Hyperchaos and Multistability in Nonlinear Dynamics of Two Interacting Microbubble Contrast Agents
Ivan R. Garashchuk, Dmitry I. Sinelshchikov, Alexey O. Kazakov, Nikolay A. Kudryashov
aa r X i v : . [ m a t h . D S ] M a r Hyperchaos and Multistability in Nonlinear Dynamics ofTwo Interacting Microbubble Contrast Agents
Ivan R. Garashchuk , Dmitry I. Sinelshchikov , Alexey O. Kazakov , andNikolay A. Kudryashov National Research Nuclear University MEPhI, Moscow, Russia National Research University Higher School of Economics, Nizhny Novgorod,RussiaMarch 12, 2019
Abstract
We study nonlinear dynamics of two coupled contrast agents that are micro-meter sizegas bubbles encapsulated into a viscoelastic shell. Such bubbles are used for enhancingultrasound visualization of blood flow and have other promising applications like targeteddrug delivery and noninvasive therapy. Here we consider a model of two such bubblesinteracting via the Bjerknes force and exposed to an external ultrasound field. We demon-strate that in this five-dimensional nonlinear dynamical system various types of complexdynamics can occur, namely, we observe periodic, quasi-periodic, chaotic and hypechaoticoscillations of bubbles. We study the bifurcation scenarios leading to the onset of bothchaotic and hyperchaotic oscillations. We show that chaotic attractors in the consideredsystem can appear via either Feigenbaum’s cascade of period doubling bifurcations orAfraimovich–Shilnikov scenario of torus destruction. For the onset of hyperchaotic attrac-tor we propose a new bifurcation scenario, which is based on the appearance of a homoclinicchaotic attractor containing a saddle-focus periodic orbit with its two-dimensional unsta-ble manifold. Finally, we demonstrate that the bubbles’ dynamics can be multistable, i.e.various combinations of co-existence of the above mentioned attractors are possible. Thesecases include co-existence of hyperchaotic regime with any of the other remaining typesof dynamics for different parameter values. Thus, the model of two coupled gas bubblesprovide a new examples of physically relevant system with multistable hyperchaos.
In this work we study nonlinear dynamics of two coupled encapsulated gas bubbles in a liquid,which are driven by an external periodic pressure field. Investigation of oscillations of suchbubbles is of interest due to their applications as contrast agents for ultrasound visualization[1–3] and future possible applications for noninvasive therapy and targeted drug delivery [4, 5].Depending on applications, different types of bubbles’ dynamics can be either beneficial orundesirable (see, e.g. [3,6]). Therefore, it is important to study the variety of possible dynamicalregimes and how the dynamics of the bubbles depend on the both control parameters and initialconditions.Typically, mathematical models of a single microbubble contrast agent are one-dimensionalnon-autonomous oscillators based on the Rayleigh–Plesset equation and its generalizations (see,1.g. [7, 8] and references therein). Later, these models were extended to the coupled Rayleigh–Plesset equations which take into account bubble-bubble interactions via the Bjerknes force(see [9–13] and references therein). From a mathematical point of view, these models aredescribed by a system of coupled nonlinear oscillators, with external periodic force, and, thus,various types of dynamics can be observed in them. Despite the great interest, there are onlyfew works devoted to studying of nonlinear dynamics of gas bubbles. For example, nonlineardynamics of a single bubble described by one of the Rayleigh–Plesset-like models was studiedin [6, 14–18], where it was shown that oscillations of a single bubble can be either regular orchaotic and routes to the corresponding attractors were studied. Some bifurcations of two and N coupled bubbles were studied in [9, 13, 16]. However, in [9] unencapsulated bubbles wereconsidered, while in [13, 16] an inappropriate model of the shell was investigated (see discussionin [8]).After some simplifications, the dynamics of two coupled bubbles is described by a systemof five ordinary differential equations. In this work we show that, in addition to regular andchaotic regimes which are typical for models of one bubble [6, 14–18], this system exhibitsquasiperiodic and, what is more interesting, hyperchaotic types of motion. While there aremany known examples of multi-dimensional nonlinear systems demonstrating quasiperiodic andchaotic dynamics with two or more positive Lyapunov exponents, to the best of our knowledge,neither quasiperiodic nor hyperchaotic oscillations of two coupled bubbles have been studiedpreviously.Moreover, in this work we propose a bifurcation scenario for the onset of hyperchaoticbehavior with two positive Lyapunov exponents. The key part of this scenario is the appearanceof a homoclinic chaotic attractor containing a saddle-focus periodic orbit with a two-dimensionalunstable manifold, i.e. such an orbit which has a pair of complex conjugated multipliers withpositive real parts while all the other multipliers have negative real part. Recall, that thechaotic attractor is called homoclinic, if it contains a saddle periodic orbit [19] together withits unstable invariant manifold.Trajectories on a homoclinic attractor can pass arbitrarily close to the saddle orbit belongingto it. The dynamics near this saddle orbits and, as a result, on the whole homoclinic attractor,significantly depends on the multipliers of the corresponding saddle orbit [20]. In particular,in the small neighborhood of a homoclinic attractor containing a saddle-focus periodic orbitwith two-dimensional unstable manifold, two-dimensional areas are expanded and Lyapunovexponents on the whole attractor “can feel” this expansion. As a result, two Lypaunov exponentsbecome positive. Note, that homoclinic attractors of this type were called in [20], [21], [19] discrete Shilnikov attractors .Another interesting property of the considered system is its multistability. It has recentlybeen shown [17, 18] that the dynamics of a single encapsulated bubble can be multistable, i.e.several attractors can co-exists at the same values of parameters. Thus, it is interesting tounderstand whether multistability persists in the dynamics of coupled bubbles or even morenew multistable states can occur. We show that the dynamics of coupled bubbles is multistableand various attractors can co-exist, including periodic and hyperchaotic attractors. Thus, wedemonstrate a possibility of hidden hyperchaos in the dynamics of two interacting gas bubbles,which is a new example of physically relevant five-dimensional dynamical system with hiddenhyperchaos.The rest of this work is organized as follows. In Sec. 2 we present the governing system This terminology arises to the paper [22] where Shilnikov, for one-parametric families of three-dimensionalflow systems, proposed a universal bifurcation scenario leading to the birth of spiral attractor containing saddle-focus focus equilibrium together with its two-dimensional unstable manifold. In the papers [21], [20] this scenariowas transferred to the case of one-parametric families of three-dimensional maps (or four-dimensional flows, ifthe corresponding Poincar´e cross-section is considered).
