Dynamic State Analysis of a Driven Magnetic Pendulum using Ordinal Partition Networks and Topological Data Analysis
DDynamic State Analysis of a Driven Magnetic Pendulum usingOrdinal Partition Networks and Topological Data Analysis
Audun Myers and Firas A. Khasawneh Department of Mechanical Engineering, Michigan State University. Email: [email protected]
Abstract
The use of complex networks for time series analysis hasrecently shown to be useful as a tool for detecting dy-namic state changes for a wide variety of applications. Inthis work, we implement the commonly used ordinal par-tition network to transform a time series into a networkfor detecting these state changes for the simple magneticpendulum. The time series that we used are obtained ex-perimentally from a base-excited magnetic pendulum ap-paratus, and numerically from the corresponding govern-ing equations. The magnetic pendulum provides a rel-atively simple, non-linear example demonstrating transi-tions from periodic to chaotic motion with the variation ofsystem parameters. For our method, we implement persis-tent homology, a shape measuring tool from TopologicalData Analysis (TDA), to summarize the shape of the re-sulting ordinal partition networks as a tool for detectingstate changes. We show that this network analysis toolprovides a clear distinction between periodic and chaotictime series. Another contribution of this work is the suc-cessful application of the networks-TDA pipeline, for thefirst time, to signals from non-autonomous nonlinear sys-tems. This opens the door for our approach to be usedas an automatic design tool for studying the effect of de-sign parameters on the resulting system response. Otheruses of this approach include fault detection from sensorsignals in a wide variety of engineering operations.
The progress in modern manufacturing operations hasgiven designers unprecedented freedom in conceptualiz-ing structural and machine components. However, oneconcern in going from a design idea to a functional prod-uct is whether the manufactured component or assemblywill behave unexpectedly under certain operation condi-tions. In order to address this concern, it is necessary to ei-ther write predictive models and simulate different scenar- ios, or manufacture prototypes and collect field data. If thedesign parameter space is large, it is possible to acquire alarge amount of data that cannot be manually analyzed byhuman operators. Therefore, it is necessary to develop andutilize analysis tools that can autonomously analyze theresulting data and classify the resulting system behavior,for example, as erratic/chaotic, or regular/periodic. Thesetools can also prove useful in the context of fault detec-tion after the part or the assembly are put into operation toguard against unexpected failures using sensory signals inthe form of time series.In this work we describe an approach for characteriz-ing a time series as chaotic or periodic based on the struc-ture of its complex network embedding. The embeddingof the time series into a network or a graph is achievedusing ordinal partitions [14]. Embedding is often neces-sary because the true underlying model of the data is un-known, and all that is available usually is a time-indexed,observed quantity such as acceleration or temperature.Therefore, embedding can help identify how the dynam-ics of the system evolve starting with a one dimensionalrecording of data. Currently, perhaps the most populartool for time series analysis is Takens’ embedding [25],where a time series is embedded into an n -dimensionalEuclidean space using a uniform subsampling by someconstant delay τ . Both the embedding dimension n andthe delay τ are parameters that need to be selected using,for example, false nearest neighbors approach [7] and thefirst minimum of the mutual information function [5], re-spectively. When the chosen parameters lead to a success-ful Takens’ embedding, the reconstructed state space canqualitatively be used to infer the underlying dynamics ofthe system that generated the time series. The above ap-proach for time series embedding assumes a deterministictime series or one with little additive noise. Additionally,the resulting embedding does not take into account the or-der of the points in the time series. Practically, Takens’embedding is often used with small n , typically less thanseven or eight, although