Homoclinic chaos in the Rössler model
Semyon Malykh, Yuliya Bakhanov, Alexey Kazakov, Krishna Pusuluri, Andrey L. Shilnikov
HHomoclinic chaos in the Rössler model
Homoclinic chaos in the Rössler model
Semyon Malykh, Yuliya Bakhanova, Alexey Kazakov, a) Krishna Pusuluri, and Andrey Shilnikov National Research University Higher School of Economics,25/12 Bolshaya Pecherskaya Ulitsa, 603155 Nizhny Novgorod, Russia. Neuroscience Institute,Georgia State University, 100 Piedmont Ave SE Atlanta, GA 30303, USA. Neuroscience Institute and Department of Mathematics & Statistics,Georgia State University, 100 Piedmont Ave SE Atlanta, GA 30303, USA. (Dated: 1 September 2020)
We study the origin of homoclinic chaos in the classical 3D model proposed by O. Rössler in 1976. Of our particularinterest are the convoluted bifurcations of the Shilnikov saddle-foci and how their synergy determines the global unfold-ing of the model, along with transformations of its chaotic attractors. We apply two computational methods proposed,1D return maps and a symbolic approach specifically tailored to this model, to scrutinize homoclinic bifurcations, aswell as to detect the regions of structurally stable and chaotic dynamics in the parameter space of the Rössler model.
This paper is dedicated to Otto Rössler on the occasionof his 80th anniversary. He, being one of the pioneers inthe chaosland, proposed a number of simple models withchaotic and hyper-chaotic dynamics that became clas-sics in the field of applied dynamical systems. The goalof our paper is to examine and articulate the pivotal roleand interplay of two Shilnikov saddle-foci in the famous3D Rössler model as they shape the topology of the chaoticattractors such as spiral, screw-type without and with fun-nels, and homoclinic, as well as determine their metamor-phoses, existence domains and boundaries. Using the sym-bolic approach we biparametrically sweep its parameterspace to identify and describe periodicity/stability islandswithin chaoticity, as well as to examine in detail a tangledhomoclinic unfolding that invisibly bounds the observabledynamics. The innovative use of 1D return maps gener-ated by solutions of the model lets us quantify the complex-ity of chaotic attractors and provides a universal frame-work for the description of a rich multiplicity of homo-clinic phenomena that the classical Rössler model is noto-rious for. I. INTRODUCTION
The Rössler model is a classic example of multi-faceddeterministic chaos occurring in many low- and high-ordersystems. Its best known feature is the onset of chaotic dy-namics due to a Shilnikov saddle-focus, that begins off witha period-doubling bifurcation cascade. The goal of this paperis to show how homoclinic bifurcations of two saddle-foci de-termine the organization and structure of chaotic attractors inthe Rössler model. We consider its following representation:˙ x = − y − z , ˙ y = x + ay , ˙ z = bx + z ( x − c ) , (1)with x , y , z being the phase variables, and a , c > b = . a) Electronic mail: [email protected] study. The convenience of the representation (1) is that oneequilibrium (EQ) state, called O , of the model is always lo-cated at the origin ( , , ) , while the coordinates of the sec-ond one O are positive and given by ( c − ab , b − c / a , − ( b − c / a )) . One can notice that when c = ab , O merges with O and passes through it as a result of a transcritical saddle-node bifurcation. This is not the case in the other form of theRössler model :˙ x = − y − z , ˙ y = x + ay , ˙ z = b + z ( x − c ) , (2)where the location of neither equilibrium is fixed, and theyemerge in the phase space through a generic saddle-node bi-furcation.To become a saddle-focus, O loses stability through asuper-critical Andronov-Hopf (AH) bifurcation. Note that thetranscritical saddle-node bifurcation makes the other EQ O a saddle-focus from the very beginning. It also undergoes anAndronov-Hopf bifurcation, though sub-critical after a saddleperiodic orbit collapses into O to convert it into a repeller. Wewill proceed with this discussion in the context of homoclinicbifurcations below.The AH bifurcation transforms the stable equilibrium O into a saddle-focus of type the (1,2), i.e., with 1D stableand 2D unstable manifolds due to a single negative eigen-value λ <
0, and a pair of complex-conjugated eigenvalues λ , = α ± i ω with a positive real part ( α > O is of the (2,1) type.The vast majority of the papers on the onset of properties ofchaos occurring in the Rössler model are focused on bifurca-tions related to the primary EQ O , while leaving the role ofthe secondary EQ O in the shadow. We will reveal its contri-bution, through homoclinic bifurcations, to the overall globalchaotic dynamics and its transformations in (1).Let us point out that in the parameter space of interestingdynamics for the model, the c -parameter is greater on the or-der of magnitude than the other parameters, a and b in the ( , ) -range, see details in .The discrepancy in the parameter values implies that theEqs. (1) have two time scales with two slow ( x , y ) - and onefast z -phase variables. Moreover the divergence of the vectorfield generated by Eqs. (1) is estimated by a + x − c or a − c < a r X i v : . [ n li n . C D ] A ug omoclinic chaos in the Rössler model 2 FIG. 1. Period-doubling bifurcations en route to spiral chaos in the 3D phase space of the Rossler model on a pathway at c = .
9. (A) Stableperiodic orbit (PO) at a = .
08 is followed by a period-2 orbit at a = .
25 in (B) and next by a stable period-4 obit at a = .
275 in (C). Period-doubling chaotic attractors at a = .
28 and 0 . O embedded at a = .
35 (F), after the shrinking hole around it fully collapses. around the origin O . This makes the Rössler model stronglydissipative around the origin of the 3D phase space. How-ever, that is not the case near the other equilibrium state O where the divergence of the vector field is small and positive: a ( − b ) ∼ o ( ) . This observation partially explains a slowconvergence to O along its 2D stable manifold as observedfrom Figs. 4B -C . Period-doubling cascade to spiral chaos
Let us recap without excessive details what is well-known:the route to spiral chaos in the Rössler model through a cas-cade of period-doubling bifurcations. It is documented inFig. 1 with fixed c = . a -parameter is increased. Thesupercritical AH-bifurcation makes O the saddle-focus andgives rise to the emergence of a stable periodic orbit (PO), seeFig. 1A. With a further increase in a , this PO loses the stabil-ity inherited by an orbit of period-2 and next by an orbit ofperiod-4 (Figs. 1B-C) through two initial period-doubling bi-furcations. Further steps lead to the onset of a longer PO anda Feigenbaum-type strange attractor discovered by O. Rösslerin the model named after him . Its interaction with a trans-verse cross-section in 3D would look like a Hénon attractor with a recognizable parabola shape as one depicted in Fig. 3.Observe the shrinking hole around O as depicted in Figs. 1D-E indicative that the saddle-focus remains yet isolated. Even-tually, at a ≈ .