2f equations for the dynamics of two coupled bubbles and discuss some of its properties. InSec. 3 we present a two-parametric chart of the Lyapunov exponents for the considered systemand discuss possible types of dynamics. In Sec. 4 we focus on scenarios leading to the onsetof both chaotic and hyperchaotic oscillations of coupled bubbles. We demonstrate that achaotic attractor can appear via either Feigenbaum’s cascade of period-doubling bifurcationsor Afraimovich-Shilnikov scenario of the destruction of invariant tori. We also propose a newphenomenological scenario for the onset of hyperchaotic oscillations as well. In Sec. 5 wediscuss a possibility of co-existence of several attractors in the systems under considerationand point out that chaotic attractor can coexist with hyperchaotic one in this system. In thelast section we briefly discuss our results specially marking that the proposed scenario of theonset of hyperchaotic attractors should be also typical for other multi-dimensional systemsdemonstrating hyperchaotic behavior.
Figure 1:
Schematic picture of two interacting bubbles oscillating in a liquid under the influence of an externalpressure field.
In this section we consider a model for the description of oscillations of two interactinggas bubbles in a liquid. Essentially, this model is formed by two generalized Raleigh–Plessetequations that are coupled via the Bjerknes forces (see, e.g. [9–13, 16]). In this work we takeinto account liquid’s compressibility in accordance with the Keller–Miksis model [23], liquid’sviscosity on the gas-liquid interface, surface tension and the impact of bubbles’ shells, whichis described by the de–Jong model [24, 25]. We also suppose that bubbles are exposed to theexternal periodic pressure field. Under these assumptions the governing system of equationsfor oscillations of two coupled bubbles is − ˙ R c ! R ¨ R + 32 − ˙ R c ! ˙ R = 1 ρ " R c + R c ddt P − ddt R ˙ R d ! , − ˙ R c ! R ¨ R + 32 − ˙ R c ! ˙ R = 1 ρ " R c + R c ddt P − ddt R ˙ R d ! , (1)where P i = (cid:18) P + 2 σR i (cid:19) (cid:18) R i R i (cid:19) γ − η L ˙ R i R i − σR i − P − χ (cid:18) R i − R i (cid:19) − κ S ˙ R i R i − P ac sin( ωt ) , i = 1 , . Here R and R are radii of bubbles, d the distance between the centers of bubbles, P stat is the static pressure, P v is the vapor pressure, P = P stat − P v , P ac is the magnitude of theexternal pressure field and ω is its cyclic frequency, σ is the surface tension, ρ is the density3f the liquid, η L is the viscosity of the liquid, c is the speed of sound in the liquid, γ is thepolytropic exponent, χ and κ s denote the shell elasticity and shell surface viscosity respectively.It can be easily seen that by means of simple transformations equations (1) can be rewrittenin the form of a five-dimensional system of ordinary differential equations in terms of thefollowing dependent variables R , R , ˙ R , ˙ R and θ ∈ [0 , π ] . We perform all our numericalexperiments exactly with this five-dimensional system. However due to its cumbersome formwe do not present it here.In what follows, we assume that P ac , ω and d are treated as the control parameters and theremaining parameters are fixed as follows: P v = 2 . kPa, σ = 0 . N/m, ρ = 1000 kg/m , η L = 0 . Ns/m , c = 1500 m/s, γ = 4 / , χ = 0 . N/m and κ S = 2 . · − kg/s. Thesevalues of the parameters correspond to the adiabatic oscillations of two interacting SonoVuecontrast agents with equilibrium radii R i = 1 . µ m [26].We suppose that the equilibrium radii of bubbles are the same (i.e., R = R = R ),because the injected ensemble of contrast agents is assumed to consist of bubbles of the samecharacteristics. Thus, system (1) is symmetric with respect to the following change of variables: (cid:16) R , ˙ R ↔ R , ˙ R (cid:17) . This symmetry leads to several conclusions.First, there always exists a family of symmetrical solutions, for which ∀ t > R ( t ) = R ( t ) if R ( t = 0) = R ( t = 0) and ˙ R ( t = 0) = ˙ R ( t = 0) , which correspond to fully synchronous in-phase oscillations of bubbles. Some of these solutions can be asymptotically stable (attractive).Since all symmetrical solutions lie in the invariant manifold R = R , ˙ R = ˙ R and system (1)restricted to this manifold is three-dimensional and volume-contracting, such regimes can be oftwo possible types: periodic and simply chaotic (with only one positive Lyapunov exponent).In other words, synchronous oscillations of two bubbles can be either periodic or chaotic, formore details see the next section.Second, asymptotically stable regimes (attractors) can exist outside the invariant manifold R = R , ˙ R = ˙ R . These regimes correspond to asynchronous oscillations of bubbles and, incontrast to synchronous ones, they can be of four possible types. In addition to periodic andsimply chaotic regimes, asynchronous oscillations can be also quasiperiodic and even hyper-chaotic. Moreover, the existence of such asymmetrical regimes lead to the trivial multistabilityin the system. Indeed, for each asynchronous regime passing through point ( R , R , ˙ R , ˙ R ) there exist the symmetrical one passing through ( R , R , ˙ R , ˙ R ) . In Sec. 5 we show that inaddition to this simple multistability, system (1) exhibits more complex types of this phe-nomenon. There we discuss coexistence of different attractors which cannot be obtained bysimple interchanging of variables R ↔ R , ˙ R ↔ ˙ R at the same values of the control param-eters. Below we will refer to the term ’multistability’ describing a situation of coexistence ofseveral substantially different attractors, that are not simply symmetrical with respect to themanifold R = R , ˙ R = ˙ R . Also note that coexistence of two synchronous periodic attractorsboth lying in the manifold R = R , ˙ R = ˙ R is possible.We perform all calculations in the following non-dimensional variables R i = R r i , t = ω − τ , where ω = 3 κP / ( ρR ) + 2(3 κ − σ/R + 4 χ/R is the natural frequency of bubbleoscillations. The non-dimensional bubbles speeds are given by u i = dr i /dτ = ˙ R i / ( R ω ) .We use the fourth-fifth order Runge–Kutta method [27] for finding numerical solutions of theCauchy problem for (1). For the calculations of the Lyapunov spectra we use the standardalgorithm by Bennetin [28]. Poincar´e cross sections are constructed at every period of theexternal pressure field. 4 Variety of dynamical regimes in the model
