Accurate predictions of chaotic motion of a free fall disk
aa r X i v : . [ n li n . C D ] F e b Accurate predictions of chaotic motion of a free fall disk
Tianzhuang Xu ( 徐 天 壮 ) , Jing Li ( 李 靖 ) ∗ , Zhihui Li ( 李 志 辉 ) , and Shijun Liao( 廖 世 俊 ) , † Center of Advanced Computing, School of Naval Architecture, Ocean and CivilEngineering, Shanghai Jiaotong University, 200240 Shanghai, China School of Physics and AstronomyShanghai Jiaotong University, 200240 Shanghai, China National Laboratory for Computational Fluid Dynamics, 100191 Beijing, China China Aerodynamics Research and Development Center, 621000 Mianyang, China
Abstract
It is important to know the accurate trajectory of a free fall object in fluid(such as a spacecraft), whose motion might be chaotic in many cases. However, it isimpossible to accurately predict its chaotic trajectory in a long enough duration bytraditional numerical algorithms in double precision. In this paper, we give the accu-rate predictions of the same problem by a new strategy, namely the Clean NumericalSimulation (CNS). Without loss of generality, a free fall disk in water is considered,whose motion is governed by the Andersen-Pesavento-Wang model. We illustrate thatconvergent and reliable trajectories of a chaotic free fall disk in a long enough intervalof time can be obtained by means of the CNS, but different traditional algorithms indouble precision give disparate trajectories. Besides, unlike the traditional algorithmsin double precision, the CNS can predict the accurate posture of the free fall disk nearthe vicinity of the bifurcation point of some physical parameters in a long duration.Therefore, the CNS can provide reliable prediction of chaotic systems in a long enoughinterval of time.
Key words free fall; chaos; numerical noises; re-entry
AMS Subject Classifications
Free fall motion widely exists in nature and industry. For example, falling leavesand feathers are the common phenomena for people to see in daily life. A very im-portant application is the spacecraft re-entry, since some objects are massive in sizeand high-temperature resistant, and the re-entry may cause structural, environmentaland safety issues on the Earth’s surface. Recently, the re-entry of China’s Tiangong-1spacecraft draws much attention to accurately predicting the uncontrolled trajecto-ries [1]. Six degrees of freedom (DoF) orbit model is the standard model describingthis problem [2, 3]. Nevertheless, accurately modeling a free-falling process is quite ∗ Email address for correspondence: lijing @sjtu.edu.cn † Email address for correspondence: [email protected] challenging by the 6 DoF model. The chaotic features inherent of this system stronglydepends on initial conditions (SDIC) [4] and, hence, is a great threat to numericalcalculation.Unfortunately, traditional numerical methods are naturally not “clean”. More orless noises (i.e., truncation errors and round-off errors) are yielded during calculation.These noises will increase exponentially in chaotic cases, due to the SDIC, which wasfirst discovered by Poincar´e [5] in 1890 and developed by Lorenz [6] in 1963, i.e., theso-called “butterfly effect”. Thus, for a chaotic dynamic system, a tiny variation ofthe initial condition can result in significant differences between numerical trajectoriesafter a long time simulation [7, 8]. Furthermore, Lorenz [9] reported that it is alsosensitive to numerical algorithm. He found that the (maximum) Lyapunov exponentalters between negative and positive values even when the time step is very small.Teixeira et al. [10] further investigated the time step sensitivity of three non-linearatmospheric models utilizing traditional algorithms in double precision. They madea somewhat pessimistic conclusion that “for chaotic systems, numerical convergencecannot be guaranteed forever.”In fact, the free fall problem itself has been experiencing a long and complicatedhistory until a widely accepted theory emerges. The qualitative discussion of it candate back to Maxwell [11]. At that time, little was known about the nature of thetransitions between different modes. Many simple shapes, such as disks, cylinders,polygons ,cones and even particles have been studied via experiments or numericalsimulations of Navier-Stokes equations, and complex falling modes were discovered,including tumbling, fluttering, steady, and chaotic postures [12–18]. Among theseshapes, the disk is the most well-studied subtopic in this field. A few analytical modelsdeveloped by researchers greatly enhanced the understanding of free fall motion influids and promised the possibility to predict the falling without solving the Navier-Stokes equations which are notorious for the high computational cost. Kuznetsov [19]organized a comprehensive summary on that topic.A direct way to reveal the rules is the experiment. As for