35 it collapses and leads to the formation ofthe primary homoclinic loop of O , see Fig. 1F. It is easy toargue that the Shilnikov bifurcation causes a homoclinic ex-plosion (hyperbolic subset) with countably many saddle POsinside the so-called Shilnikov whirlpool formed nearby O in the phase space. It follows from the above arguments concerning the divergence that ∑ i = λ i = c − a > λ − α >
0, or the saddle index ρ = λ / α >
1, is fulfilled. Saddle orbits remain saddle in for-ward time too. This explains the nature of chaotic dynam-ics near the Shilnikov saddle-focus. A historic remark: thepaper by Arneodo, Collet and Tresser was the first numericalevidence of strange attractors associated with the Shilnikovtheorem in the Rössler model (1).Recall, however, that a 3D dissipative system with theShilnikov saddle-focus cannot produce a purely chaotic at-tractor but a quasi-attractor , due to homoclinic tangen-cies that give rise to stable POs unpredictably emerging in itsphase-space through saddle-node bifurcations and followedby period-doubling ones. To detect such stable POs and todetermine the corresponding stability windows in the bipara-metric sweeps, we propose and develop a symbolic approachcombined with a phase space partition. With this approach,one can generate long binary sequences whose Lempel-Zivcomplexity indicates whether they correspond to regular orchaotic dynamics in the model. On the contrary, short binarysequences let us detect homoclinic bifurcations of saddle-fociin the phase and parameter spaces.The paper is organized as follows: firstly, we scan the pa-rameter plane to detect the regions of regular and chaotic dy-namics in the Rössler model. That is done by partitioning the3D phase space and introducing a symbolic description forlong-term solutions of Eqs. (1). We next analyze the collectedbinary sequences to determine whether they are periodic oraperiodic and what the Lempel-Ziv complexity of chaotic se-quences is. This fast and effective approach is an alternativeto a less speedy method of Lyapunov diagrams, see detailsin Sec. II. The following Section III is focused on the pro-omoclinic chaos in the Rössler model 3 FIG. 2. The ( c , a )-parameter [5000 × shrimp -shaped regions, hosting stable periodic orbits inside (some sampled in panels A-G), while the regions in the grey-ish color are associatedwith chaotic dynamics (H) Solutions of (1) start escaping to infinity above some demarcation line (discussed in detail below). posed algorithms for computationally devising 2D and 1D re-turn maps to examine the structure of the chaotic attractors inthe model, and what makes them morph from spiral to multi-funnel shapes and how to quantify the transition in the pa-rameter space. We argue that the number of critical pointsin such 1D map can be used to reasonably categorize homo-clinic chaos associated with the saddle-foci as spiral, screw,and multi-funnel attractors, see details in Sec. III. Next wediscuss an efficient approach to search for a small family ofhomoclinic bifurcations of the primary saddle-focus O in bi-parametric sweeps. Having found these bifurcation curves,we employ the developed machinery of the 1D maps to giveinsights into a fine structure of special cases – hubs where ho-moclinic curves form the edges of the chaotic attractors in thephase space. In addition, Section III highlights how 1D mapscan predict the homoclinic bifurcations before they occur inthe Rössler model. Section IV is focused on the organizationof a typical hub and how several periodicity hubs shape thestability islands in the parameter space. Section V discusseshow the solution of the Rössler model becomes unboundedand what is the pivotal role of the secondary saddle-focus inshielding them from escaping to infinity. We reveal that thecrisis is due to invisible homoclinic bifurcations of the saddle-focus O and how this is related such that the Rössler modelbecomes no longer dissipative but expanding. Using the shortsymbolic dynamics we find the first basic homoclinic bifurca-tions of O and demonstrate how they arrange and demarcate the existence region of deterministic chaos in the given model.In Section VI, we will speculate about a nested organizationof homoclinic bifurcation curves in the interior of the primaryU-shaped one. II. BIPARAMETRIC SWEEP WITH LZ COMPLEXITYAND DETERMINISTIC CHAOS PROSPECTOR
In the given section, we discuss the symbolic approachcombined with the partition of the phase space to detect theregions of chaos and stability islands in the parameter spaceof the Rössler model. This is the first step prior to ap-plying more dedicated tools for examining a variety of ho-moclinic bifurcations. We previously developed a symbolictoolkit, code-named deterministic chaos prospector (DCP),running on graphics processing units (GPUs) to perform in-depth, high-resolution sweeps of control parameters to dis-close the fine organization of characteristic homoclinic andheteroclinic bifurcations and structures that have been univer-sally observed in various Lorenz-like systems, see andthe reference therein. In addition to this approach capitalizingon sensitive dependence of chaos on parameter variations, thestructural stability of regular dynamics can also be utilized tofast and accurately detect regions of simple and chaotic dy-namics in a parameter space of the system in question . The Z -symmetry is exploited to generate periodic or aperiodic bi-omoclinic chaos in the Rössler model 4 FIG. 3. (A) Spiral attractor ( c = . a = . O , and its intersectionpoints with the 2D cross-section (blue plane) given by y = x ≤
0. (B) Computationally interpolated 1D return map F ∶ x n → x n + of an x -interval spanning from O through the edge of the chaotic attractor. Forward and backward iterates, { , , , } , of the critical point on the x n -axis, approach/converge, resp., to the repelling fixed point (FP) at the origin O , corresponding to the homoclinic orbit of O in (A). nary sequences that can be associated, respectively, with sim-ple or chaotic flip-flop patterns of solutions of Lorenz-like sys-tems. The application of the symbolic techniques for studiesof neural dynamics in cell models and small neural networksis demonstrated in .Our practical approach tailored specifically for homoclinicchaos in the Rössler model uses particular events when the z -variable reaches consequently its maximal values on the at-tractor, see Fig. 1F. The binary sequence ( k n ) representing atrajectory is computed as follows: k n = { , when z max > z th ,0 , when z max ≤ z th . , z th = . ( c − ab )/ a . (3)Here, the z -threshold is set relative to the location of thesaddle-focus O . The choice of partition is motivated by itssimplicity and may possibly be different, yet sufficient for ourpurpose. We note that various partitioning approaches may beemployed for the same end result - effective detection of sta-bility windows corresponding to regular and structurally sta-ble dynamics in the parameter space of the Rössler model.With this simple algorithm we map the maxima values ofthe z -variable to the binary symbols 0 and 1, based on somethreshold value z th . Biparametric sweeps such as those shownFigs. 2, 6A, 8A are obtained by computing long trajectoriesstarting from identical initial conditions (near the origin), andby skipping some initial transients, as two control parameters, c and a , are varied across a 5000 × ( ... ) by simply ( ) . Aperiodicstrings representing chaotic trajectories, are processed using the Lempel-Ziv (LZ) compression algorithm to measure theircomplexity . As a string is continuously scanned, new wordsare added to the LZ vocabulary. The size of the vocabularytowards the end of the string, normalized with its length isused as the complexity measure, which is low for periodic se-quences and high for chaotic ones. Chaotic dynamics are as-sociated with the region painted in biparametric sweeps in theshades of gray, so greater LZ-complexity means darker gray,thus indicating higher instability. For trajectories mapped toperiodic strings, the shift symmetry of the sequences must beconsidered. For example, depending on the length of the ini-tial transient omitted, the same periodic orbit could be repre-sented by either { } ) or { } . We normalize such shift sym-metric periodic sequences to the smallest binary valued cir-cular permutation. Thus, both { } and { } are normalizedto { } , while the periodic sequences { } , { } , { } are normalized to { } . Let us re-iterate that we deliberatelytailor this approach only to examine and detect chaos and sta-bility islands due to saddle-node bifurcations in the parame-ter sweep. As such, period-doubling bifurcations are beyondthe scope of our consideration, even though the symbolic ap-proach can be further enhanced to detect such bifurcations aswell.In order to identify various stability windows in the para-metric plane, we define a formal convergent power series P for the normalized periodic sequence { k i } qi = p given by: P = q ∑ i = p k i q + − i . (4)By construction, the P -value ranges between 0 and 1. Theboundary values are set by the periodic sequences { } ) and { } , respectively, for infinitely long sequences. Here, we em-ploy p = q = { k i } (after normalization of shift symmetry), andthus, have the same P values, which are then projected onomoclinic chaos in the Rössler model 5 FIG. 4. Changes in the topology of the chaotic attractor in the Rössler model as it shifts closer to the secondary saddle-focus O along thepathway at c = .