In this section we demonstrate the diversity of possible dynamical regimes in system (1) andshow that depending on the values of the control parameters d/R and P ac two bubbles canexhibit periodic, quasiperiodic, chaotic or hyperchaotic oscillations. Moreover, due to the sym-metry of the system, periodic and chaotic regimes can be either synchronous or asynchronous.Hyperchaotic and quasiperiodic oscillations cannot be synchronous.Fig. 2a shows a chart of two maximal Lyapunov exponents λ ≥ λ on ( d/R , P ac ) parameterplane for fixed ω = 2 . · s − . This value of ω belongs to the range of frequencies which isrelevant for biomedical applications. Also, the system demonstrates quite rich dynamics at thisfrequency, and typical attractors and bifurcation scenarios are presented in physically relevantinterval of pressures and distances between the bubbles.Depending on values of λ and λ the corresponding pixel on the chart is painted with acertain color using the following scheme: • λ < , λ < – periodic regime – blue color; • λ = 0 , λ < – quasiperiodic regime – green color; • λ > , λ ≤ – simple chaotic regime (strange attractor with one positive Lyapunovexponent) – yellow color; • λ > , λ > – hyperchaotic regime (strange attractor with two positive Lyapunovexponents) – red color;Big blue region on the left in the chart of Lyapunov exponents corresponds to the stableperiodic regimes (see Fig.-s 2d, i), when two bubbles perform in-phase synchronous oscillations.At the top part of the chart this blue region adjoins yellow domain corresponding to a simplestrange attractor with one positive Lyapunov exponent (see Fig. 2h), corresponding to syn-chronous chaotic oscillation of two bubbles. In Sec. 4.1.1 we show that the transition to chaosin this part of the chart occurs via Feigenbaum’s cascade of period-doubling bifurcations.At the middle and bottom parts of the diagram from Fig. 2a the big blue region adjoins thegreen-colored region corresponding to quasiperiodic regime, when torus (invariant curve on thePoincar´e map) is an attractor in the system, see Fig. 2f. The invariant torus in Fig. 2f is bornfrom the asynchronous periodic regime shown in Fig. 2g via the Neimark–Sacker bifurcation.It is worth noting that the asynchronous periodic limit cycle in Fig. 2g is (likely) born via asaddle-node bifurcation, which is not reflected in the two-dimensional chart. Thus if we moveleft (decrease d/R ) from the point corresponding to Fig. 2g to the border of the plot, there willbe a switch from asynchronous limit cycle to a synchronous one (the point where the saddle-nodebifurcation occurs), and this switch cannot be observed in the two-dimensional chart Fig. 2a.Moving through the region corresponding to the quasiperiodic regime from left to right one canobserve transition to chaotic (in the middle part of the chart) and even hyperchaotic (in thebottom part of the cart) attractors. Simple chaotic attractors occurring after the destructionof an invariant torus as well as hyperchaotic attractors, correspond to asynchronous oscillationsof the bubbles, see Fig. 5c. In Sec. 4.1.2 we show that in the middle part of the chart chaoticattractors appear in accordance with Afraimovich–Shilnikov scenario of the destruction of aninvariant torus. In Sec. 4.2 we study the scenario for the onset of hyperchaotic oscillations inthe bottom part of the chart.From Fig. 2a one can see that yellow- and red-colored regions corresponding to chaotic(see e.g. Fig. 2c,h) and hyperchaotic (see e.g. Fig. 2b,e) attractors alternate with the so-called stability windows inside which stable periodic orbits are observed (see Fig. 2a). Someof these stability windows have shrimp-like form [29]. Such stability windows indicate the5igure 2: (a) Charts of Lyapunov exponents for ω = 2 . · s − and (b)–(i) projections of phase portraitsof steady state regimes (attractors) for some representative points of this chart. Projections of attractors onPoincar´e map are pained in black color, projections of phase trajectories for periodic attractors – in blue color.The following attractors are shown here: (b) hyperchaotic attractor at d/R = 32 , P ac = 1 . MPa with largestLyapunov exponents of λ = 0 . , λ = 0 . ; (c) synchronous chaotic attractor at d/R = 30 , P ac = 1 . MPa with λ = 0 . , λ = − . ; (d) syncronous 12-periodic limit cycle at d/R = 28 , P ac = 1 . MPa with λ = − . , λ = − . ; (e) hyperchaotic attractor at d/R = 22 , P ac = 1 . MPa with λ = 0 . , λ =0 . ; (f) quasiperiodic attractor at d/R = 14 . , P ac = 1 . MPa with λ = 0 , λ = − . ; (g) asynchronous4-periodic limit cycle at d/R = 10 , P ac = 1 . MPa with λ = − . , λ = − . ; (h) synchronous chaoticattractor at d/R = 10 , P ac = 1 . MPa with λ = 0 . , λ = − . ; (i) synchronous 2-periodic limit cycleat d/R = 6 . , P ac = 1 . MPa with λ = − . , λ = − . . existence of the specific homoclinic bifurcations (cubic homoclinic tangencies or symmetricalpairs of homoclinic tangencies) in the system [30]. All these stability windows indicate thatchaotic dynamics in system (1) are not hyperbolic and even not pseudohyperbolic [31–33]. Inother words, in the accordance with PQ-hypothesis [33] strange attractors in the system underinvestigation belong to a class of quasiattractors introduced by Afraimovich and Shilnikovin [34]. Stable periodic orbits of high periods and with narrow absorbing domains exist insidesuch attractors or appear with arbitrarily small perturbations. However, from a physical point ofview in most cases quasiattractors do not differ from genuine (pseudohyperbolic) attractors dueto narrow absorbing domains of periodic orbits belonging to them. It is important to note thatcurrently there are no known systems which demonstrate hyperbolic or even pseudohyperbolichyperchaotic behavior. 6 Transition to chaos and hyperchaos