visualizing any embeddings with1 a r X i v : . [ phy s i c s . d a t a - a n ] A ug > G = ( E , V ) where V are the vertices of the graphand E are its edges. In contrast to Takens’ continuousrepresentation of the data, Graph embedding leads to adiscrete representation which can be more easily visual-ized even for high-dimensional data. The idea is that theshape of the graph can provide information on the under-lying structure of the dynamical system.A commonly used network embedding based on phasespace reconstruction is the recurrence network [4]. Thenodes in a recurrence network are formed by each of theembedded vectors, e.g., from from Takens’ embedding.An edge is added between two nodes if the Euclidean dis-tance between the corresponding embedded vectors is lessthan a user-specified threshold ε ∈ [ , ∞ ) . Although re-currence networks embedding can represent the underly-ing structure of the phase space, it also introduces a vari-able ε that needs to be selected. Khor et al. used a k -nearest neighbor network [9] where ε was replaced withanother user parameter k ∈ Z + . Another option for net-work embedding that we mention here is the visibilitygraph [10, 13]. However, recently, a network embeddingapproach based on ordinal partitions was described [14].This method does not require a distance threshold likerecurrence networks, but rather forms a networks basedon the permutation transitions within a time series. Insection 3.2 below a description of the basic idea of thisapproach is provided in more detail, which is the methodwe chose to work with in this manuscript.Ordinal partitions embedding provides a framework forembedding the time series into a graph; however, the chal-lenge becomes in identifying the system state (periodicor chaotic) using the structure or shape of the resultinggraph. More specifically, we would like to have a quanti-tative measure to classify the visual differences we see inordinal partition embeddings of periodic and chaotic sig-nals, see Fig. 1 for an example.Figure 1: Example ordinal partition networks generatedfrom a (a) periodic time series and (b) chaotic time series.One tool that has been shown to successfully quantifythe shape of ordinal partitions network comes from Topo-logical Data Analysis (TDA). Specifically, in [16], per- sistent homology, the flagship tool from TDA was suc-cessfully used to distinguish between periodic and chaoticsignals. However, previous work on the ordinal partitionand TDA pipeline only considered simulated signals ofautonomous nonlinear systems; therefore, extrapolatingthe validity of the findings to physical systems, especiallywith time dependent forcing, has not been previously in-vestigated.In this work, we study the applicability of the methoddescribed in [16] to detect transitions from periodic tochaotic dynamics in a magnetic, single pendulum underbase excitation,see Fig. 2 and Section 2. This physicalsystem provides rich, non-linear dynamics over a widerange of parameter values and demonstrates both periodicand chaotic dynamics. Further, it ties in with other appli-cations where similar systems are used such as as energyharvesting [2, 11] and mass dampers [12].The paper begins by introducing the experimental setupand model in Section 2. A quick introduction to ordinalpartition networks and persistent homology is then pro-vided in Section 3. In Section 3.3 we introduce point sum-maries for quantifying the shape of the networks downto a single statistic. Then, in Section 4, an example re-sponse from the experimental magnetic pendulum withrelatively complex periodic dynamics is used to demon-strate the functionality of the method for analyzing thetime series through the shape of the resulting networks.Finally, in Section 5 we provide results for the variationof the base excitation amplitude to detect dynamic statetransitions over a wide variety of dynamic responses withvarying complexity. Note : a Computer Aided Design (CAD) model and de-sign document for the pendulum used for the experimen-tal section of this manuscipt is available through
