteristic Eponents (LCE) to be (+, 0, 0, ), to find them we can diagonalize the Jacobian and then find the Eigen values or use another method. v v,,4 ± ( ) ± ( ) = ± ( + ) ( + ) = (.) Using the conditions from (.7) Det J = + + with = Det J = ( ) 4 4, 0 ( ) Det J = with = E = ( ) 4, /6 (.) For = 0 Det( J ) 0, Det = for =, 0 E = / For = Det( J ) 0and Det = for =± E = /6

4 I did observe the epected Det = at the fied points (0,0),(0, ) but not at (, /).This solution is actuall a fied line along which onl point ields Det ( J ) =.I didn t figure out what is the meaning of that reasult. For the fied point = we get the energ level which is the boundar of the potential region that allows orbits and when = 0 we are out of that region. I want to note two interesting points in equation (.), E = /6is the limit of the energ above which we get no orbit. And the solution of E =, =0 indicates the range of possible values of and at those points we get elliptical orbit, elliptic in phase space oscillating through the center in space. GENERAL VISUALIZATION APPROACH M plan was to create graphical analsis tools and then probe the sstem. The graphical tool I wanted to build should have produced Poincaré plots for different intersecting planes, D phase space plots, capabilit to plot a single path and an evolution of arra of initial conditions, to plot D vs. plots, to calculate Lapunov eponents, Entrop rate h μ = λi λi > 0 and finall to evaluate what percentage of the solution space is chaotic. I started to work on that plan but had to significantl reduce m goals in order for this project to be completed on time. (The mid wa change of plan shows in the pthon program flow.) COMPUTATIONAL STRATEGY After choosing some initial conditions, I will use equations(.6), integrate them using Runga Kutta method and use equation (.) to make sure that we keep the condition of energ conservation, making sure not to accumulate error due to the numerical calculation. = p = p p = p = E = ( p + p + + ) + To get the Poincaré I will look at an intersection of the chosen D phase space solution with the plane (In the actual program I reduced it to the plane = 0 onl.) a + b cp d = 0 (.4) 4

5 p = E+ p p = E+ p (.5) The plan for error correction was to check the error of a certain number of iterations and randoml appl correction to one of the four phase space coordinates. I planned to do it this wa in order to avoid over correcting one coordinate. In practice, due to time limitations, I just monitored the numerical error and since it never went over e 6I never applied corrections (the correction functions, though not use are in the program). LOOKING AT OUR POTENTIAL Potential equipotentials lines, V(, ) = ( + + ) the straight lines are for E = and the 6 smaller closed curves are for a smaller E. Note the ± π /smmetr of the potential. Recall the results of the Jacobian determinant. Finding etremums b finding where the first derivatives are zero and then using the descriminant st derivatives : V = (+4), V = ( +- ) (.6) V = ( + 4 ), V =V =, V = ( - 4 ) (.7) The Descriminant: D = V V -V If D>0 and V (, )>0, the point is a relative minimum.. If D>0 and V (, )<0, the point is a relative maimum.. If D<0, the point is a saddle point. 4. If D=0, higher order tests must be used. (.8) The etremum points I found were: (0, 0), (0, ), ( /, -/), (- /, -/) 5

6 Using the descriminant point (0, 0) is a relative minimum and the rest are saddle points. So we can epect orbital motion to eit onl in the area within the triangle created b the points (0, ), ( /, -/), (- /, -/). When going out of that area the trajector just runs awa. (This can be verified b the program.) 6

7 CHAOTIC REGION The Poincaré plots together with the D phase space picture are ver useful in understanding what is going on. B plaing with the initial conditions it looks like the onset of chaos is around E = /9. When the energ was E < /9 we got non chaotic orbits, though some were ver comple. It looks like we have an elliptical orbit on the edge of possible solutions for the equations E = =0. (I did not investigate it enough to see, when we get period doubling). But, we could qualitativel observe how the chaotic region, in the solution space, is getting bigger with E when E > /9. FIGURE : POINCARÉ PLOT OF 5 INITIAL CONDITIONS DIMENTIONLESS ENERGY, E=/, NON CHAOTIC ORBIS FIGURE : E = /9. MORE COMPLEX PICTURE BUT STILL NO CHAOTIC REGIONS 7

8 FIGURE : CHAOTIC REGIONS STARTING TO APPEAR FIGURE 4: VERY CHAOTIC FIGURE 5: MOSTLY CHAOTIC 8

9 SCREEN SHOTS OF SOME D PHASE SPACE PLOTS FIGURE 6: E=/, = 0, = , p =,0 ELIPTICAL ORBIT IN PHASE SPACE, = 0.5 IS A BIFURACATION POINT THE COLOR INDICATES THE AGE OF THE CURVE FIGURE 7: E=/, = 0, = 0.4, p =,0 STILL NON CHAOTIC REGION THE COLOR INDICATES THE AGE OF THE CURVE 9

10 FIGURE 8: E=/6, = 0, = 0., p =,0 CHAOTIC ORBIT FIGURE 9: E=/6, = 0, = 0., p =,0 NON CHAOTIC ORBIT IN MOSTLY CHAOTIC LANDSCAPE PROJECT CONCLUSIONS The onset of chaos for this sstem was approimatel E = /9. I found it b looking at the Poincaré maps and tring different energies. Poincaré is a great tool to look at a sstem that has three independent coordinates; it is simple, and together with the D plots, ver helpful in getting some intuition and basic analsis for the sstem dnamics. When planning a project, if ou have more realistic goals ou can achieve more. I spent too much time thinking how to write the program to be more general and fleible, and at the end had to cut it in order to get results which left the program mess and left man interesting quantities not eplored. 0

11 PROGRAM ALGORITHMS I DID NOT INCLUDE IN THE PROGRAM Multi plane intersection The wa I thought to allow intersecting the D phase space with an plane was, to rotate the coordinate sstem in a wa that the intersecting plane will be at ' = 0, this, I thought, will make it much easier to determine when the curve crosses the intersection plane. The algorithm: The plane: a + b + cz = 0 The normal to the plane: nˆ = ( abc,, ) The normal to the plane in the final orientation: n = (0,0,) The two rotation angles are: θ = n nˆ i And φ =n ˆ i ( nˆ n ) Rotating to the new coordinate sstem Checking and collecting the points ( ', p', ') where ' changes sign Printing ( ', p' ) Lapunov Eponents Following the idea that the orthonormal vectors tangent to the curve are the quantities we need to follow Define all the derivative equations, d i, 4 D phase space 4 equations Initialize the vectors (,0,0,0...),(0,,0,0...)... Iterate the di equations and find the new vector length, di,(this is an indication of stretching or contracting in a certain direction) LCE ( ) i = LCEi + Log di Now use the Gramm Schmit process to make the new set of vectors orthonormal again. Repeat for as man iterations as desired and at the end do LCE = LCE /( iteration #) This process is taken from the eample pthon programs. BIBLIOGRAPHY. Jim Crutchfield lecture notes for the course Modeling Chaos & Compleit 008. Strogatz, Nonlinear Dnamics and Chaos. James D. Meiss, Differential Dnamical Sstems 4. Goldstein Poole & Safko, Classical Mechanics 5. HeilesEquation.html