As it can be clearly seen from the chart of Lyapunov exponents (Fig. 2a) the dynamics insystem (1) can be either regular (periodic or quasiperiodic) or chaotic and even hyperchaotic.As a rule, chaotic attractors appear from simple (regular) attractors as a result of the im-plementation of some bifurcation scenario. The most known examples of such scenarios are:1) the Feigenbaum’s cascade of period-doubling bifurcations [35] according to which chaoticattractor appears from a stable periodic orbit via infinite sequence of period doubling bifur-cations; 2) destruction of an invariant torus by Afraimovich-Shilnikov scenario [36]; 3) theShilnikov scenario [22] due to which spiral attractor containing a saddle-focus equilibrium witha two-dimensional unstable invariant manifold appears from a stable equilibrium as a result ofcertain local and global bifurcations. It is worth noting that all the above mentioned scenarioscan be observed in flow system with dimension N ≥ or in diffeomorphisms – discrete system(except for the Shilnikov scenario) with dimension N ≥ and generally lead to the appearanceof chaotic attractors with only one positive Lyapunov exponent.An important class of chaotic attractors of multi-dimensional flows ( N ≥ ) and maps ( N ≥ ), namely so-called homoclinic attractors containing saddle periodic orbits with its homoclinicstructure, was introduced in [21], [20], where the classification of such attractors and alsophenomenological scenarios of their appearance were proposed (see, also [19] for more examplesof such attractors in three-dimensional H´enon maps). The classification of homoclinic attractorsis based on the type of a saddle orbit belonging to an attractor. Two main attractors of thistype are the discrete Lorenz and figure-eight attractors. They contain a saddle fixed point witha one-dimensional unstable manifold forming a homoclinic structure resembling a butterfly andfigure-eight, respectively. As it was recently shown in [32] these two attractors belong to a classof pseudohyperbolic [31], [37] (“genuinely” chaotic) attractors. Shortly speaking, each orbit ona pseudohyperbolic attractor has a positive Lyapunov exponent and, what is important from aphysical point of view, this property persists after small perturbations (changing in parameters).However, both the discrete Lorenz and figure-eight attractors cannot be hyperchaotic.Another important example of a homoclinic strange attractor proposed in [21], [20] is a discrete Shilnikov attractor . In contrast to the all above mentioned examples of strange attrac-tors, the discrete Shilnikov attractor contains a saddle-focus fixed point with two-dimensionalunstable invariant manifold. In all small neighborhoods of such fixed point, two-dimensionalareas are expanded. Lyapunov exponents on this attractor as a whole ’can feel’ this expansion,and thus, two Lyapunov exponents can be positive. Unlike the discrete Lorenz and figure-eightattractors, the discrete Shilnikov attractor cannot be pseudohyperbolic. This attractor belongsto another class of strange attractors called by Afraimovich and Shilnikov in [34] as quasiattrac-tors . Homoclinic tangencies inevitably arise in quasiattractors and lead to the birth of stableperiodic orbits inside such attractors. Thus, these attractors either contain a stable periodicorbit with large periods and narrow absorbing domains or such orbits appear after arbitrarysmall perturbations (parameter changing).Discrete Shilinikov attractors were found in different dynamical systems such as the gener-alized three-dimensional H´enon maps, nonholonomic models of Chaplygin top [38] and Celticstone [45], the model of coupled identical oscillators [39].However, not in all cases they were iden-tified as hyperchaotic attractors. Apparently, in some cases the expansion of two-dimensionalareas near a saddle-focus orbit with a two-dimensional unstable manifold is compensated by thevolume contraction near some other saddle periodic orbits that also belong to the attractor, butwhich have only a one-dimensional unstable manifold. Since Lyapunov exponents are averagecharacteristic of an attractor, only one Lyapunov exponent can become positive in this case.In Sec. 4.2 we show that for system (1) discrete Shilnikov attractors containing a saddle-focusperiodic orbit are hyperchaotic. We also propose a new phenomenological scenario which leads7o the appearance of hyperchaotic attractor and demonstrate that exactly due to this scenariohyperchaos appears in the system under consideration. However first of all we describe scenariosof transition to simple chaotic (with only one positive Lyapunov exponent) attractors in themodel. Feigenbaum’s infinite sequence of period-doubling bifurcations [35] is one of the typical scenariosof the chaos onset in one- and two-dimensional maps and three-dimensional flows. However,such scenario is also found in multi-dimensional maps ( N ≥ ) and flows ( N ≥ ) (see, e.g.