Fractional Fourier detection of Lévy Flights: application to Hamiltonian chaotic trajectories
FFractional Fourier detection of L´evy Flights: application toHamiltonian chaotic trajectories
Fran¸coise Briolle , † , Xavier Leoncini , Benjamin Ricaud CReA, BA 701 Salon de Provence 13300, France. Centre de Physique Th´eorique, CNRS-Aix-Marseille Universit´e,Campus de Luminy, Case 907, F-13288 Marseille cedex 9 France. Institute of Electrical Engineering, EPFL,CH-1015 Lausanne, Switzerland.
October 17, 2018
Abstract
A signal processing method designed for the detection of linear (coherent) behaviorsamong random fluctuations is presented. It is dedicated to the study of data recorded fromnonlinear physical systems. More precisely the method is suited for signals having chaoticvariations and sporadically appearing regular linear patterns, possibly impaired by noise.We use time-frequency techniques and the Fractional Fourier transform in order to make itrobust and easily implementable. The method is illustrated with an example of application:the analysis of chaotic trajectories of advected passive particles. The signal has a chaoticbehavior and encounter L´evy flights (straight lines). The method is able to detect andquantify these ballistic transport regions, even in noisy situations.
The analysis of chaotic signals and the detection of particular patterns inside them are impor-tant issues for physics and nonlinear science. The presence of special patterns such as intermittentdeterministic behaviors reveal important information on a given physical system. We propose inthis work a signal processing method able to detect regular behaviors occurring in chaotic signals.To demonstrate its efficiency we apply it to signals where L´evy flights occur: particles displayintermittent behavior with almost random motion succeeded with periods of ballistic motion.This method possesses several key properties required for the study of experimental data. It isrobust, not influenced by the nature of the random fluctuations of the signal nor by a reasonableamount of noise which may be present all over the signal (due to experimental measurements).Secondly, it relies on the Fractional Fourier transform. Several numerical implementations ofthis transform are available among the scientific community which makes the method relativelyeasy to use for non-expert in signal processing. In addition, this transform can be implementedusing fast algorithms.The robustness of our method relies on an uncertainty principle which is reminiscent ofquantum mechanics. It can be shown that one can not measure exactly both frequency and † Corresponding author, email address: [email protected] a r X i v : . [ n li n . C D ] O c t ime of a given signal. We use this latter relation to our advantage. Through an elementarytransformation we turn random fluctuations of the signal amplitude into random fluctuationsof the frequency of a new signal. When these frequencies are rapidly varying, case of randombehavior or noise in the signal, the uncertainty principle makes it impossible to have preciseinformation on these variations. In the meantime, coherent behavior is emphasized since it isless fluctuating. As a consequence it eases the detection process and makes it more robust.This work follows the preliminary results presented in [7] where the use of the uncertaintyprinciple was first stated. We focus here on the signal processing method and integrate it intothe more general framework of the Fractional Fourier transform.In section 2 the signal processing method is presented and in section 3, we present an illustra-tion of the method with its application to simulated data from a physical phenomenon: advectedparticles composed of random motion and L´evy flights. The technique presented here is dedicated to the analysis of signals being made of two ingredients: • parts with random fluctuations, e.g. (fractional) Brownian motions, Gaussian or othertypes of noises. • parts with a linear behavior with respect to the variable, often embedded with a reasonableamount of noise (”reasonable” will be made precise in the following).This kind of intermittent signals is typically found in nonlinear physics experiments, for instancein fluid mechanics at the transitions between regular and chaotic/turbulent regimes. An