5. (A ) the spiral attractor at a = . ) the corresponding 1D unimodal map; (B ) a screw-like attractor at a = . ) the corresponding 1D map with two critical points; and (C ) multi-funnel attractor at a = .
455 and (C ) the associated 1D map withmultiple critical points. As the saddle-focus O is isolated from the non-homoclinic chaotic attractor in each phase projection, all maxima inthe corresponding maps are located below the repelling FP at O representing O . to the biparametric sweep using a colormap. This colormaptakes P values into 2 discrete bins of RGB-color values as-signed from 0 through 1 for each of red, green and blue indecreasing, random and increasing order, respectively. Theresults of the sweep are demonstrated in Figs. 2, 6A and15A. Stability windows with distinct periodic orbits can bepainted with different or same solid colors in the sweeps,while chaotic regions of structural instability are shown ingray; moreover we re-emphasize the darker grey pixels areassociated with more developed chaos in the Rössler model.To conclude this section, let us point out at the mysteriousdemarcation line separating the region (here in white) wherethe solutions of Eqs. (1) start escaping to infinity. The rest ofthe paper is basically dedicated to disclosing what this bound-ary is and how it influences the evolution of chaotic dynamicswith underlying homoclinic bifurcations of both saddle-foci.We will proceed with the discussion of these results in detailbelow. III. FROM SPIRAL TO SCREW-TYPE CHAOS WITHFUNNELS
The goal of this section is to demonstrate how chaotic at-tractors initially developed through period doubling cascadesevolve further, specifically as they approach the demarcationline in Fig. 2. We will demonstrate the role and applicabilityof L.P. Shilnikov saddle-focus bifurcation and its unexpecteddevelopments for the Rössler model.It was well-reported and discussed that chaotic attractors in the Rössler model can be of various shapes and complexitybased on the number of nested sub-funnels, see Fig. 4, thatincreases as the demarcation line is approached from below.Alternatively, the transition after which the solutions becomeunbounded is often associated with the crisis occurring in themodel on the demarcation line . It is also well-known thatthe number of funnels can be effectively exposed with the useof 1D return map due to the strong contraction, as was donefor the first time for the given system in . 1D maps hap-pened to be instrumental in determining the topology of theattractors in question and the complexity of saddle periodicorbits embedded into an attractor. Depending on the numberof branches in the corresponding maps, chaotic attractors canbe categorized as spiral, screw, and multi-funnel attractors . Computational derivation of 1D maps
Let us itemize our construction procedure for 1D returnmaps tailored for the Rössler attractors. First, the half-plane { y = , x ≤ } is chosen as a cross-section transverse to spi-raling trajectories of (1) near O to define a 2D Poincaré map ( x n + , z n + ) = F ( x n , z n ) , as illustrated in Fig. 3A. Due to thestrong contraction near the saddle-focus O , the intersectionpoints of the chaotic attractor with the cross-section appearto lie on a 1D densely populated curve, as depicted by theinset of Fig. 3A. Next, we parameterize this curve by theLagrange polynomial z ( x ) using four points: the first point ( , ) corresponds to the saddle-focus O , the second pointomoclinic chaos in the Rössler model 6 FIG. 5. Chaotic attractors (gray color) morphing from (A ) the spiral one with a homoclinic orbit of the saddle-focus O at ( c , a ) ≃ ( . , . ) to the attractor with emerging funnels at ( . , . ) with a unidirectional heteroclinic connection from O to O in (B ), and through the multi-funnel attractor at ( . , . ) in (C ). The spirals in the orange color reveal the ways the attractors hit sometransverse cross-sections ( y = − , -4) in the phase space. Blue lines in panels B and C represent the location of the 2D stable manifold W s of the saddle-focus O that makes the chaotic attractor wraps around the left unstable separatrix W u + of O the more, the closer it shifts to thesecondary saddle-focus, whose manifold W s shields the solutions of Eqs. (1) from escaping to infinity, see Fig. 2H. ( x min , z ( x min )) is a point with the minimal x -coordinate be-longing to the attractor, and two middle points on it. Finally,we evaluate the corresponding 1D map x n + = F ( x n , z ( x n )) using 5000 x -samples on the interval [ x min , ] . The resultingone-dimensional map for the chaotic attractor embedded withthe primary homoclinic orbit to the saddle-focus O is pre-sented in Fig. 3B. By construction, such a 1D map has a fewpivotal features: (a) it inherits a unimodal shape (with a singlepoint) from a period-doubling cascade; (b) 0 = F ( ) is a re-pelling fixed point (FP) representing the saddle-focus O ; (c)the far-right critical point always touches the x n -axis; (d) inthe homoclinic case, the forward iterates of the critical pointterminate at the repelling FP, while its backward iterates con-verge to it along the right increasing branch of the map (corre-sponding to the homoclinic loop in Fig. 3A); (e) otherwise, theforward iterates of the critical point cannot reach the FP as theleft branch(es) and all local maxima are lowered below, priorto or after the homoclinic bifurcation. This can be interpretedthat the saddle-focus O becomes (temporarily) isolated fromthe chaotic attractor in both super- and sub-critical homocliniccases, like the one depicted in Fig. 1E.Having developed the proposed 1D map methodology, wecan better elaborate on the classification of attractors by thecomplexity of the funnel proposed in . Namely, when-ever the corresponding 1D map has only two branches, theattractor is called spiral, see Fig. 4A. It can be homoclinic ifthe critical point is mapped onto the FP at O , like in Fig. 3B,or not (when the left branch is not as high as the right one,see Fig. 4A ). The chaotic attractor is called of screw-typewhen the map gains an additional (third) branch, see Fig. 4B ,and multi-funnel if the map has four or more branches, seeFig. 