GitHub at https://github.com/Khasawneh-Lab/simple_pendulum .The driven magnetic pendulum is a well known sys-tem to exhibit chaos [22, 26, 8]. Therefore, we designedand built a magnetic pendulum apparatus, and utilized theordinal partition embedding and TDA to characterize thedynamics of the resulting signals.In this section we derive a simplified equation of mo-tion using Lagrange’s approach. The design, manufac-turing, and equipment used for the experiment are alsoexplained. Additionally, we describe our methods for es-timating and measuring the constants that appear in theequation of motion.2 .1 MODEL We begin by deriving the equations of motion for thephysical system shown in Fig. 2. Let the total mass ofthe rotating components be M , the distance from the rota-tion center O to the mass center of the rotating assembly r cm , and the mass moment of inertia of the rotating com-ponents about their mass center be I cm . Further, assumethat the magnetic interactions are well approximated by adipole model with m = m = m representing the magni-tudes of the dipole moment. To develop the equation of Base Excitation
Datum
Figure 2: Rendering of experimental setup in compari-son to reduced model, where b ( t ) = A sin ( ω t ) is the baseexcitation with frequency ω and amplitude A , r cm is theeffective center of mass of the pendulum, d is the mini-mum distance between magnets m = m = m (modeledas dipoles), and (cid:96) is the length of the pendulum.motion, we use Lagrange’s equation (Eq. (9)), so the po-tential energy V , kinetic energy T , and non-conservativemoments R are needed. In this analysis the damping mo-ments and the moments generated from the magnetic in-teraction are treated as non-conservative. The potentialand kinetic energy are defined as T = M | (cid:126) v cm | + I cm ˙ θ , V = − Mgr cm cos ( θ ) , (1)where (cid:126) v cm is the velocity of the mass center given by (cid:126) v cm = r cm ˙ θ [ cos ( θ ) ˆ ε x + sin ( θ ) ˆ ε y ] + A cos ( ω t ) ˆ ε x . (2)In Eq. (2), A cos ( ω t ) is introduced from the base ex-citation b ( t ) = A cos ( ω t ) in the x direction with A as theamplitude and ω as the frequency and ˆ ε x and ˆ ε y are theunit vectors in the x and y directions, respectively.The non-conservative moments are caused by the en-ergy lost to damping. For our analysis, we consider threepossible mechanisms of energy dissipation: Coulombdamping τ c , viscous damping τ v , and quadratic damping τ q . We chose to use all three mechanisms of damping dueto previous work on damping estimation for a pendulumsimilar to the one we used [18]. These three moments aredefined as τ c = µ c sgn ( ˙ θ ) , τ v = µ v ˙ θ , τ q = µ q ˙ θ sgn ( ˙ θ ) , (3)where µ c , µ v , and µ q are the coefficient for Coulomb, vis-cous, and quadratic damping, respectively.To begin the derivation of the torque induced from themagnetic interaction τ m , consider two, in-plane magnetsas shown on the left side of Fig. 3. The red side of themagnet in the figure represents its north-pole. From thisrepresentation, the magnetic force acting on each magnetis calculated as F r = µ o m π r [ c ( φ − α ) c ( φ − β ) − s ( φ − α ) s ( φ − β )] , F φ = µ o m π r [ s ( φ − α − β )] , (4)where m and m are the magnetic moments, µ o is themagnetic permeability of free space, and c ( ∗ ) = sin ( ∗ ) and s ( ∗ ) = sin ( ∗ ) . Equation (4) assumes that the cylindri-cal magnets used in the experiment can be approximatedas a dipole. We later show that this assumption is satis-factory in Fig. 5 of Section 2.3. These magnetic forcesFigure 3: A comparison between a generic, in-planemagnetic model in global coordinates and the equivalentmagnetic forces in the pendulum model F r and F φ (seeEq. (4)).are then adapted to the physical pendulum as shown onthe right side of Fig. 3, with α = π / β = π / − θ .Additionally, φ and r are calculated from θ , d , and (cid:96) from3ig. 2 as φ = π − arcsin (cid:18) (cid:96) r sin ( θ ) (cid:19) , and (5) r = (cid:113) [ (cid:96) sin ( θ )] + [ d + (cid:96) ( − cos ( θ ))] . (6)The moment induced by the magnetic interaction is then τ m = (cid:96) F r cos ( φ − θ ) − (cid:96) F φ sin ( φ − θ ) . (7)Using τ m from Eq. (7) and the non-conservative torquesfrom Eq. (3), R is defined as R = τ c + τ v + τ q + τ m . (8)Finally, the equation of motion for the base-excited mag-netic single pendulum is found by substituting the aboveexpressions into Lagrange’s equation and noting that L = T − V ∂∂ t (cid:18) ∂ L ∂ ˙ θ (cid:19) − ∂ L ∂ θ + R = . (9)Equation (9) was symbolically manipulated to expressit in state space format using Python’s Sympy package.Then, the system was simulated at a frequency of f s = odeint function from the Scipy library.