illus-tration of such signal is shown on Fig. 1 (left). Several regions can be distinguished: randomfluctuation zones associated to a Brownian motion and some linear regions (of different lengthand slope) corresponding to a different behavior (L´evy flights). Note that the linear parts maycontain small fluctuations. Our technique is able to detect these linear parts, even embedded innoise, and to measure their length and slope. In order to follow a rapidly varying signal the measurements must be precise both in the vari-able value and in the measured quantity depending on it. In some configurations, where theuncertainty principle holds, this is not possible. This principle prevents for example the preciseevaluation of the frequency of an oscillating signal when this frequency is evolving with time(non-stationary signal). It is often a problem in physics but we propose here to use it to ourbenefit: we need to emphasize the low fluctuation components of our signals among the randomvariations. For that, we turn our signal into a time-frequency measurement problem.The first step of our analysis is to interpret the signal s , a vector of R N , depending on thevariable t ∈ [1 , , · · · , N ], as the phase derivative (the fluctuation of the “frequency component“)of a new signal S depending on time t . The oscillating signal S is made of an single non-stationaryfrequency component in the following way: S ( t ) = e iϕ ( t ) , (1)where ϕ is related to the signal s by: ϕ ( t ) = t (cid:88) τ =1 s ( τ ) . (2)2n order to see the image of s through this transformation, we compute the short-time Fouriertransform of S : V S ( t, f ) = N (cid:88) τ =1 S ( τ ) e iπfτ g ( τ − t ) , with a Gaussian window g ( τ ) = exp( − τ ). The modulus of this representation V S of the function S gives the spectrogram (see e.g. [1] for more details on time-frequency techniques). As anexample, for s given on Fig. 1 (right), |V S | is plotted on Fig. 1 (left). One can guess the signal − − − s F r equen cy ( * F s ) Time (nb samples)0 200 400 600 800 1000 − Figure 1: Left: tracer trajectory s with fluctuating regions and linear regions (L´evy flights).Right: spectrogram of S (absolute value of the short-time Fourier transform of S ). Darkerregions are associated to high values of |V S | . s from V S but important differences can be seen. First the thin line of the graph of s is nowa thick line in the time-frequency plane. Secondly, regions of high fluctuations appear blurredand diffuse. Indeed, the short-time Fourier transform can be seen as a convolution betweenthe modulated signal S ( τ ) e iπfτ and a Gaussian window, which has a blurring action. As aconsequence random behavior is blurred even more, spread over a neighborhood zone, whereaslinear parts remain relatively sharp. We now turn to the method able to detect and quantifythese time-frequency patterns. For the detection of linear behavior in chaotic signals, we need a method able to detect thesestraight line patterns. In a 2 dimensional image, one would use techniques such as the Houghtransform. In our case, we need a similar tool retrieving straight lines which would appear whena time-frequency decomposition is done (such as the short-time Fourier transform, the Gabortransform, or the Wigner-Ville transform). The appropriate tool for this purpose is based on theFractional Fourier transform. Let us first introduce it. There are two definitions in the discretesetting (where signals are sampled and of finite length) [5], which are not equivalent. The firstone involves Hermite functions and the second one relies on the discretization of the integral.We choose the second definition as it is computationally faster than the first one, since it can bedone with a fast Fourier transform. The discrete Fractional Fourier transform F θ f of a signal f