4C . We re-emphasize that either attractors become ho-moclinic only if the images (forward iterates) of the far rightcritical point on the x n + = O occurs or not for the given parameter values. Funnels stirred by saddle-focus O Let us highlight an invisible role of the second saddle-focus O implicitly regulating the formation of funnels in thechaotic attractors in the Rössler model. Recall that this saddle-focus is of the (2,1)-type. Its 2D manifold W s , locally dividingthe 3D phase, forms an umbrella shielding the domain of thechaotic attractor that prevents nearby solutions from escapingits attraction. Its 1D unstable manifold W u includes O itselfand two outgoing separatrices, say W u + and W u − , such that W u + converges to an attractor (Fig. 5B ), while W u − tend toinfinity (Fig. 2H).Figures 5B and 5C illustrate the role of O in the funnelformation. Both depict the chaotic attractor in the phase spaceas the demarcation line in Fig. 2 is approached from belowin the parameter space. The attractor is superimposed witha local portion of the stable manifold (shown in blue) of thesaddle-focus O ; blue spirals also reveal the slow rate of con-vergence to O . The (orange) spiraling line is composed ofa large number of intersection points of the chaotic attractorwith a transverse cross-section ( y = − − O , the more it becomes wrapped around W u + . Figure 5B also shows W u + unidirectionally connecting O with O . Thisexplains the origin of the funnels observed in the chaotic at-tractors in the Rössler model, which are reflected in addingnew branches in the 1D return maps (Figs. 4A -C ).omoclinic chaos in the Rössler model 7 FIG. 6. (A) Fragment of the biparametric sweep. Solid, colored regions S , S , S ,... are associated with stability islands corresponding tothe attracting POs, emerging within chaos-land painted with the gray-ish colors. The line h with the end-point H corresponds to the primaryhomoclinic bifurcation of the saddle-focus O (see Fig. 1F). (B) Sketch of the self-similar bifurcation diagram near h . Outer borderline of ashrimp-shaped region S i are due to saddle-node (SN) bifurcation curves SN i − , i (in red) bridging with a successful one S i − and originatingfrom the Belyakov cod-2 point corresponding to a homoclinic saddle with double real eigenvalues λ = λ <
0. Inner demarcation line of S i is due to period-doubling (PD) bifurcations (blue lines): the first curve PD i − , i connecting successive S i and S i − regions also starts off theBelyakov point, and so forth. Note the cusps on SN-curves such that SN i , i + links S i and S i + islands, while SN i , ∞ terminates at the Belyakovpoint. (C) Evolutions of single and double homoclinic orbits to O transitioning from a saddle to a saddle-focus and back along the U-shapedbifurcation curve h , with the tip H being a turning point (simulations due to MatCont ). IV. HOMOCLINICS, HUBS AND SHRIMPS
Here we will discuss phenomena that underlie a multi-plicity of periodicity windows (well seen in Fig. 2). It isvery well-known that there are countably many sad-dle POs near the Shilnikov homoclinic saddle-focus in thephase space. The stable and unstable manifolds of such sad-dle POs can as cross each other transversally as become tan-gent; moreover, systems with such homoclinic tangenciesfill in open regions of the parameter space. Specifically,homoclinic tangencies give rise to the emergency of stablePOs in the 3D phase space through forthcoming saddle-nodebifurcations see a few stable OPs sampled in Fig. 2A–2G. One can see from the sweep in 2 that the parameter spaceof the Rössler model, which is mostly populated with chaoticdynamics also embeds solid-colored regions called period-icity windows or stability islands that are populated by at-tracting orbits. Strange attractors of such intermediate nature,where hyperbolic subsets may coexist with stable POs therein,were termed quasi-attractors by V.S. Afraimovich and L.P.Shilnikov , see also . Below, we will elaborate on the bifur-cation unfolding of those self-similar stability windows, la-belled by S i in Fig. 6A, that have the specific shrimp-shape.The gray curve h in Fig. 2A corresponds to the primary ho-moclinic bifurcation of the saddle-focus equilibrium O . Thestudy was the first where the bifurcation analysis of stabilitywindows emerging near this curve was carried out in detail. Its unfolding is outlined in Fig. 2B, following the original paper.Specifically, its authors showed that each periodicity windowis bordered by a saddle-node bifurcation curve SN i − , i on oneside such that a stable PO emerges upon inward-crossing ofthis curve, along with a saddle one. The stable orbit undergoesa cascade of period-doubling bifurcations. Curves PD i − , i andPD i , i + in Fig. 2B represent the first bifurcations in such cas-cades. FIG. 7. 1D return map generated by the trajectories of the chaoticattractors sampled along the pathway a = .
35 near the primary bi-furcation curve h : (A) the local minimum is below the repelling FPat the origin at c = . ), (B) the local minimumis taken to the origin after one iterate at c = .