The setup of the experiment was manufactured by extend-ing the capabilities of a previously manufactured simplependulum [18]. To increase the non-linearity, in-planemagnets on the base as well as at the end of the pendu-lum were added. To assume a permeability of free space µ , any ferromagnetic material within the vicinity was re-moved, which made the use of 3D printed componentscritical. In Fig. 4 an overview of the utilized, 3D-printedcomponents are shown. Specifically, Figs. 4 (a) and (b)show exploded views of the end mass of the pendulum,and the linear stage for controlling the distance d , respec-tively. The magnets used are two, approximately identi-cal, rare-earth (neodymium) N52 permanent magnets witha radius and length of 6.35 mm (1/4").Table 1 provides a list of the item, description, andmanufacturer for all of the experimental equipment usedto collect the rotational data from the magnetic single pen-dulum under base excitation. To estimate the magnetic dipole moment m of the cylin-drical magnets used (see Fig. 4), we performed an exper-iment similar to the one described in [6]. When the dis-tance between the magnets is less than a critical value r c , N52 Magnet10-32 Nylon Nut3D PLA PrintN52 Magnet10-32 Nylon Nut3D PLA Print10-32 Nylon Bolt10-32 Nylon Nut
Figure 4: Manufacturing overview with experimentalsetup. In Fig. (a), an exploded view of the end mass(100% infill 3D printed PLA components) is shown withthe magnet press fit into end of pendulum. In Fig. (b), anexploded view of the linear stage controlling the verticalposition of the lower magnet.Item Description ManufacturerShaker 113 Electro-Seis APSDC Power Supply Model 1761 BK PrecisionAccelerometer Model 352C22 PiezotronicsRotary Encoder UCD-AC005-0413 PositalData Acquisition USB-6356 Nat. Inst.PC OptiPlex 7050 DellTable 1: Equipment used for experimental data collection.modeling the magnets as dipoles can lead to large errorssince the dipole model does not accurately approximatethe repulsive force between the magnets. This distancewas estimated as r c = .
035 m (see Fig. 5). Additionally,in the region where r > r c , the force curve, a function ofscale r − , was fit to the curve to estimate the magneticdipole moment as m = .
85 Cm.The other parameter values as well as their uncertain-ties (when applicable) are provided in Table 2, which arein reference to Fig. 2. Most of these parameters wereeither estimated using
SolidWorks or by multiple directmeasurements.To validate the parameters, an experiment and simula-tion of a free drop of the pendulum are compared. Theresulting angle θ ( t ) is shown in Fig. 6, which shows avery similar response between simulation and experiment.4igure 5: Measured repulsion force as a function of dis-tance compared to theoretical force in Eq. (4) with θ = F theory is based on dipole model witha dipole moment m = .
85 cm, which was estimated us-ing a curve fit to the region where the magnetic thickness T (cid:28) r . Region of poor fit is marked for r < .
035 m.Parameter (units) Value Uncertainty ( ± σ ) d (m) 0.36 0.005 (cid:96) (m) 0.208 0.005 g (m / s ) 9.81 - M (kg) 0.1038 0.005 r cm (m) 0.188 - ω (rad/s) 3 π - µ (Cm) 1 . × − - m (Cm) 0.85 - µ c (-) 0.002540 0.000020 µ v (-) 0.000015 0.000003 µ q (-) 0.000151 0.000020Table 2: Equation of motion parameters to simulated pen-dulum with associated uncertainty.Additionally, the simulation is within the bounds of uncer-tainty of the encoder σ data = ◦ as shown in the zoomedin region of Fig. 6. This section introduces the tools needed to form ordinalpartition networks as well as analyze their shape usingpersistent homology.