[40]). On the other hand, since in multi-dimensional systems period-doubling bifurcationscompete with Neimark-Sacker bifurcations, the transition to chaos via Feigenbaum’s cascadefor multi-dimensional systems is more rare than in the case of two-dimensional maps and three-dimensional flows.Here we show that strange attractors in system (1) can appear via Feigenbaum’s cascade ofperiod-doubling bifurcations. Let us fix P ac = 1 . and move along the path KL : ( P ac , d/R ) =(1 . · , → ( P ac , d/R ) = (1 . · , . . We start from d/R = 8 because periodicattractors at d/R = 6 . (see Fig. 2i) and d/R = 8 (see Fig. 3b) look identical and we do notlose any information by starting from this point. Fig. 3a shows the corresponding bifurcationtree in this case. Phase portraits of attractors at some values of parameter d/R are presentedin Figs. 3b–d, where in blue color we show the projection of the phase curves onto the ( r , u ) plane, and in black color – projections of the corresponding Poincar´e map (at t = 2 πk, k ∈ N )on the same plane. At the starting point of the path a stable periodic orbit (point of period2 on the Poincar´e map) is an attractor of the system, see Fig. 3b. When the parameter d/R increases this periodic orbit undergoes cascade of period doubling bifurcations, see Figs. 3c,dand finally, at d/R ≈ . , Feigenbaum-like strange attractor emerges, see Fig. 3e where thecorresponding Poincar´e map is presented. The set of Lyapunov exponents for this attractor at ( d/R, P ac ) = (9 . , . is λ = 0 . , λ = − . , λ = − . , λ = − . .Figure 3: Transition from synchronous periodic to synchronous chaotic oscillations on the path: P ac = 1 . MPa, . < d/R < . via the Fegeinbaum‘s cascade. (a) bifurcation tree; (b) projection of the phase portraitof the 2-periodic limit cycle at d/R = 8 ; (c) 4-periodic limit cycle at d/R = 9 ; (d) 16-periodic limit cycle at d/R = 9 . ; (e) projection of the Poincar´e section of the chaotic attractor at d = 9 . · R with two largestLyapunov exponents: λ = 0 . , λ = − . . In the general case Feigenbaum’s cascade gives rise to the onset of a strange attractor8ith only one positive Lyapunov exponent. However, it is important to note, that at somespecific cases such scenario can lead to the onset of hyperchaotic behavior. For example,if we take two identical oscillators demonstrating transition to chaos via cascade of period-doubling bifurcations and make them interact through a very weak coupling, both elementswill demonstrate chaotic behavior, which will lead to two positive Lyapunov exponents in thecoupled system. Such transition to hyperchaos was observed e.g. in [41]. Since we take twoidentical elements the transition to hyperchaos in accordance with this scenario is also possiblein our system if we suppose that the bubbles are quite distant (i.e. d/R ≪ ). However, thiscase is less interesting from a physical point of view and, therefore, we do not consider it here. As one can see from the chart of Lyapunov exponents (Fig. 2) some regions of parameterscorresponding to stable periodic regimes adjoint the region with stable qusiperiodic regimes– invariant tori. The boundary between these regions is formed by the curve of supercriticalNeimark-Sacker bifurcation. Passing through this curve stable limit cycle loses its stability,becomes of a saddle-focus type with a two-dimensional unstable invariant manifold and a stableinvariant torus appears.From another side of the regions of the existence of quasiperiodic regimes the dynamicsof system (1) can be chaotic. This means that in system (1) chaotic attractors can appearafter destruction of a torus. There are a few typical scenarios of transition to chaos throughthe destruction of an invariant torus. One of such scenarios was proposed by Afraimovich andShilnikov in [36].Figure 4: (a) Sketch of the bifurcation diagram illustrating bifurcation of an invariant torus. P , Q , P i and C – regions of the existence of (b) stable periodic orbit, (c) stable invariant torus, (d) resonant periodic orbits,and (e) torus-chaos attractors, respectively. M N – some path along which torus-chaos attractor appears inaccordance with Afraimovich-Shiknikov scenario.
Here we show that the Afraimovich–Shilnikov scenario is the second typical scenario (the9rst is Feigenabum’s cascade of period-doubling bifurcations) for the onset of chaos in system(1). But first of all, let us recall some important details concerning typical organization of abifurcation diagram inside region Q of an invariant torus existence for two-dimensional maps,see Fig. 4a. Resonance regions P i – the so-called tongues of synchronization alternate withquasiperiodic regions Q above the curve of the Neimark-Sacker bifurcation N S and with chaoticregions in the upper part of the diagram. Notice that in the tongues of synchronization resonantstable and saddle periodic orbits appear on torus (through the saddle-node bifurcations SN ),while this torus still exists, but now it is formed by the closure of the unstable invariant manifoldof the resonant saddle orbit , see Fig. 4d. Moving along arbitrary path in the parameter planeFigure 5: Afraimovich–Shilnikov scenario for the onset of torus-chaos attractor along the path OP: P ac = 1 . MPa, d/R ∈ [13 , . (a) bifurcation tree and (b) map of the two largest Lyapunov exponents associated withthis path; (c)–(h) Projections of attractors onto ( r , r ) -plane: (c) stable limit cycle (point of period 4 on thePoincar´e map) at d/R = 13 ; (d) four-component torus after the Neimark-Sacker bifurcation at d/R = 14 . ; (e)resonance on the torus, d/R = 16 . ; (f) resonance after the first period-doubling bifurcation, d/R = 16 . ;(g) d/R = 16 . , Feigenbaum’s cascade continunues leading to chaos ; (h) – torus-chaos at d/R = 17 . withthe following Lyapunov spectre λ = 0 . , λ = − . , λ = − . , λ = − . . one can observe sequences of alternated regular, quasiperiodic and chaotic regimes. Thus, itis important to note that bifurcations of an invariant torus and, in particular, transition tochaos depend on the path in the bifurcation diagram. Moreover, the parts of this path from For multi-dimensional maps ( N ≥ ) the structure of bifurcation diagrams is similar but slightly differ-ent from two-dimensional case. Inside tongues of synchronization, together with period-doubling bifurcations,Neimark-Sacker bifurcations are also possible, see Fig. 6a. M N one can