the free fall motion ofdisks, it mainly focuses on the relationships between physical parameters and fallingmodes. Willmarth et al. designed a series of experiments and measured a phasediagram of fluttering, tumbling and steady descent according to six related physicalparameters [20]. By dimensional analysis, three similarities were obtained, and, withsmall ratio between the thickness and the diameter of the disk, falling modes onlydepend on the dimensionless moment of inertia and the Reynolds number. Field etal. further found a chaotic transition region between fluttering and tumbling [21].Zhong et al. recently conducted the most comprehensive experimental research onfree-fall thin disks. They closely studied the relationships between fluttering (zigzagin their words) free-fall motion and Reynolds number. They found a critical Reynoldsnumber Re cr ≈ Re below Re cr while invariant beyond Re cr [22]. Zhong et al also researched the mechanism howtwo-dimensional fluttering modes transform to three-dimensional spiral modes [23,24].Their experimental results have been numerically confirmed with immersed boundary-lattice Boltzmann flux solver [25, 26]. Based on the three-dimensional spiral modes,Kim et al. further studied the free-fall motion of a pair of rigidly linked disks. Theydiscovered a mutative falling mode with two disks falling in helical and conical motions[27]. Recently, Lee et al also studied the bristled disk where they found that thebristled structure of disks could strengthen the stability of free-fall motion [28].Naturally, analytical models were built to explain the various modes and the bi-furcation. Kirchhoff made a pioneering distribution by deriving finite-dimensionalgoverning equations [29], called Kirchhoff equations, based on an important fact thatvelocities of the solid body moving in an ideal incompressible fluid can be decou-pled from the field equations of the fluid itself. Nevertheless, Kirchhoff equationsare only related to ideal fluid corresponding to conservative systems without consid-ering dissipation. It can describe the steady fall regimes and to sustain regular orchaotic oscillations and rotations [30, 31], but still far away from the real motion.Inherited from Kirchhoff equations, many researchers tried to introduce appropriateamendments to build a better model, especially on replicating the fluttering, tum-bling, stable and chaotic falling modes and also manifesting the bifurcation structurebetween fluttering and tumbling. There are two models with significant importance,and we introduce them briefly. The first is the Tanabe–Kaneko model [32]. This modelfirstly introduces the Joukouvsky theorem to introduce the effect of circulation andimplements a sign function to relate the lift term with the kinematic information of adisk. Tanabe and Kaneko explained that the introduction of Joukouvsky theorem andexpressed the circulation with a sign function may give rise to complex dynamics andchaos during falling in a fluid due to gravity. Though there was some incorrectnessof the Tanabe–Kaneko formulation, including that they omitted the effect of addedmass and Archimedean buoyancy and there was some contradiction between the co-efficients, which was criticized after publication [33, 34], the Tanabe-Kaneko modelqualitatively gives a reasonable picture of possible regimes of complex dynamics fora disk falling in a fluid. Andersen, Pesavento, and Wang proposed a more elaboratemodel to describe the fall of a flat disk or a body with an elliptic profile in a fluidthrough a finite-dimensional model [35, 36]. The Andersen-Pesavento-Wang modelconsiders the problems in Tanabe–Kaneko model and is more coherent with the ex-perimental results and the numerical results from the direct numerical simulation ofthe Navier-Stokes equations.However, Andersen et al. paid more attention on the phenomenology of theirmodel but lacked close research in the simulation part. Without reliable numericalsimulation near the heteroclinic bifurcation region, they only mentioned the possibilitythat the chaotic transition region found in experiments could be the heteroclinic bifur-cation in their model. Moreover, it turned out in our simulation that the model cannotprovide meaningful prediction in chaos by traditional numerical methods. Thus, weaim to conduct reliable numerical simulations to closely study the chaotic cases andheteroclinic bifurcation regions. The motivation of this paper is to implement a rad-ical numerical strategy to empower the Andersen-Pesavento-Wang model with theability to accurately predict trajectories in extremely sensitive cases.In the present paper, the impact of numerical noises are eliminated by a novelapproach. Liao [37] suggested a numerical strategy in 2009, namely the “Clean Nu-merical Simulation” (CNS) [38, 39], to overcome the limitations mentioned above