Ordinal partitions embedding uses the permutation transi-tions within the time series [14, 27, 20]. Permutations asa time series analysis tool were first popularized by Bandtand Pompe through permutation entropy [1]. However, Figure 6: Free drop test between collect angular positiondata θ data with encoder uncertainty σ data and the simulatedresponse θ sim . As shown in the zoomed-in region, the sim-ulated response is within the bounds of uncertainty of theactual response.this relatively simple statistical summary of entropy doesnot capture any information about the time ordering ofthe permutation transitions. A natural way to then capturethe frequency and ordering of the permutation transitionsis through a complex network or graph. For ordinal par-tition networks, the vertices V are the collection of per-mutations found within the time series x ( t ) and the edges E are formed from the transitions between permutations.To elucidate how these networks are obtained, considerthe example shown in Fig. 7. The example begins withthe simple time series in Fig. 7-(a), which is defined as x ( t ) = sin ( t ) and was sampled at a rate of f s ≈ x = x , x , x , x , . . . , x k ,where k is the total number of samples. In order to definethe permutations, we need to set two parameters: a delay τ and the dimension n . The delay parameter τ representsa uniform subsampling of the time series, while the di-mension n determines the size and the possible number ofthe used permutations. Specifically, a dimension n meansthat there is a total of n ! possible permutations. Note thatthese two terms are synonymous to the ones used in Tak-ens’ embedding; however, there is not yet a theory thatconnects the two sets of terms.We select τ and n using multi-scale permutation en-tropy as suggested in [15, 19]. The formation of thesevectors is shown for our example time series in Fig. 7-(b), where n = τ =
1. Let us now consider (cid:126) s =[ x , x , x ] to demonstrate the permutation assignment.First, an ordinal ranking of (cid:126) s results in x < x < x .This forms the permutation of form π from s as shownin Fig. 7-(c). Continuing this permutation assignmentfor the rest of the embedded vectors yields the repeatingsequence of permutations π → π → π → π → π . . . (ignoring repeated permutations) as shown in Fig. 7-(c).This permutation sequence can be represented as a net-work (see fig 7-(d)) where edges are formed from the per-5 ... ... ... ... Figure 7: Example ordinal partition network embedding of a time series x = x , x , x , x , . . . , x k shown in Fig. (a). InFig. (b), subsamples (cid:126) s i of the time series of dimension n = τ . In Fig. (c)the ordinal transformation of the vectors (cid:126) s i into their corresponding permutation π j with j ∈ [ , ] is shown. Figure (d)records the permutations and their transitions in the form of a network where π → π → π → π → π → . . . → π .mutation transitions with the nodes being the visited per-mutations. It should also be mentioned that for permuta-tions of dimension n , there are n ! possible permutations.Therefor, as the dimension increase, the complexity of thetime series is better captured. However, with exceedinglyhigh dimensions ( n > This section briefly describes computing the persistent ho-mology of complex networks; a more detailed descriptionis provided in [16]. It begins by introducing simplicialcomplexes and their filtration, followed homology groupsand persistent homology.
SIMPLICIAL COMPLEXES:
Simplicial complexesare one of the backbones of persistent homology. A sim-plicial complex K is a collection of simplices, where asimplex σ is a collection of vertices from the full set ofvertices V such that σ ⊆ V . In our application the ver-tices that compose the simplicies are the vertices of thegraph itself. The dimension d of a simplex is based on thenumber of vertices in the simplex or as dim ( σ ) = | σ | − d =
0, an edge is d =
1, aface is d =
2, and so on.A filtration is a collection of simplicial complexes suchthat K ⊆ K ⊆ K , . . . , K N , where each simplex is gener-ated at a specific filtration level. Filtrations are usually ac-complished by incrementing a threshold parameter α thatincreases the number of connected vertices, thus formingmore simplices and growing the overall simplicial com-plex. To illustrate a filtration for our application, Fig. 8shows a simple network example with 9 nodes as shown inthe bottom left. At the top of the figure we show the filtra-tion over multiple scales of α . The vertices are connected,i.e., and edge is added, when the shortest (unweighted)distance between them is less or equal to the scale α or d ( u , v ) ≤ α , where u and v are two vertices in the graphand d ( ∗ ) is the shortest path distance. HOMOLOGY:
A homology group can be geometri-cally understood as simple structures of dimension d ,where a point is a d = d =
1, and avoid is d =
2. In this work we will only use loops ( d = PERSISTENT HOMOLOGY:
The main idea of per-sistent homology is to track the formation and collapse ofcertain homology groups throughout the filtration of thesimplicial complex K . We can think of the formation of afeature at a filtration level α B as its birth and the collapseat a filtration level α D as its death. The lifetime L of afeature is then calculated as L = α D − α B . Let us now re-turn to our simple network example in Fig. 8. The bottomright of the figure shows the persistence diagram, whichis used to track the births α B and deaths α D of the d = ( α B , α D ) . At afiltration level of α = α =
1. At the nextfiltration level, α =
2, the smaller of the two loops dies,which is tracked in the persistence diagram as the point ( , ) . Then, at the final filtration level α =
3, our largerloop also dies, which is again recorded in the persistencediagram as the point ( , ) . We can then calculate the life-times by taking the difference between the death and birthfiltration levels of the two loops for lifetimes of 1 and 2for the small and big loop, respectively. Our next goal is to develop summary statistics of the re-sulting persistence diagrams of the unweighted and undi-rected ordinal partition networks. This will be done6 etwork Persistence Diag. D e a t h Birth
Figure 8: An example filtration of α ∈ [ , , , ] showingthe nodes when α =
0, both loop structures being bornat α = α =
2, and the death ofthe larger loop at α =
3. This filtration and the associatedbirths and deaths are recorded in a persistence diagram,which summarizes the loops in R with the coordinates ofa feature as ( α birth , α death ) .through two statistics: the periodicity score and persistententropy. PERIODICITY SCORE:
The first summary statisticwe develop is the periodicity score, which summarizeshow periodic a network is based on a comparison to anunweighted cycle graph G (cid:48) with n vertices. If we are us-ing the distance metric of the shortest path with an un-weighted graph, then all loops will form at α B = α D = (cid:100) n (cid:101) . This results in the persis-tence diagram D (cid:48) from G (cid:48) with exactly one point with alifetime of L n = maxpers ( D (cid:48) ) = (cid:108) n (cid:109) − . (10)Let us now assume we are given another unweightedgraph from our ordinal partition network G with n ver-tices. This results in the persistence diagram D , where themaximum lifetime of D is used to calculate the network’speriodicity score as P ( D ) = − maxpers ( D ) L n . (11)This peridoicity score is similar in nature to that devel-oped in [17], but applied to unweighted networks. Addi-tionally, it is normalized in such a way that P ( D ) ∈ [ , ] ,with P ( D ) = G is a cycle graph. NORMALIZED PERSISTENT ENTROPY:
Persis-tent entropy was first developed by Chintakunta et al. [3]as an implementation of the original definition of infor-mation entropy by Shannon [21]. Persistent entropy is calculated as the entropy of the lifetimes from a persis-tence diagram. This summary statistic is defined as E ( D ) = − ∑ x ∈ D pers ( x ) L ( D ) log (cid:18) pers ( x ) L ( D ) (cid:19) , (12)where L ( D ) = ∑ x ∈ D pers ( x ) is the sum of lifetimes ofpoints in the diagram. To make it possible to make com-parisons across multiple persistence diagrams, we nor-malize E according to E (cid:48) ( D ) = E ( D ) log (cid:0) L ( D )) . (13) To demonstrate the method, we will be using a time se-ries obtained from the angular position θ ( t ) of the mag-netic pendulum experiment shown in Fig. 2 with base ex-citation amplitude A = .
08 m and frequency ω = . [ , , , , ] . Additionally, a his-togram is used to show the lifetime multiplicity, i.e., howmany points are overlaid in each location of the persis-tence diagram. The periodicity score was calculated as P ( D ) ≈ .
61 and the persistent entropy was calculated as E (cid:48) ( D ) ≈ .
45 using the lifetimes in Fig. 9-(f).To make a fair comparison, the same process as shownin Fig. 9 is applied to a time series generated from abase excitation with A = .