89 (on h ), and (C)the map at c = . ) is similar to the one inpanel A. Note a cusp point c i inside each shrimp-like stability win-dow S i , where the sub-criticality of SN-bifurcations changesdue to such cusps, the dynamics inside the shrimp-shaped re-omoclinic chaos in the Rössler model 8 H H H h h h H H H H H h m m m m c-parameter a - pa r a m e t e r c-parameter
4 4.2 4.4 4.6 4.8 5.0 . . . . . distance on Q R -pathdistance on Q R -path A BC
A B C
FIG. 8. (A) Biparametric sweep with superimposed curves: m i , ( i = ,.., ) corresponding to the emerging ( i + ) -th branches and new criticalpoints in the 1D return maps, see examples in Figs. 4 and 9; specifically, above m the spiral attractor transforms into a screw-type . Curvesh i with tipping points H ii on m i stand for homoclinic bifurcations of O and primary periodicity hubs H ii , resp. Shrimp-like windows (somecircled) form what looks like nested spirals around the periodicity hubs. (B, C) the distance between the saddle-focus O and the chaoticattractors on the P Q - and P Q -pathways, resp., plotted against the c -parameter. Its zeros correspond to the intersection points of thesepathways with the indicated curves h i . gion can be bistable. As was established in that one curve,SN , originating from the cusp c inside the shrimp S con-nects to a subsequent cusp c within the other shrimp S ,containing the same stable POs. Furthermore, the bifurcationcurve, SN , forming the left-bottom boundary of S coalesceswith the SN-curve representing the right-bottom boundary ofthe third shrimp S , and so forth. Due to such connectivity,these stability windows, alternating with regions of chaoticdynamics, appear as some nested spirals winding around whatis code-named periodicity hubs , associated with a faux cod-2 bifurcation, if any. Note that this interesting structure wasfirst reported in the original paper . Hubs in the Rösslermodel, as well as in other systems with homoclinic saddle-foci, were numerically studied using Lyapunov exponents invarious diverse applications, for example see , and the ref-erences therein.It is interesting to note that, apparently, the SN and PDbifurcation curves edging the shrimp-shaped windows nearthe homoclinic bifurcation curve all branch out from a spe-cific cod-2 point corresponding to a transition from a ho-moclinic saddle to a homoclinic saddle-focus. In virtue ofthe Belyakov theorem , such a point gives rise to a count-able number of SN-curves and double homoclinic bifurca-tion curves. It is shown in , with the aid of the numericalparameter-continuation approach, that this is the case for thetritrophic food chain model with similar dynamical and bifur-cational properties. As for the Rossler model, it is shown numerically that several SN and PD bifurcation curves dooriginate from the Belyakov point, as sketched in the top-chart in Fig. 2b.Another interesting result reported in is that all primaryand secondary homoclinic bifurcation curves in both foodchain and Rössler models have the specific U-shaped formin the parameter space. Due to strong contraction and slow-fast dynamics, two close (10 − ) branches of the curves h and others look like a single one, see a schematic representa-tion in Fig. 2C. The fold or turning point H on h representsthe primary periodicity hub. Observe that that the left branchof h corresponds to double homoclinic orbits, in contrast toorbits of the right branch . Primary homoclinic curves and periodicity hubs
It was shown in that on the left from the primary homo-clinic bifurcation curve h , there exist several subsequent onesh i , i ∈ [ , , . . . ] corresponding to double, triple and longerhomoclinic orbits, see Figs. 6A and 8A. All these curves areof the U-shaped form with fold points at the correspondingperiodicity hubs H ii . In Fig. 8, such curves can be typicallydetected and computed using numerical continuation toolk-its such as AUTO and MatCont (in backwards time, making O of (2,1)-type not (1,2)). We stress that in between curvesh i and h i + , there are no other homoclinics to the saddle-focus O . This fact can be simply verified using the pro-posed 1D maps. Let us consider three particular points inthe ( c , a ) bifurcation diagram: the points A ( . , . ) andC ( . , . ) are chosen on the opposite sides from the bifur-omoclinic chaos in the Rössler model 9cation curve h of the primary homoclinics of O , whereas thepoint B ( . , . ) is placed right on it, see Fig. 8A. At thepoint A, as well as the point C, the images of the local maxi-mum in the 1D map are below the repelling FP at the origin O ,which is interpreted as the saddle-focus O is isolated, outsideof the chaotic attractor, see Fig. 7A and 7C. This is not thecase for the point B, where the local maximum is taken to theFP O after a single iterate (see 7B), which indicates that thesaddle-focus O is the intrinsic component of this homoclinicattractor, in contrast to the other two cases.In this work, we applied an alternative approach for find-ing the curves h i which is better suited for systems with var-ious saddles of (1,2)-type, . Our search strategy is basedon the fact that a homoclinic attractor (the definition intro-duced in )) embeds homoclinic loops of O , whose simu-lated trajectories will come close by the saddle-focus O even-tually. By performing one- and two-parameter sweeps, we es-timate the distance to O from a long typical transient trajec-tory on the attractor. Whenever the distance becomes less thansome threshold value d tr (we use d tr = . O is meant to belong to the attractor.Figure 8B is a de-facto demonstration of the efficiencyof the described approach for two one-parameter pathways:P Q given by { a = .
45; 4 ≤ c ≤ } ) and P Q given by { a = − / c + .
55; 3 . ≤ c ≤ . } . Along the first path-way P Q , the distance graph has two local minima belowthe threshold: the first minimum corresponds to the homo-clinic bifurcation value on h and the second one occurs onthe curve h . Along the second pathway P Q , there are foursuch local minima in the graph, the distance plotted againstthe c -parameter, corresponding to the curves h , h , h , andh .Next, let us describe the curves m , m , and m that are alsosuperimposed with the DCP-sweep in Fig. 8 A . These curveswere found using the 1D return maps, namely: below m j , 1Dmaps consist of j connected parabolas, whereas above it, themaps gain the additional ( j + and 4B depicting the 1D maps below and abovethe curve m , respectively . When the image of the new lo-cal maximum (on the far left branch) on the curve m j is therepelling FP O , this corresponds to the occurrence of the j -round homoclinic orbit to the saddle-focus O . For example,see the 1D map at the hub H on the primary curve h throughwhich the curve m passes in Fig. 3.Figure 9 illustrates the homoclinic phenomena in otherfound hubs H , H , and H . The first row in this figurepresents the 2D Poincaré maps computed at these points. Notethat these maps contrast significantly from the map at the pri-mary hub H (Fig. 3A), as they include 2, 3, and 4 branches,correspondingly, on the right-hand side.Next, we parameterize these 2D maps by 1D ones. To dothat, we take four points on the bottom branch of each 2D mapto compose the Lagrange polynomial, as previously describedin Sec. III. The resulting 1D maps are shown in the middle rowin Fig. 9. One can see that the 1D map at H ii -hub indeed con-sists of i parabolas whose far left point is taken at the repeller O , after one iteration. The dotted lines representing homo-clinic obits in these Lammerey (coweb) diagrams correspond FIG. 9. Homoclinic bifurcations at three primary hubs: (A) H lo-cated at ( c ≈ . , a ≈ . ) , (B) H at ( . , . ) , and (C)H around ( . , . ) (see the bifurcation diagram in Fig. 8A.)The corresponding 2D cross-sections (top panels), and 1D returnmaps (middle panels) for h i -curves with H ii -tuning points reveal i branches and parabolas, resp. with the saddle-focus O placed in thetop-right corner of the Lammerey (coweb) digram (A –C ). Bottompanels illustrate the homoclinic orbits demarcating the “edge” of thechaotic attractor.FIG. 10. The separatrix loop of the saddle-focus O at ( c = . , a = . ) on the h -curve is not fully maximized to edge the homoclinicattractor (shown in the background), see the contrast with Figs. 9 and11 for the primary and secondary hubs, resp. to the homoclinic orbits (bottom row) to the saddle-focus O shown in the phase space of the Rössler model.It is important to underline another feature of periodicityhubs. In each hub the separatrix loop to the saddle-focus ismaximized in its size so that it becomes the “edge” of thehomoclinic attractor, see Figs. 3B, 9A , 9B , and 9C . Thisis also documented in the corresponding 1D maps. Indeed,below H ii on m i the left branch of the 1D map remains lowerthan the FP O . Above H ii , a newly emerging branch is yetlower than the FP O , so the homoclinic orbit used it only topass through before landing in the repelling FP at the origin.In contrast, Fig. 10 illustrates the homoclinic attractor with thesuperimposed homoclinic orbit at the point B on the curve h ,which is far from forming its edge.omoclinic chaos in the Rössler model 10 FIG. 11. Homoclinic bifurcations at three secondary hubs: (A) H located at ( c ≈ . , a ≈ . ) ; (B) H at ( . , . ) ; and H near ( . , . ) , see Fig. 8A. The corresponding simulated 1Dmaps depict the homoclinic orbits to the repeller FP O located at theorigin and associated with the saddle-focus O . Bottom panels depictthe homoclinic loops edging the superimposed chaotic attractors. A. Hubs inside the U-shaped h -curve By carefully examining the bi-parameter sweeps, one canobserve a hierarchy of secondary hubs (e.g. H , H , H )forming organization centers for shrimp-like stability win-dows, see Fig. 8A. As was shown in , these secondary hubslie inside the U-shaped curves h i corresponding to the primaryhomoclinic loops. Note that the distinctive U-shape of the bi-furcation curves is a common feature of diverse applicationswith a homoclinic saddle-focus, see for example and thereferences therein. In this subsection, we show how 1D mapshelp to predict the form of homoclinic loops originating fromthe most visible secondary hubs located inside the curve h .Let us examine the fragment of h between the curves m and m moving upward along h . While searching for the lo-cations of secondary hubs in the diagram (Fig. 8A), let us an-alyze the corresponding 1D maps, while keeping in mind theimportant feature of all hubs, namely: in each hub the homo-clinic orbit of the saddle-focus O is maximized to “edge” thehomoclinic attractor outwardly in the phase space, see Figs 9and 11 and contrast them with Fig. 10The hubs H is the most visible secondary hub along thecurve h (we mark the biggest shrimps around it by whitecircles in Fig. 8A). The corresponding 1D map is presentedin Fig. 11A . The dotted lines represent homoclinic obitsin the Lammerey (coweb) diagram. The corresponding ho-moclinic loop superimposed with the attractor is presented inFig. 11A . Moving further along the curve h , the correspond-ing 1D map gains an additional decreasing branch. In the hubH , the far-left point of this branch matches with the repellingFP O giving birth to a new homoclinic loop, see the corre-sponding 1D map with the Lammerey diagrams and phaseportrait of the attractor with the homoclinic loop in Figs. 11B and 11B , respectively.In the H -hub below the curve m , the far-left decreasingbranch gives rise to another homoclinic orbit passing on the edge of the attractor including, see the corresponding 1D mapsuperimposed with the Lammerey diagram in Fig. 11C , andthe phase projection of the corresponding homoclinic attractorpresented in Fig. 11C .Above the curve m , the 1D map gains an additional in-creasing branch and then, an additional decreasing one. Mov-ing towards m , on the base of these branches, new homo-clinic loops appear according to the described scenario. Theseloops make an additional global passage with respect to theloops in the hubs H , H , and H . Moreover, we believethat the same evolution of homoclinic loops is also observedalong all curves h i corresponding to the primary i-round ho-moclinic loops.Finally, we would like to emphasize that each periodic-ity hub H i j is a fold point for the corresponding homocliniccurve h i j . Each such curve has a U-shape, with two ex-tremely close to each other branches. All these curves resideinside the curve h i corresponding to the primary i-round ho-moclinic loop. The possible structure of the correspondingbi-parameter diagrams will be discussed in last section of thispaper. V. BEYOND THE BOUNDARY OF RÖSSLERATTRACTOR
This section reveals what the invisible role of the secondarysaddle-focus O is in the organization of the existence regionfor observable chaotic attractors in the Rössler model. We willargue that the corresponding demarcation line above whichthe solutions start running to infinity (Fig. 2) can be well ap-proximated by the curves of homoclinic bifurcations of O .The associated bifurcation curves are detected using the sim-ulations combining the symbolic descriptions for homoclinicorbits of O with biparametric sweeps.It was shown in Sec. I that on the demarcation line theRössler attractor merges with the 2D stable manifold W s ofthe saddle-focus O of (2,1)-type, see Fig. 5. Recall that O has two 1D unstable separatrices: W u + fills in the chaotic at-tractor, while W u − always runs away to infinity. For the pa-rameter values slightly above the demarcation line, the solu-tions start escaping along W u − (see Fig. 2H), because the 2Dmanifold W s of O no longer shields them. This lets one saythat the given crisis of the chaotic attractor is associated withan “infinitely long” homoclinic orbit of the saddle-focus O .As there is no way that such long homoclinics can be wellcomputed, so we would limit ourselves to finding shorter ho-moclinic orbits and corresponding bifurcation curves in the ( c , a ) -diagram. While sweeping ( c , a ) -plane, we calculatethe number N of global passages (or turns) of W u + aroundthe primary saddle-focus O , before it crosses over W s of O to escape to infinity.Specifically, for each parameter pair in the sweep, we gen-erate a binary sequence S according to the following algo-rithm. Whenever the unstable separatrix W u + of O completesa global passage (or a sizable turn) around the equilibrium O , the symbol 1 is added to S . After that there are two op-tions: (i) if W u + makes another turns around O , the secondomoclinic chaos in the Rössler model 11 FIG. 12. Geometric idea behind the symbolic algorithm for the detection of homoclinic loops to the saddle-focus O with 2D stable manifold W s (shown in light blue) and 