oftraditional algorithms in double precision. Employing the CNS, reliable/convergentnumerical simulations of chaotic dynamical systems can be obtained in a controllableinterval of time 0 ≤ t ≤ T c , where T c is called the “critical predictable time”. Com-pared with the traditional validated numerics methods like interval arithmetics [40],the CNS is a practical numerical strategy. The implementation of MP makes it easierto use and computationally cheaper than interval arithmetic while the convergencechecks to determine T c still practically ensure the reliability of computational re-sults. This method has been proved effective to calculate reliable trajectories in manychaotic systems, including Lorenz equation [37], three-body system [42–44], and alsospatio-temporal chaos [45, 46]. For example, by implementing the CNS, Li et al. suc-cessfully found more than 2000 new periodic orbits of the three-body problem whichwas pointed out by Poincar´e [5] as a classic chaotic system. Most of these periodicsolutions are inaccessible by traditional means [42–44], which illustrates the useful-ness of the CNS as a powerful tool for reliable investigation of chaotic systems inphysics with high fidelity. As for the spatio-temporal chaos, Lin et al. [45] used theCNS to control numerical noises smaller than the micro-level thermal fluctuations, bywhich it was proved that the inherent micro-level thermal fluctuations are the rootsource of macroscopic randomness of Rayleigh-B´ernard turbulent convection flows.Hu et al. [46] developed a more efficient algorithm of the CNS to simulate the one-dimensional complex Ginzburg-Landau equation(CGLE). It further exhibits that CNSmethod can accurately maintain both the statistical features of spatio-temporal sys-tems and the symmetric characteristics of the solutions in which traditional numericaltreatment has failed.We organize this manuscript as follows. The Andersen-Pesavento-Wang modeland the CNS strategy are briefly introduced in section 2. In the third section, wedemonstrate the sensitivity of the free fall problem to numerical noises and the ad-vantage of the CNS method by comprehensive comparisons from both chaotic andperiodic simulations. Finally, we close with discussions and concluding remarks in thelast section. As shown in Fig. 1, the Andersen-Pesavento-Wang model is a comprehensive analyticalmodel to predict the trajectories of a freely falling two-dimensional disk driven bygravity: I ∗ ˙ V x ′ = ( I ∗ + 1) ˙ θV y ′ − Γ V y ′ − sin θ − F x ′ , ( I ∗ + 1) ˙ V y ′ = − I ∗ ˙ θV x ′ + Γ V x ′ − cos θ − F y ′ , (cid:18) I ∗ + 12 (cid:19) ¨ θ = − V x ′ V y ′ − τ, (1) Γ y' x' g θ xy Figure 1: The local reference frame ( x ′ , y ′ ) fixed with the disk and the global referenceframe ( x, y ) of a freely falling two-dimensional desk, where θ denotes the rotation angleof the disk, g is the acceleration due to gravity, Γ is the circulation, respectively.with the coordinate transformation ( ˙ x = V x ′ cos θ − V y ′ sin θ, ˙ y = V x ′ sin θ + V y ′ cos θ, (2)where the dot denotes the derivative with respect to the time t , ( x ′ , y ′ ) is the localcoordinate fixed with the disk, ( x, y ) is the global (inertia) coordinate, ( V x ′ , V y ′ ) isthe velocity of disk in the local coordinate, θ is the rotation angle of the disk, thecirculation Γ is given by Γ = 2 π − C T V x ′ V y ′ q V x ′ + V y ′ + C R ˙ θ , (3)the viscous forces ( F x ′ , F y ′ ) and torque τ are given by( F x ′ , F y ′ ) = 1 π A − B V x ′ − V y ′ V x ′ + V y ′ ! q V x ′ + V y ′ ( V x ′ , V y ′ ) , τ = (cid:16) µ + µ | ˙ θ | (cid:17) ˙ θ, (4)respectively. All variables are dimensionless. This model has seven dimensionlessparameters I ∗ , C T , C R , A, B, µ , µ , where I ∗ = ( ρ s b ) / ( ρ f a ) with ρ s , ρ f being the den-sities of disk and fluid and a, b the length of the semi-major and semi-minor axis ofthe elliptical disk, other parameters are related with the geometry of the falling disk.For an elliptical disk, [35, 36] gave C T = 1 . , C R = π, A = 1 . , B = 1 . , µ = 0 . , µ = 0 . C T , C R , A, B, µ , µ . In this paper we only adjustthe parameter I ∗ to produce different falling modes of the disk. Generally, the CNS is able to obtain long-term reliable results thanks to its strategyto control the “numerical noises”, say, decrease the truncation errors to a requiredlevel by implementing numerical schemes with extremely high precision and controlthe round-off errors within a required range with all physical/numerical variables andparameters in multiple-precision.Truncation errors come from the discretization of continuous systems. The nu-merical methods have the following general form: f ( t + h ) = f ( t ) + h × RHS ( t ) (6)where RHS ( t ) denotes the right hand side. It varies according to the numericalmethods. For N-th order Runge-Kutta family method, N-step multi-step method