observe the following sequence of regimes: stable periodic orbit (Fig. 4b), stable torus(Fig. 4c), resonance torus existing in resonance region P j (Fig. 4d), and, finally, torus-chaosattractor (Fig. 4e).According to Afraimovich and Shilnikov [36], in two-parametric families of two-dimensionalmaps, destruction of an invariant torus inside resonance regions can happen due to the followingscenarios: 1) period-doubling bifurcation with a stable resonant orbit (e.g. if to move upwardsinside a resonance region); 2) homoclinic bifurcation, when unstable invariant manifold of theresonant saddle periodic orbit touches (and than intersects) its stable manifold (e.g. if to moveinside a resonance region through the curve SN to the chaotic region, see path M N in Fig. 4a);3) more complex and difficult to observe scenario associated with the increasing of oscillationsof the unstable manifold of a resonant saddle orbit, see details in [36].What is important, in all these cases, when leaving a resonance region, one can observe thechaotic regime associated with the previously existed invariant torus. Such chaotic attractorswere called as torus-chaos attractor in [36].Fig. 5 gives an illustrative example of the onset of torus-chaos attractor in system (1) inaccordance with Afraimovich-Shilnikov scenario. Here we fix P ac = 1 . MPa and move alongpath OP from the chart of Lyapunov exponents (Fig. 2): < d/R < . Fig.-s 5a and 5bshow the corresponding bifurcation tree and the graph of the two largest Lyapunov exponentsin this case. Portraits of some attractors along this path are presented in Fig.-s 5c–h, wherein blue color we show the projection of phase portraits onto ( r , r ) plane, and in black color –projections of the corresponding Poincar´e map on the same plane ( r , r ) .The stable limit cycle (stable fixed point of period four on the Poincar´e map) existing at d/R = 13 (see Fig. 5c) undergoes the Neimark-Sacker bifurcation at d/R ≈ . after whicha stable invariant torus (four-component invariant curve in the Poincar´e map) appears, seeFig. 5d. Then, at d/R ≈ . we get into a resonance region where stable periodic orbitappears, see Fig. 5e. Moving inside this resonance region the stable periodic orbit under-goes Feigebaum’s cascade of period-doubling bifurcations, see Fig.-s 5e–g for details.Finally, at d/R ≈ . , we get out of the resonance region and torus-chaos attractor appears, see Fig. 5g.In the next subsection we will demonstrate that some paths out of resonance regions lead tothe onset of hyperchaotic attractors. We also will give a bifurcation scenario of such transition. (1) Starting from three-dimensional maps (four-dimensional flows), in addition to Afraimovich-Shilnikov scenarios, some other ways of the destruction of an invariant torus become possible.Here we would like to mention the following two scenarios: 1) cascades (finite) of period-doubling bifurcation of an invariant torus [42], [43], [46]; 2) secondary Neimark-Sacker bifurca-tion with a stable resonant periodic orbit inside a tongue of synchronization.Thus, a typical bifurcation diagram near the Neimark-Sacker bifurcation for three-dimensionalmaps differ from the corresponding diagram in two-dimensional case, see Fig. 6a. The maindifference is that a resonant periodic orbit in three-dimensional case can undergo the secondaryNeimark-Sacker bifurcation
N S instead of a typical for two-dimensional maps period-doublingbifurcation, see right-top part in Fig. 6a.Secondary Neimark-Sacker bifurcation is the first (but not main) step in our scenario forthe onset of hyperchaotic attractors. After this bifurcation multi-round stable invariant torus(multi-component invariant curve in the Poincar´e map) is born in the system, while, what isvery important, resonant periodic orbit becomes a saddle-focal with a two-dimensional unstablemanifold, see Fig. 6e, where the saddle-focus periodic orbit is denoted as SF i .The next step in the framework of this scenario is associated with the destruction of a11igure 6: Sketch of the bifurcation diagram illustrating the scenario of the onset of a hyperchaotic attractor inmulti-dimensional maps ( N ≥ ). P , Q , P i , C and H – regions of the existence of (b) stable periodic orbit, (c)stable invariant torus, (d) resonant periodic orbits, (e) stable torus after secondary Neimark-Sacker bifurcation N S , (f) torus-chaos attractors, and (g) hyperchaotic Shilnikov attractor, respectively. M N – some path alongwhich hyperchaotic attractor appears. multi-round stable invariant torus. It does not matter how it happens through the Afraimovich-Shilnikov scenario, cascade of torus period-doubling bifurcations or even via the tertiary Neimark-Sacker bifurcation. But we suppose, and it is quite natural, that after the corresponding bi-furcations torus-chaos attractor with one positive Lyapunov exponent appears, see Fig. 6f. Itis worth noting that in this case, immediately after transition to chaos, saddle-focus SF i doesnot belong to the torus-chaos attractor. It means that orbits of this chaotic attractor do notattend some neighborhood of SF i , see Fig. 6f.The final, and the key step in the scenario is the inclusion of the saddle-focus periodicorbit SF i which appeared after the secondary Neimark-Sacker into the chaotic attractor. Afterthis inclusion, saddle-focus orbit SF i together with its two-dimensional unstable manifold andits homoclinic structure starts to belong to the attractor, i.e. discrete homoclinic Shilnikovattractor based on this saddle-focus orbit emerges, see Fig. 6g. Orbits on this attractor can passarbitrary close to SF i , where two-dimensional areas are expanded. As a result, two Lyapunovexponents become positive i.e. a hyperchaotic attractor is born.The inclusion of a saddle-focus periodic orbit to the chaotic attractor can occur in differentways. It depends on the transition from the stable multi-round