1D outgoing separatrices: returning W u + and W u − escaping to infinity. (A) Primary homoclinic orbit Γ makinga single turn around the saddle-focus O at µ = µ = { } , or as { } if it makes it twice (B), or { } forthree turns in (C), before it runs to infinity alongside W u − with a sequence { } . The notions Γ j and Γ jk for multiple double and triplehomoclinic loops occurring at different values of µ j and µ jk , resp., are used to indicate the number(s) j , k of smaller rounds of W u + on thesecond and third turns before it returns to O . The further the system gets away from the primary loop, the longer the homoclinic orbits Γ N with increasing index N become, with a greater multiplicity.FIG. 13. (A) ( c , a )-parameter sweep capitalizing on the symbolic approach with binary sequences up to three symbols long: the red-coloredregions is where the 1D unstable separatrix W u + of the saddle-focus O runs to infinity after its first turn around O ; the correspondingsequence is { , , } . The regions in blue/yellow colors correspond to the sequences { , , } and { , , } , resp. The borderline L betweenthe red and yellow painted regions stands for the primary homoclinic loop Γ of O , while boundaries (curves L j and L j merging in the somepoints) between the blue and yellow painted regions correspond to double homoclinic loops Γ j , making j -turns around O on the secondpassage. (B) Homoclinic orbits Γ , Γ , and Γ with one, two and three turns around the saddle-focus O at specific points (white dots) inpanel A. (C) Bifurcation diagram, courtesy , in the ( µ , ρ ) -parameter plane (here, µ measures the distance between a saddle-focus and itsreturning separatrix, and ρ is the saddle-index introduced above). Horizontally stretching U-shaped curves L , j for double homoclinic orbitswith increasing j -index accumulate to the curve L (given by µ =
0) corresponding to primary homoclinic orbit. symbol 1 is appended to the sequence S = { , } and so forth S = { , , , . . . } ; (ii) if W u + runs to infinity, the second sym-bol is 0 ( S = { , } ), and the calculations are halted for theseparameter values. The latter case corresponds to the occur-rence of the only primary homoclinic orbit of the saddle-focus O , see Fig. 12A. Unlike it, secondary, tertiary and longer ho-moclinics correspond to multiple bifurcation curves in the pa-rameter plane.Let us observe that, for example, between any ( c , a ) -parameter pair corresponding to sequences { , , } andomoclinic chaos in the Rössler model 12 FIG. 14. (A) Biparametric sweep revealing multiplicity of homoclinic orbits to the saddle-focus O and the corresponding binary sequences upto 4 symbols long. The color scheme is borrowed from Fig. 13: the sequences { , , , } correspond are the borderlines of the regions paintedin the magenta color. The borderlines between the magenta and yellow colored regions (curves L jk and L jk bending at folds) correspond to thetriple homoclinic loops Γ jk making j -rounds around O on the second turn/passage and k -rounds on the third turn. The boundary between themagenta and blue painted regions is not a bifurcation, as associated with transformations of double loops to the triple ones due to the integerarithmetics issue. (B) Triple homoclinic loops Γ , Γ , and Γ (with 1 round on the first turn and 1, 2, and 3 rounds on the third turn) in thethree points marked in panel A. (C) Schematic diagram, courtesy , revealing self-similar, nested organization of triple homoclinics curveslocated within a pair of curves L j and L j + , corresponding to double homoclinics. { , , } , there are always the pairs on some borderline, reada bifurcation curve, corresponding to the double homoclinicorbits Γ j , see Fig. 12B, wheres between sequences { , , , } and { , , , } there exists a triple homoclinic orbit Γ jk , seeFig. 12C, and so on.Finally, the pixels of the regions with unitary sequences S = { , , . . . , } are painted in the yellow color, while re-gions corresponding to different sequences ending with 0, like( S = { , , . . . , , } ), are painted in contrast colors dependingon the number of 1’s in S , see the color bars at the top inFig. 12 for sequences of length N = { , , } .We begin our consideration with primary and double ho-moclinic orbits of the saddle-focus of O . Figure 13A repre-sents the corresponding biparametric sweep in the dedicatedregion 2 ≤ c ≤ . ≤ a ≤ .
75. The red-colored regionis associated with the same sequence S = { , } , meaning thatthe unstable separatrix W u + runs to infinity after a single turnaround the saddle-focus O . In the blue-colored regions asso-ciated with the same sequence S = { , , } , W u + makes twoturns around O before escaping. In the yellow-painted re-gion, it makes at least three turns around O to generate thesequence S = { , , } . Hence, the borderline between the red-colored region and the yellow-colored region corresponds tothe primary homoclinic orbit Γ of O .The boundaries of the multiple blue-colored regions cor-respond to the double homoclinic orbits Γ j , where the index j indicates the number of small rounds around the outgoing separatrix W u + of the saddle-focus O on the second turn.As one can see from Fig. 13A, these bifurcation curves havea U-shaped form . We differentiate their branches with la-bels L j and L j . It is known from that these U-shapedcurves of double loops accumulate to the primary curve L ,see Fig 13C. The closer these curves approach to L , the greaterthe index j becomes. One can see from the symbolic sweeppresented in Fig. 13 that this agrees well with the theory. Afew double homoclinic orbits Γ j are sampled in Fig. 13B forthe indicated points on curves L j , j ∈ { , , } in Fig. 13A.Note that within the blue-colored regions, the unstable sep-aratrix W u + always escapes after two turns around O . Thisimplies that no other homoclinic bifurcations can populatethese regions. However, that is not the case for the yellow-colored region in which triple- and longer homoclinic orbitscan occur. Fig. 14A represents the homoclinic bifurcationdiagram for the sequences of length up to four symbols. Init, there are multiple magenta-colored regions associated withthe same sequence S = { , , , } that populate the space be-tween the blue-colored regions. Their boundaries, L jk and L jk ,correspond to triple homoclinic orbits Γ jk , where the indices j and k stand, respectively, for the numbers of rounds on thesecond and third turns around O .Bifurcations of triple homoclinic orbits Γ jk were studiedin . The idea of bifurcation unfolding is sketched in Fig. 14C.In the Rössler model, such bifurcations occur between curvesomoclinic chaos in the Rössler model 13 L j + and L j for double homoclinic orbits marked with bluehorizontal lines in Fig. 13C. Depending on the relationshipbetween j and k , these curves can be of either the U-shapedform if j < k (as for double loops provided j ≤ k ), or of theform of horseshoe if j > k .We would like to comment that specific boundaries betweenthe magenta- and blue-colored regions in the symbolic sweepin Fig. 14A are not bifurcation curves, but artifacts caused byinteger-based number of turns of W u + flying around O . Threetriple homoclinic orbits Γ jk are sampled in Fig. 14B for themarked parameter values on the curves L jk , j = k ∈ { , , } in the bifurcation diagram in Fig. 14A. Stitched bifurcation diagram