andN-th order Taylor series method, the right hand side have the general forms: RHS ( t ) = N X i =1 k i f ( t k i ) + O ( h N ) RHS ( t ) = N X i =1 k i f ( t i − N − ) + O ( h N ) RHS ( t ) = N X i =1 f ( i ) ( t ) i ! h i − + O ( h N ) (7) O ( h N ) is the order of global truncation errors. The CNS is aimed to reduce thetruncation errors so small that it would not damage the long term prediction, either byreducing time steps h or increasing N with high order methods. The round-off errorsinvariably arise with data are stored in computers in finite digits. We implementedthe multiple-precision libraries (MP, called the MPFR library in the C language) [41]to also reduce the round-off errors to a small enough level.By that, the numerical noises of the simulation are controlled arbitrarily small. Todetermine the critical predictable time T c , one would conduct an additional simulationwith even smaller numerical noises. In a temporal dynamic system, we assume thatnumerical noises grow exponentially within an interval of time t ∈ [0 , T c ]: E ( t ) = E exp( κt ) , t ∈ [0 , T c ] (8)where the constant κ > E . It denotes the level of initial noises (i.e.,truncation and round-off errors), and E ( t ) is the level of evolving noises of numericalsimulation. Theoretically, a critical level of noise E c determines the critical predictabletime T c by the equation: E c = E exp ( κT c ) (9)It is obvious to tell from the above equation that the smaller initial noise E promises alonger T c . Since the true orbits are impossible to get, the CNS implements a practicalNo. angle of attack rotation ( x, y, θ, V x ′ , V y ′ , ˙ θ )1 (0 , , , , . , √ (0 , , , , . , √ (0 , , , , . , √ √ (0 , , , , . , T c . Let Φ( t ) be a numerical simulation reliable in t ∈ [0 , T c ]with the initial noise E , and Φ ′ ( t ) be another simulation (with the same physicalparameters and the same initial conditions) in t ∈ [0 , T c ] with the initial noise E ′ that is several orders of magnitude smaller than E . According to the hypothesis thatnumerical noises grow exponentially, there sure is T ′ c > T c and Φ ′ ( t ) in t ∈ [0 , T c ]should be much closer to the true orbit than Φ( t ). Therefore, we use the Φ ′ ( t ), abetter simulation with less numerical noises, to decide the T c of Φ( t ). After obtainingthe T c of Φ( t ), we can name safely that Φ( t ) is “clean” numerical simulation (CNS)in t ∈ [0 , T c ]. Those, as mentioned earlier, provide us a heuristic explanation of thestrategy of the CNS. The CNS also applies to non-hyperbolic chaotic systems.We implement the strategy above to obtain the CNS results of the Andersen-Pesavento-Wang model. Given the discontinuous term in equation (4), we use a fixedstep fourth order Runge-Kutta method with a strict time-step in multiple precision.To determine T c , an additional simulation with even smaller time steps and more digitsto store data in computer to guarantee that the extra simulation contains even lessnumerical noises. For the formal definition of T c , we follow the form in the paper [37].With formula that (cid:12)(cid:12)(cid:12) − u u (cid:12)(cid:12)(cid:12) > δ, at t = T c where δ = 1% in this paper, we determinethe exact T c of the Andersen-Pesavento-Wang model. In the following manuscript,we would like to demonstrate where and how numerical noises can harm the fidelityof trajectory prediction in the Andersen-Pesavento-Wang model. It was regraded that chaotic free-fall motion of disks in the water is long-term un-predictable by numerical simulation. In this section, we address that the CNS isable to provide long-term reliable prediction. We study the chaotic case with I ∗ =2 . x, y, θ, V x ′ , V y ′ , ˙ θ ) in Table. 1 with their corresponding schematic diagram shown inFig. 2.Traditional methods are powerless as a prediction tool with chaos. In Case 1, we Figure 2: The schematic diagram of the four types of initial conditions considered.The four cases are correspondent to disks that fall from a static state with no angle ofattack, that fall from a static state with an angle of attack, that fall from a rotationalstate with no angle of attack and that fall from a rotational state with an angle ofattack divergence happens within this region
Figure 3: (a) The trajectories of the same initial condition computed by the fourth or-der Runge-Kutta method in double precision with different time steps. All trajectoriesare divergent from each other, making these trajectories of little value as prediction.(b) The trajectories of the same initial condition but computed by the CNS. Theblack dotted trajectory is computed with smaller time steps and data stored in moredigits, which proves the red trajectory is “clean”.computed trajectories by the fourth order Runge-Kutta method in double precisionwith different time steps from 1e − −