torus to chaotic regime. In allknown models demonstrating the onset of the discrete Shilnikov attractor (in three-dimensionalH´enon maps [19], nonholonomic models of Chaplygin top [38] and Celtic stone [45]) this inclu-sion happens in a soft manner by a smooth transformation of a torus-chaos attractors. However,we also suppose that a saddle-focus orbit can be included to the chaotic attractor sharply dueto the crisis of multi-round torus-chaos attractor.In order to support the proposed scenario we show that the transition to hyperchaos alongpaths AB and GH (see Fig. 2a) happens in full compliance with this scenario. Firstly, letus consider the path AB , corresponding to the following parameters interval: P ac = 1 . MPaand < d/R < . The bifurcation tree, corresponding to this route is shown in Fig. 7a12igure 7: The implementation of the proposed scenario of the onset of a hyperchaotic attractor along the path AB : P ac = 1 . , < d/R < . (a) and (b) bifurcation tree and the graph of two largest Lyapunov exponentsassociated with this path with the enlarged area for . < d/R < ; (c)–(h) projections of the Poincar´emaps for different attractors on the ( r , u ) plane: (c) four-component torus at d/R = 16 ; (d) resonance orbitat d/R = 17 . ; (e) multi-component torus after the secondary Neimark-Sacker bifurcation, d/R = 17 . ;(f) high-period resonance orbit emerges on the secondary torus and starts to go through the period-doublingbifurcations, d/R = 18 . ; (g) – chaotic attractor after the period-doubling cascade at d/R = 18 . with twolargest Lyapunov exponents λ = 0 . , λ = − . , one can see the gaps in the chaotic attractor existingaround the saddle-focus orbit ; (h) hyperchaotic Shilnikov attractor containing saddle-focus periodic orbit witha two-dimensional unstable manifold at d/R = 18 . . and the graph of two largest Lyapunov exponents is presented in Fig. 7b. We also showthe enlarged part for both these graphs at the right panels in Fig.-s 7a,b in order to explore13ome important details concerning secondary Neimark-Sacker bifurcation and the transition tohyperchaos. Projections of the Poincar`e sections for some representative attractors are shownin Fig.-s 7c–h.From Fig. 7 one can observe the following bifurcations sequence happening tothe asynchronous 4-periodic limit cycle existing at d/R = 13 : it undergoes the Neimark-Sacker bifurcation at d/R ≈ . and the 4-component torus arises, see Fig. 7c. Then ahigh-periodic resonance occurs on the torus, see Fig. 7d. With further increasing of d/R the multi-component torus emerges after the secondary Neimark-Sacker bifurcation while theformer resonance orbit becomes saddle-focus with two-dimensional unstable manifold, see 7e.Soon the multi-component torus gives rise to the torus-chaos attractor (see Fig. 7g), whichappears via the cascade of period-doubling bifurcations happening with some stable resonantorbit emerging on this torus (see the long-periodic orbit emerging after few period-doublingbifurcations in Fig. 7f). It is important to note that saddle-focus orbit occurring after thesecondary Neimark-Sacker bifurcation does not belong to this torus-chaos attractor. Gazingat Fig. 7g, one can see some visible gaps existing around the saddle-focus orbit. Finally, at d/R ≈ . , the saddle-focus orbit starts to belong to the chaotic attractor. As a result thehyperchaotic attractor, containing this saddle-focus orbit appears, see Fig.7h. The Lyapunovexponents at d = 18 . are λ = 0 . , λ = 0 . , λ = − . , λ = − . .Further we demonstrate several more routes illustrating the same scenario of the onsetof a hyperchaotic attractor. Let us fix P ac = 1 . and move along the path GH from thechart of Lyapunov exponents in Fig. 2a. Fig.-s 8a,b show the corresponding bifurcation treeand a the graph of the largest Lyapunov exponents along this path. From these figures onecan see that at d/R ≈ . two Lyapunov exponents become positive, i.e. hyperchaoticattractor appears. Portraits of some represntative attractors for several values of parameter d/R are presented in Fig.-s 8c–h, where projections of the Poincar´e map onto the ( r , r ) plane are shown. The beginning of the route GH corresponds to an asynchronous 4-periodiclimit cycle. At d/R ≈ . it undergoes the Neimark-Sacker bifurcation, after which a stableinvariant torus (four component invariant curve on the Poincar´e map) appears, see Fig. 8c.With increasing d/R we pass through few resonance regions (see Fig. 8d). Soon after this theinvariant torus starts to loss its smoothness, see Fig. 8e. Then, leaving one of the resonanceregions, torus-chaos attractor with one positive Lyapunov exponent and the following spectreappears: λ = 0 . , λ = − . , λ = − . , λ = − . , see Fig. 8f. With furtherincreasing d/R we again pass through a resonance region, but now the resonant orbit undergoesthe secondary Neimark-Sacker bifurcation at d/R ≈ . after which a multi-round invarianttorus (multi-component invariant curve on the Poincar´e map) appears, while the resonanceorbit becomes saddle-focus SF with a two-dimensional unstable manifold, see Fig. 8g. Not longafter this multi-round invariant torus gives rise to a chaotic attractor and finally the discretehyperchaotic Shilnikov attractor containing the saddle-focus orbit SF appears (see Fig. 8g).The set of Lyapunov exponents for this hyperchaotic attractor at ( d/R, P ac ) = (10 . , . is λ = 0 . , λ = 0 . , λ = − . , λ = − . .The third route we breifly discuss here is the route EF: d/R = 27 . , . MPa < P ac < . MPa. Unlike for the previous routes, for this one we fix d/R and vary P ac . In Fig. 9a,bwe present the bifurcation diagram and the graph of two largest Lyapunov exponents associatedwith the route. At the beginning of this path there is an asynchronous 6-periodic limit cycle.Increasing P ac we can observe the first Neimark-Sacker bifurcation occuring at P ac ≈ . MPa(see Fig. 9b). The bifurcation scenario proposed earlier is very hard to notice, unless we study asignificantly enlarged picture, see 9c. Further increasing P ac leads to emergence of a resonance.Soon, at P ac = 1 . MPa it undergoes the secondary Neimark-Sacker bifurcation at whichthe former resonance orbit becomes saddle-focus with two-dimensional unstable manifold anda multi-component torus emerges, see 9d. This secondary torus is very hard to notice becauseit exists in a very narrow range of pressures. Further increasing of P ac leads to inclusion of the14igure 8: The implementation of the same scenario of the onset of hyperchaotic attractor along the path GH : P ac = 1 . MPa, . < d/R < . (a) and (b) bifurcation tree and graph of two largest Lyapunov exponentsassociated with this path; (c)–(h) projections of the Poincar´e map of several attractors on the ( r , r ) plane.