Figure 15 completing our case study of the Rössler modelrepresents two bifurcation diagrams stitched together into one:the bottom section reveals the bifurcations mostly due to theprimary saddle-focus O in the phase space where the modelis strongly dissipative, while the top chart includes the bifur-cation curves corresponding to homoclinic orbits (up to thesymbolic length 6) of the secondary, invisible saddle-focus O in the phase subspace, with a positive divergence of the vectorfield. One can see from the stitched diagram that the U-shapedhomoclinic curves well approximate from above, the demar-cation line of the existence region of the Rössler attractors,regardless of whether they are regular or chaotic. For the sakeof visual clarity, we did not use the longer homoclinic orbitsin this chart.We underline that without knowing the the homoclinicstructures due to the secondary saddle-focus O , the com-plete perception of the origin and driving forces of the ob-servable chaotic dynamics in the Rössler model and the meta-morphoses of its attractors stirred by the primary saddle-focus O , would be barely possible. VI. h -INTERIOR HYPOTHESIS In this last section, we would like to hypothesize about whatway and order the h j -curves of the secondary homoclinicsof the saddle-focus O can be organized inside the primarybifurcation h -curve in the parameter space. The distance( ∼ − ) between the folded branches of h is too infinites-imal to apply implicit computations at this scale, such as theparameter continuation. Furthermore, like any strong dissipa-tive system, solutions including 1D separatrices of O withoutexception, of the Rössler model (1), while integrated in back-ward time, become highly sensitive to the smallest perturba-tions and tend to escape quickly to infinity in no time.So, let us try to re-visit and examine further, the other notfully exploited options. Fig. 5 is one such example. Recallthat the orange spirals in it represent the image of the homo-clinic attractor transversely cut through by a 2D cross-section.In other words, by construction, spirals flattened in Fig. 16Arepresent the computational point-wise reconstruction of the FIG. 15. Stitched ( c , a ) -bifurcation diagram for the Rössler model.The top b/w chart is due to the primary (top line) and subsequenthomoclinics of the saddle-focus O of (2,1)-type, whose correspond-ing bifurcation curves terminate on the sub-critical AH-bifurcationcurve (that makes O a repeller); the terminal points on the AH-curverepresent the cod-2 Belyakov bifurcation nicknamed as Shilnikov-Hopf. The bottom part reflects the complex organization of regularand chaotic dynamics stirred up by the primary saddle-focus O of(1,2)-type.
2D unstable manifold W u of the saddle-focus O in the samecross-section on which some 2D return map is thereby empiri-cally defined. Note that the intersection(s) of the 1D W s of O is some point(s), which are close to the spirals representing the W u ( O ) -image the h -curve in the parameter space. The pri-mary homoclinic orbit of O emerges when W s ( O ) touches W u ( O ) , or in the 2D map reduction, when the W s ( O ) -image– the first intersection point, namely the p -point in Fig. 16A,touches the spiraling W u ( O ) -image.Let us make a virtual experiment: locally cross the h -curvetwice by following the PQ -pathway from the right to the left,see Fig. 16B. By construction, crossing h only or severalh j -curves (within h ) along the PQ -pathway in the ( c , a ) -plane makes the W s ( O ) -image trace down a pq -pathway inthe 2D map as well that crosses twice (in and out) one orconsecutively several nested arches of the spiraling W u ( O ) -image. Therfore, the first crossing at the point, p , in the map(Fig. 16A) occurs when the h -curve is inward-intersectedfrom right to left. The corresponding homoclinic orbit is pre-sented in Fig. 16C. The second intersection point, p corre-sponds to the secondary loop occurring on the h -curve, seeFig. 16D, and so forth.A similar reasoning is also applicable to disclose the struc-ture of the periodicity hubs: whenever the parametric path-way passes right through a hub, then the W s ( O ) -trace (readomoclinic chaos in the Rössler model 14the pq -pathway) becomes only tangent to some spiral of the W u ( O ) -image in the corresponding 2D map. Now we can ar-gue to justify a nested organization of the h j -curves. The as-sertion is that the W s ( O ) -trace typically (not in hubs) crosseseach spiral or arch of the W u ( O ) -image twice. Each pair ofsuch crossing points corresponds to the two branches of eachU-shaped h j -curve nested successfully within a similar outerU-shaped h j − -curve, and so forth, as sketched in Fig. 16B.Panel E in Fig. 16 presents our vision how a global recon-struction of h j -curves linking the Belyakov cod-2 points andthe periodicity hubs in the parameter plane may possibly looklike. Let us make it clear again that this is so far our best in-terpretation of this homoclinic puzzle based upon performedsimulations and bifurcation logic. It would be helpful to com-pare our hypothesis with theoretical and computational find-ings from other systems featuring both periodicity hubs andBelyakov cod-2 bifurcations of the given type. CONCLUSION
In this paper, we presented a case study of essential ho-moclinic bifurcations two Shilnikov saddle-foci that governthe shape and the intrinsic structure of developed determinis-tic chaos observable in the Rössler model. We combine twocomputational approaches – the first one capitalizing on 1DPoincaré return maps and the second utilizing the symbolicdescription to provide both evident qualification and properquantification of underlying global bifurcations of the Rösslerstrange and periodic attractors. The map-based approach letsus accurately detect the location of the homoclinic bifurcationcurves and their turning points – the periodicity hubs in theparameter space. The symbolic approach with long transientsis the new development in the computational apparatus in non-linear sciences that aids to identify the regions of chaotic andperiodic dynamics in biparametric sweeps. The use of shortbinary sequences is a powerful instrument to locate and inves-tigate homoclinic saddle-focus bifurcation in the given model.The generality of our computational toolkit makes it univer-sal and applicable to other systems of diverse origins, rangingfrom mathematics through life sciences.
ACKNOWLEDGEMENTS
We thank the Brains and Behavior initiative of GeorgiaState University for the B&B fellowship awarded to K. Pusu-luri. We are grateful to D. Turaev and J. Scully for inspiringdiscussions, as well as H.G.E. Meijer for MatCont tutoring.The Shilnikov NeurDS lab thanks the NVIDIA Corporationfor donating the Tesla K40 GPUs that were actively used inthis study. A. Kazakov and A. Shilnikov acknowledge a par-tial funding support from the Laboratory of Dynamical Sys-tems and Applications NRU HSE, grant No. 075-15-2019-1931 from the Ministry of Science and Higher Education ofRussian Federation. Yu. Bakhanova and A. Kazakov acknowl-edge the RSF grant No. 19-71-10048 for the funding supportrelated to the results presented in Sec. IV and Sec. VI. S. Ma- lykh acknowledges the RSF grant No. 20-71-10048 for thefunding support related to the results presented in Sec. V.
DATA/CODE AVAILABILITY
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