7. The trajectories are plotted in Fig. 3, It isfound that all trajectories are divergent from each other after about 250
U T . Thoughtraditional numerical methods can obtain qualitatively correct chaotic trajectories,these trajectories are of little use from the aspect of prediction: it’s impossible to tellwhich one should be used as the predicted trajectory. The CNS, on the other hand,is able to give true trajectories: the trajectory with h = 1e − h = 1e − T c and numericalnoises in case1. We have demonstrated that with h = 1e − K T c T c -log hT c =-42.81log h+112.56T c = 10.95K+96.11 Figure 4: (a) The linear relationships between T c and decimal digits K ; (b) Thelinear relationships between T c and the logarithm of steps.designed to contain only truncation errors: thus all data are stored in 60-digits whilethe time steps are larger than h = 1e −
6. The second set is designed to contain onlyround-off errors: thus the same time step h = 1e − T c we could compute the T c of these cases by comparing the trajectorieswith the “clean” trajectory computed with h = 1e − T c pf the Andersen-Pesavento-Wang modelalso applies to the exponential growth law. There exist linear relationships between T c and the logarithm of time steps h , T c and the digits of data K , respectively. Thequantitative results are given by the following equations: T c ≈ min { . K + 96 . , − .
81 log h + 112 . } (10)which provide a rough estimation of the T c of case1. The exact T c is decided by theminimum of the T c with truncation and round-off errors. Generally, with smaller timesteps h and larger digits K , longer reliable prediction could be obtained. Also it isable to estimate the time steps and digits needed for a certain T c for the purpose ofprediction.In all four cases, the CNS is able to provide convergent prediction of the ex-act falling trajectories of disks. Let us compare the differences between the fallingtrajectories computed by the traditional methods and the CNS. The details of howtrajectories computed by traditional methods diverges from the true trajectories ofall four cases are demonstrated in Fig. 5. The trajectories in red are computed bytraditional methods with h = 1e − θ . Then the phase differencecontinuously increases and finally develops into a different falling path, marking thefailure of prediction. Among the four cases, the divergence happens at different times.The first case that disks fall from a static state with no angle of attack has the latest0Figure 5: Comparison between trajectories computed by traditional methods versusthat computed by the CNS. (a) (b) (c) (d) correspond to case 1 to 4 respectively.divergence, because it takes the longest time for the initial condition to develop intochaos. Except for that, these four cases have the same qualitative phenomenon. In this section, we study the heteroclinic bifurcation region reported by Andersenet al. in their paper on the analysis of transition between fluttering and tumblingstate [36]. They described it as a sharp transition while the bifurcation region aresensitive to noises. In their work, Andersen et al. succeeded in studying the hetero-clinic bifurcation to the precision of 1e −
5. With the help of CNS, we approach theheteroclinic bifurcation region in even higher precision of 1e −
11 and provide reliableprediction which was never reported before. It is worth noting that only the CNS canpredict the falling modes in that high precision as shown in the following example.Given the dynamical characteristics of the Andersen-Pesavento-Wang model, ithas four steady solutions, corresponding to two fixed points where disk falls vertically,1
Figure 6: Two characteristic trajectories near the I ∗ c . The trajectory with I ∗ = 1 . I ∗ = 1 . v x ′ v y ′ θ ˙ θ = ∓ p πA − B π , π = ∓ V π , π (11)and two other fixed points where the face of disk is normal to the direction of motion: v x ′ v y ′ θ ˙ θ = ∓ p πA + B , π = ∓ W , π (12)Andersen et al. discovered that the fluttering modes transform to tumbling modesvia a heteroclinic bifurcation at a specific I ∗ c . That is with I close to I ∗ c , all disks with I ∗ < I ∗ c fall in fluttering modes while disks with I ∗ > I ∗ c fall in tumbling modes,as shown in Fig. 6. So mathematically, I ∗ c is the critical parameter that decidesdisks’ falling modes. Noticing that the periodicity of disks grows exponentially longeras I ∗ approaches I ∗ c . Andersen et al. explained that the phenomenon results fromheteroclinic bifurcation like the famous Silnikov’s phenomenon.With the help of the CNS, it is able to obtain an I ∗ which is extremely close to I ∗ c by a dichotomization process based on the fact that I ∗ c must lie between flutteringand tumbling modes. For example, I ∗ = 1 . I ∗ c withprecision 1e −