085 and frequency ω = . Hz ,which results in a chaotic response. The resulting net-work from the permutation sequence is shown in Fig. 10-(a). It is clear that the network from the chaotic timeseries shows significantly more loops with, in general,smaller loop sizes. The size and quantity of these loopsare shown in the persistence diagram of the network withthe lifetimes (with multiplicity) shown in Fig. 10-(b) and(c), respectively. The periodicity score was calculated as P ( D ) ≈ .
95 and the persistent entropy was calculated as E (cid:48) ( D ) ≈ . A of the base excitation.7 t − θ ( t ) (a)0 1000 2000 3000 4000 i π i (b)(c) 0 500204060 (d) 0200 5 10Birth0510 D e a t h (e) Count L i f e t i m e (f) Figure 9: Example of method applied to experimental datawith a periodic response Fig. (a). In Fig. (b) the sequenceof permutations are shown for n = To show that the persistence diagrams from ordinal parti-tion networks can distinguish periodic from chaotic timeseries over a wide range of parameters, we use a bifurca-tion analysis. This was done by simulating the magneticpendulum with the variation of the base excitation ampli-tude A from 0.001 meters to 0.025 m by increments of5 × − m with the frequency held constant at ω = π rad/s. At each amplitude, a time series was simulated for400 seconds at a sampling frequency of 200 Hz, whereonly the last 100 seconds were used to avoid the transientresponse. The maxima and minima from each time se-ries were recorded as a qualitative method for detectingdynamic state changes. These extrema are shown in thetop figure of Fig. 11 with the red and blue data points rep-resenting maxima and minima, respectively. Moreover,regions with a periodic response are highlighted (lightgreen). In addition to the local extrema, the periodicityscore P ( D ) and persistent entropy E (cid:48) ( D ) were calcu-lated from the resulting persistence diagrams of the or- (a)0 1 2 3 4 5Birth024 D e a t h (b) Count L i f e t i m e (c) Figure 10: Example of method applied to experimentaldata with a chaotic response Fig. (a). In Fig. (b) the se-quence of permutations are shown for n = P ( D ) and E (cid:48) ( D ) , respectively. Addition-ally, for the periodic responses, the complexity is capturedby the value of each score. This paper described a novel method for analyzing a timeseries from a mechanical system through ordinal partitionnetworks and TDA. The example that we designed andbuilt to experimentally validate the developed approachis a magnetic simple pendulum with base excitation. Inaddition to the experimental model, we also derive thegoverning differential equation and fit the correspondingparameters. The generated time series for the physical ex-periment and the numerical simulation of the model werethen analyzed using ordinal partition networks and per-sistent homology from TDA. This was done by showinghow a time series can be transformed into an ordinal par-tition networks, which captures a summary of the phasespace reconstruction through the permutation transitions.As shown in Fig. 1, ordinal partition networks of periodictime series result in a relatively simple structure, whilethose from a chaotic response have an irregular shape.8igure 11: Bifurcation analysis of magnetic pendulumthrough numeric simulations with variation of base ex-citation amplitude A ∈ [ . , . ] meters with a stepsize of 5 × − meters. The top figure shows the lo-cal extrema in the generated time series with red beingmaxima and blue being minima. The middle and lowerfigures shows the periodicity score and normalized per-sistent entropy, respectively. Additionally, regions whereboth P ( D ) < . E (cid:48) ( D ) < . P ( D ) and 0.75 for persis-tent entropy E (cid:48) ( D ) successfully separated periodic fromchaotic time series with a value above the threshold signi-fying a chaotic response. We remark that in contrast to thework in [16], this paper represents the first application ofthe described approach to time series obtained from non-autonomous systems. Therefore, we believe that one of itsstrengths in contrast to the delay-reconstruction approachis that it does not require any special embedding proce-dures for forced systems, see [23, 24]. ACKNOWLEDGMENTS
This material is based upon work supported by the Na-tional Science Foundation under grant nos. CMMI-1759823 and DMS-1759824 with PI FAK.
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