3f length N is defined as (see [5, 6]): F θ f ( µ ) = A θ √ N e iπ ( µ /N tan θ ) N (cid:88) t =1 e i [ ( π/N tan θ ) t − (2 πµ/N sin θ ) t ] f ( t ) , (3)where A θ = (cid:114) − i tan θ = e − i [ π sgn( θ ) / − θ/ (cid:112) | sin θ | , (4)The angle θ parametrizes the transform such that: for θ = 0 we obtain the signal in time (notransform) and for θ = π/ f . The variable µ is thetime for θ = 0, the frequency for θ = π/ θ . Since t ∈ [1 , N ], µ = 2 πn/N with n ∈ [1 , N ]. A connection can be made with the previousstudy [7] in the following way. Up to a phase factor exp( iµ / θ ) and a normalization constant (cid:113) − i tan θ , the FRFT is the projection of the signal f on a basis of chirp signals: ψ θ,µ ( t ) = e i [ ( π/N tan θ ) t − (2 πµ/N sin θ ) t ] . Notice that applying the FRFT is not strictly equivalent to the calculation done in [7]. Theadditional phase factor has no importance since only the magnitude of the transform is used todetect the presence of L´evy flights. However, the normalization factor A θ which depends on theangle is important when comparing projections at different angles. Hence the present version(which includes it) is more natural and accurate. Also, one has to change µ into ( − µ ).As pointed out in [7] and discussed in [6], numerical instabilities may arise when calculatingthe Fractional Fourier transform for small values of θ . This correspond to the detection process ofL´evy flights with steep slopes. To cope with this problem, the property F θ − π/ F π/ = F θ is used:for θ ∈ [ π/ , π/
4] or θ ∈ [ − π/ , − π/
4] the FRFT is directly computed and for θ ∈ [ − π/ , π/ θ − π/ |F θ f | will increase whenever a chirp of slope1 /N tan θ is present. Hence searching for linear patterns in the time-frequency plane is reducedto looking for peaks of the FRFT in the ( θ, µ ) space. Suppose a peak is present at ( θ , µ ), thenthe slope s of the linear part can be deduced from θ , its shift d from the frequency origin with µ and its length l is proportional to |F θ f ( µ ) | : s = 1 N tan θ , d = µ N sin θ , l = |F θ f ( µ ) | (cid:112) N | tan θ | + 1 . (5)The FRFT can be reversed and it is possible to detect a linear part with slope 1 / tan θ insidethe signal then erase it in the ( θ, µ ) space and to re-synthesize the signal without this linear partby applying a FRFT of angle ( − θ ).In order to detect the different slopes of the L´evy flights it is necessary to apply the FRFTfor different θ regularly spaced. The number of selected θ is fixed by the user depending on howaccurate he wants to be and is independent of the length of the signal N . The fast implementationof the FRFT is of complexity O ( N log N ), hence the overall complexity is of the same order.4 .3 Third step: detection and characterization of L´evy flights On the signal shown in Fig. 1, one can see several L´evy flights (left) which have been turnedinto linear chirps in the frequency-time plane (right). For a specific angle θ m , the FractionalFourier transform will produce one sharp peak corresponding to the presence of a chirp. It isillustrated in Fig. 2 (left), where |F θ m ( µ ) | is plotted. For µ m ∼ |F θ m ( µ m ) | gives evidence that there is a L´evy flight with a particular slope and length given by the Eq. ((5)].This search for maxima is the process that detects linear parts in the time-frequency plane. − µ (nb samples) M agn i t ude ( a . u . ) F r equen cy ( * F s ) Time (nb samples)0 200 400 600 800 1000 − Figure 2: Left: for θ m , signal projections | C ( θ m , µ ) | . Right : short-time Fourier transform of thesignal S , partial reconstruction of S . The longest L´evy flight has been removed.Since the Fractional Fourier transform is invertible, we can re-synthesize the signal back tothe initial representation after setting the values of the transform in red region of Fig. 2 (left) tozero. This result is illustrated on Fig. 2 (right), which represents the short-time Fourier transformof the newly recreated signal S . The largest frequency slope of S has been completely removed,the rest remaining untouched. This shows that indeed the peaks in the FRFT correspond toL´evy flights.In order to detect all the L´evy flights, a search of the peaks in the ( θ, µ ) plane has to bedone. In [7] the suggestion to use a matching pursuit has been proposed. This fits well with theirapproach i.e a projection of the signal on a set of vectors, the chirp signals. The same resultmay be obtained with the Fractional Fourier transform as it is unitary. We have to proceed asfollows. Each time a peak as been detected, say at ( θ m , µ m ), F θ m s ( µ m ) is set to zero as well asa small neighborhood (user defined) around µ m . It is illustrated on Fig. 2 where the red regionis the selected neighborhood to be set to zero. Doing the inverse Fractional Fourier transformwill