11. Its true trajectory is in fluttering modes computed by the CNS,plotted in Fig. 7. The CNS results can predict the falling modes even in extremelyhigh-precision near the heteroclinic bifurcation point.However, the trajectories computed by the traditional methods with differenttime steps (1e −
2, 1e − −
4) behave chaotic as a Computational Chaos (CC)phenomenon, shown in Fig. 7. From the aspect of prediction, that means traditional2 h Figure 7: The computational chaos resulted from numerical noises. The black lineis the trajectory computed by the CNS, which falls in the fluttering modes. On theother hand, the trajectory computed by the fourth order Runge-Kutta method indouble precision is neither fluttering nor tumbling. Instead they are in a special typeof chaos resulted.methods cannot predict the falling modes of trajectories near bifurcation point. TheCC addresses a loose similarity between heteroclinic bifurcation region and chaotictransition region [21]. However, noting that when I ∗ is close to I ∗ c , the periods areelongated. It is easy to find that the “chaotic” trajectories in Fig. 7 have the charac-teristics of periodicity, compared with the chaos discussed in the previous section inFig. 3. The obvious differences confirm that the heteroclinic bifurcation region andthe chaotic transition region are different.Considering the special characteristics of the CC here, it is worth for us to dis-tinguish it from the classic CC proposed by Lorenz [47] as a new type. They havetwo main differences. First, this type of CC behaves characteristics of periodicity asmentioned. The uncertainty only happens near the heteroclinic bifurcation point thatdecides the disk would either flutter or tumble in the next period. Therefore, we refersuch chaos as heteroclinic bifurcation chaos. Second, the CC proposed by Lorenz areonly caused by too large time steps and can be eliminated when traditional methodswith smaller time steps are used. On the other hand, the CC found in the Andersen-Pesavento-Wang model is unavoidable when computed by traditional methods. Nomatter what time steps are picked, the CC always happens. That can be explainedby that such CC can be caused both by truncation errors and round-off errors, sotraditional methods is powerless to control these errors small enough to ensure thesimulation “clean”.Given the characteristics of heteroclinic bifurcation chaos, experimental noises,as pointed out by Andersen et al. [36], could also trigger that heteroclinic bifur-cation chaos in that region. We can only conclude the heteroclinic bifurcation ofAndersen-Pesavento-Wang model is actually different from the chaotic transition re-gion. However, we cannot preclude the existence of heteroclinic bifurcation chaos3discovered here though it is strange that it was never reported before in any experi-ments of disks falling in the water. Whether this type of falling modes exists or theAndersen-Pesavento-Wang model is not correct as describing the transition betweenfluttering and tumbling is still an open question worth further study. In this paper, we show that the uncertainty of the numerical simulation of free fallmotion in the water can be overcome by a novel but powerful tool, the CNS. In thechaotic cases, numerical noises grow exponentially. Hence, even though they are assmall as 1E −
16, they are amplified exponentially. It is impossible to obtain long-termreliable prediction of chaotic free fall motion in the water by means of traditionalnumerical algorithms in double precision unless the CNS is applied. Besides, in theheteroclinic bifurcation region, numerical noises can accumulate during the long periodand destroy the bifurcation structure, behaving like a new type of computationalchaos. Under this circumstance, traditional methods cannot even predict the disk’sfalling modes. Although numerical noises can ruin the validity of this problem, a goodway to control them at a small enough level can still give us a trustful and reliableresult in a long enough interval of time.In the past, the role of numerical noises was usually neglected. It is magnificentthat we implement physical models to simulate and predict versatile phenomena inthe nature, but it is not recommended to draw conclusions without awareness ofthe fidelity of the simulation. Take the re-entry problem as an example. In thiskind of big engineering project, no doubt a better prediction of the landing spot willbring tremendous economic and safety benefits. Hence, this work shows people acorrect direction to radically solve chaotic problems by numerical methods, and it isof practical importance.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (No.91752104) and the project “Development of large-scale spacecraft flight and reentrysurveillance and prediction system” of manned space engineering technology (2018-14).
Data Availability Statement
The data that support the findings of this study are available from the correspondingauthor upon reasonable request.4