lead to a signal containing all but the chirp component associated to the peak. This processis to be repeated for all peaks. Let { M i = ( θ i , µ i ) } i be the coordinates of the set of peaks inthe FRFT domain, and let Ω i be a small neighborhood around each M i . Denote by F θ i s | µ i =0 the transformed signal where µ i and its neighborhood have been set to zero. At iteration i , thesuppression of the i th peak in s i is given by: s i +1 = F − θ i ( F θ i s | µ i =0 ) . (6)The effect of this process is shown in Fig. 3 (left), which represents the short-time Fouriertransform of S . The two longest chirps have been erased from the signal, the rest has been5 r equen cy ( * F s ) Time (nb samples)0 200 400 600 800 1000 − − − − A m p li t ude ss2 Figure 3: Left: short-time Fourier transform of the signal S where two L´evy flights have beenremoved. Right : original signal s (black) and partial reconstruction s
2, without L´evy flights(red).preserved. A reconstruction of the signal s , which is obtained by direct time derivative of thephase of S , is plotted on the right in Fig. 3. Remark that this method do not affect the randompart of the signal, only the L´evy flight parts are removed. We then could use the remainingrandom part to perform other analysis. In experimental conditions, measurements are always impaired by noise coming from varioussources. Hence a method dedicated to the analysis of experimental data must still perform itsdetection despite a relatively high level of noise. We show here that even with an additionalGaussian noise, our method is still efficient.The signal to noise ratio (SNR), is defined as:
SN R = 10 . log P s P n = 10 . log (cid:80) s i (cid:80) n i , where n is the noise. We have added a Gaussian noise with different levels of amplitude to oursignal. The method manages to characterize and extract the two main L´evy flights for signalto noise ratios down to 17 dB. Fig. 4 shows an example of noisy signal (left) and the resultingsignal S after the extraction of the two longest linear behaviors (right).We shall now apply our method to a specific example as a proof of concept. Namely we shallconsider data originating from chaotic advection. Before doing so we shall briefly present thephenomenon and the physical context. In this section we briefly discuss the phenomenon of stickiness that occurs in low-dimensionalHamiltonian systems. Stickiness occurs in the vicinity of some islands of regular motion, inducingmemory effects and L´evy flights. In order to be more explicit we shall consider a specific examplewhere this occurs, namely the phenomenon of chaotic advection of passive tracers. In order to6
200 400 600 800 − − − s F r equen cy ( * F s ) Time (nb samples)0 200 400 600 800 1000 − Figure 4: Left: tracer trajectory s with Gaussian noise (SNR ∼
17 dB). Right: short-time Fouriertransform of the signal S where two largest L´evy flights have been detected and removed.generate a specific flow we shall consider the one generated by three vortices (see for instance[32]). For chaotic advection we consider a flow v ( r , t ) of an incompressible fluid ( ∇ · v = 0). Thenotion of a passive particle corresponds to an idealized particle which presence in the fluid hasno impact on the flow. This is usually not true, but if the particle is small enough this can be agood approximation. The particle is then just transported by the fluid and its motion is givenby the equation: ˙ r = v ( r , t ) , (7)where r = ( x, y, z ) corresponds to the passive particle position, and the ˙ to a time derivative.For a two-dimensional flow, Eq. (7) corresponds actually to Hamiltonian equation of motion.Since the flow is incompressible, we can define a stream function which resumes to a scalar field,meaning that v = ∇ ∧ (Ψ z ), where z is the unit vector perpendicular to the considered twodimensional flow. The equations governing the motion of a passive tracer Eq. (7) become˙ x = ∂ Ψ ∂y , ˙ y = − ∂ Ψ ∂x , (8)where the space coordinates ( x, y ) correspond to the canonical conjugate variables of the Hamil-tonian Ψ.When Ψ is time independent, the system is integrable, and particles follow stream lines.When the stream function Ψ becomes time-dependent, we end up with a Hamiltonian systemwith 1 − degrees of freedom. These systems generically exhibit Hamiltonian chaos. Thisphenomenon was dubbed chaotic advection [21, 22, 23]. As a consequence chaotic advection canenhance drastically the mixing properties of the flow, in the sense that mixing induced by chaoticmotion is much more rapid than the one occurring naturally through molecular diffusion. This iseven more important when the flow is laminar [24, 25, 26]. When dealing with mixing in micro-fluid experiments and devices chaotic advection becomes crucial. Indeed since the Reynoldsnumber are usually small, chaotic mixing becomes, de facto, an efficient way to mix. There are7lso numerous domains of physics, displaying chaotic advection-like phenomena, for instance ingeophysical flows or magnetized fusion plasmas [11, 12, 13, 14, 15, 16, 17, 18, 19].In order to test the L´evy detection protocol we established, we will consider data originatingfrom passive particles which have been advected by a two-dimensional flow generated by threepoint vortices. We shall thus briefly recall the notion of a point vortex. As mentioned before moving on to advection, let us discuss briefly the flow generated by pointvortices. For this purpose we start with the equation governing the vorticity of a perfect two-dimensional incompressible flow (the Euler equation): ∂ Ω ∂t + { Ω , Ψ } = 0 , Ω = −∇ Ψ , (9)where {· , ·} corresponds to the Poisson brackets. To get the point vortex dynamics we considera vorticity field given by a superposition of Dirac functions:Ω( r , t ) = N (cid:88) i =1 Γ i δ ( r − r i ( t )) , (10)where, Γ i designates the strength (vorticity) of a point vortex located in the two-dimensionalplane on the point r i ( t ). One then finds that this singular distribution becomes an exact solu-tion (in the weak sense) of the equation (9) when the the N point vortices have a prescribedmotion[27]. To be more specific the dynamics has to reflect the one originating from N -bodyHamiltonian dynamics. And when considering no boundary condition, meaning allowing the flowto live on the infinite plane, the Hamiltonian becomes H = 12 π (cid:88) i>j Γ i Γ j ln | r i − r j | , (11)where the the canonically conjugate variables of the Hamiltonian are Γ i y i and x i , and are thusstrongly related to the actual vortex position r i ( t ) in the plane.When actually computing the equation of motion originating from the Hamiltonian (11), (andthis how they actually make sense and were computed) we can notice that each vortex is movingaccording to the velocity generated by the other vortices but himself. Having the evolution ofthe positions of the point vortices we have as well access to the stream function (the Hamiltoniangoverning passive tracers) Ψ( r , t ) = − π N (cid:88) i =1 Γ i ln | r − r i ( t ) | . (12)Finally we would like to point out that the Hamiltonian of the vortices (11) is invariant bytranslation and by rotation. as a consequence of these symmetries and the associated conservedquantities, the motion of point vortices becomes chaotic when N >
We have discussed chaotic mixing in a flow generated by three point vortices. In these systems,transport can be anomalous. To be more precise, the type of transport is defined by the valueof the characteristic exponent of the evolution of the second moment.In summary, transport is said to be anomalous if it is not diffusive in the sense that (cid:104) X −(cid:104) X (cid:105) (cid:105) ∼ t µ , with µ (cid:54) = 1:1. If µ < µ = 1 transport is Gaussian and we have diffusion.3. If µ > s i ( t ) = (cid:90) t | v i ( τ ) | dτ , (13)where v i ( τ ) is the speed of particle i at time τ . Then to characterize and study transport wecompute the moments M q ( t ) ≡ (cid:104)| s ( t ) − (cid:104) s ( t ) (cid:105)| q (cid:105) , (14)where (cid:104) . . . (cid:105) corresponds to ensemble averaging over different trajectories. In order to characterizethe transport properties we compute the evolution of the different moments, from which weextract a characteristic exponent, M q ( t ) ∼ t µ ( q ) . (15)9ransport properties are found to be super-diffusive and multi-fractal [32], and this is theresults of the memory effects engendered by stickiness: in the vicinity of an island, trajectoriescan stay for for arbitrary large times mimicking the regular trajectories nearby, these islands actthen as pseudo-traps. This stickiness generates a slow decay of correlations (memory effects),which results in anomalous super-diffusive transport.To illustrate the phenomenon the Poincar´e section of passive tracers motion and the stickingregions are represented in Fig. 5, (see [32] for details). Once a trajectory sticks around an island,its length grows almost linearly with time, with an average speed around the island genericallydifferent from the average speed over the chaotic sea. This implies the presence of of L´evyflights in the data corresponding to trajectories lengths. In Fig. 5, four sticking regions havebeen identified, these regions are naturally expected to give rise to four different typical averagespeeds, one therefore expect to identify four different types of L´evy flights in the advected data. We now consider blindly data obtained from the advection of 250 tracers in the point vectorflow described in the previous subsection. That is to say, we analyze with our method 250signals dislaying similar properties as the one presented in section 2. We set up a threshold onthe modulus of the projection coefficients ((3)), in order to select only the most relevant L´evyflights. Similar transport data was was analyzed in [32], with traditional tools and found to beanomalous and super diffusive. As mentionned, the starting point of the anomaly was tracedback to a multi fractal nature of transport linked to stickiness on four different regular regions.One would thus expect four different type of L´evy flights in the data (see Fig. 5).In the present case, the method described in part 2 has been applied to the data set. Ourgoal is to detect the multi-fractal nature of the transport resulting from the sticky islands, whichwould serve as a proof of concept and pave the way to apply the method to numerical andexperimental data. The results are presented in Fig. 6. − −
100 0 100 200 300 400 5000100200300400500600700 velocity du r a t i on o f t he f li gh t − − − nb samples s velocity : 510velocity : − Δ l Δ h Figure 6: Left: duration of the L´evy flights as a function of the velocity. Right: the velocity ofthe main L´evy flight is plotted for each trajectory.For each trajectory, L´evy flights have been detected and characterized by their length in time,10 l , and velocity, ∆ h/ ∆ l = s . The process describe in detail in part 2.3, will give, for each flight,its slope (related to the velocity) and length.The Fig. 6 (left) is an illustration of the duration of the flights as a function of the velocity:four different values have been estimated ( ∼ − , ∼ , ∼
190 and ∼ The first part of the signal processing technique makes use of the uncertainty principle. Thishas a ”dilution effect” on the rapidly varying chaotic parts of the signal while coherent patternsare only slightly affected. This part is critical for the robustness of the detection. Numericalsimulations shows that our technique is indeed extremely robust.The second part of the signal processing technique belongs to the framework of sparsity basedanalyses. We present a transformation (namely the Fractional Fourier transform) which givesa sparse representation of the data of interest: L´evy flights become sharp peaks in the FRFTrepresentation. The key point is that we know the pattern we want to detect and choose thetransformation in consequence.The door is open to further extension and generalization of our method. Suppose one knows apriori the patterns to detect which may not be linear but curved or of some other slowly varyingshape (slowly varying with respect to the chaotic fluctuations). A different representation thanthe FRFT should be used based on the shape information. One may use a basis or a set of vectorsdifferent from the set of linear chirps. Possible alternatives may be found in e.g.[8, 9] where whatthey call ”tomograms” are bases of bended chirps and other more general time-frequency forms,associated to one or more parameters (equivalent of θ in the FRFT case). One may also thinkof Gabor frames made of chirped windows[10]. Once the representation in which the relevantinformation is sparse has been found, the peak detection process remains the same. References [1] Flandrin, P. (1993), Temps-Fr´equence, Herm`es.[2] Almeida, L. B. (1994), The Fractional Fourier transform and time-frequency representa-tions,
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