(c) four-component torus at d/R = 10 . ; (d) resonance orbit at d/R = 10 . ; (e) torus starts losing itssmoothness, d/R = 10 . ; (f) torus-chaos attractor at d/R = 10 . ; (g) multi-component torus afterthe secondary Neimark-Sacker bifurcation, d/R = 10 . ; (h) hyperchaotic attractor containing saddle-focusperiodic orbit with a two-dimensional unstable manifold at d/R = 10 . . saddle-focus orbit with two-dimensional unstable manifold into the attractor and it becomeshyperchaotic.Finally, we would like to raise an open question. In which cases a discrete Shilnikov attractoris hyperchaotic and in which it has only one positive Lyapunov exponent and what exactlyresponse for it. For example, the discrete Shilnikov attractors from the nonholonomic model ofChaplygin top [38] and Celtic stone have only one positive Lyapunov exponent. On the otherhand, such attractors are hyperchaotic in three-dimensional H´enon maps [44], [19], the modified15igure 9: Transition to hyperchaos along the route EF: d = 27 . · R , P ac : . MPa < P ac < . MPa. (a),(b) bifurcation tree and graph of two largest Lyapunov exponents associated with this path; (c) enalarged partof the graph of the Lyapunov exponents, corresponding to . MPa < P ac < . MPa; (d) projection ofthe Poincar´e map of the multicomponent torus after the secondary Neimark–Sacker bifurcation on the ( r , r ) plane. oscillator of Anishchenko-Astakhov and the model under consideration. We suppose that insome cases two-dimensional area expansion near the saddle-focus orbit with two-dimensionalunstable manifold is compensated by the volume contraction near some other saddle periodic(quasiperiodic) orbits that also belongs to the attractor but which has only one-dimensionalunstable manifold, and thus, two-dimensional areas are contracted in their neighborhood. SinceLyapunov exponents are average characteristic of the attractor, only one Lyapunov exponentcan become positive in this case. Now let us consider several routes, mentioned in Fig. 2a, for which multistability is presented,namely, routes AB, CD, MN. First of all, we provide one more two-dimensional map in Fig. 10,corresponding to the second leaf of the two-dimensional map presented in Fig. 2a. Comparisonof Fig.-s 2a and 10 shows that there a lot of substantially multistable areas in this parametersregion (not even taking into consideration the possibility of coexistence of different attractorsof the same type, for example two different periodic attractors, which are not reflected in thiskind of maps).Let us start the one-parameter analysis from the path AB lying in the following parametersinterval: P ac = 1 . MPa, R < d < R . We have already shown that one of the attractorsexisting at the point A (the asynchronous one) goes through the scenario described in the pre-vious section and becomes hyperchaotic, see Fig. 7. However there exists also a synchronousperiodic attractor at the point A, which, on the same path, goes through a Feigenbaum’s cas-cade of the period-doubling bifrucations and becomes synchronous chaotic with one positiveLyapunov exponent. In Fig. 11 we present the bifurcation tree and the graph of two largestLyapunov exponents corresponding to the transition of the synchronous periodic to the syn-chronous chaotic oscillations on the route AB. This also provides an example of coexisting of a16igure 10: Another leaf of the chart of Lyapunov exponents for ω = 2 . · s − . Comparsion to Fig. 2demonstrates signifacnt presence of multistability. synchronous chaotic oscillations with a hyperchaotic attractor (compare Fig.-s 7 and 11).Route CD corresponds to P ac = 1 . MPa and the following d/R interval: < d/R < . ,see Fig. 12. No hyperchaotic attractors are presented on this route, but several examples ofmultistability: coexistence of quasiperiodic oscillations with periodic, quasiperiodic and chaotic,asynchronous periodic and synchronous chaotic, quasiperiodic and synchronous periodic.A lot of complicated behavior can be observed on the path MN. For this route we fix P ac = 1 . MPa and vary d in the following interval: . < d/R < . For this pathwe provide two bifurcation trees which correspond to different attractors in Fig.-s 13a,b andpresent the merged picture of the Lyapunov spectra for both attractors in Fig. 13c. We do thatin order to explicitly show coexistence of attractors of different types. In Fig. 13c, Lyapunovexponents λ and λ are two largest exponents of the attractor corresponding to Fig. 13 a, and λ and λ are the largest exponents of the attractor corresponding to Fig. 13c. λ f == 0 is thereferent exponent, which is always zero and the same for both attractors. On the right side ofthe diagram in Fig.-s 13a,c one can observe an abrupt shift from hyperchaos to chaos occuringat d/R ≈ . . This represents the moment when the chaotic attractor, corresponding to theone in diagram Fig. 13b, remains the only attractor in the system, and the system switchesto it. In the remaining interval . < d/R < , the Lyapunov spectra overlap and thebifurcation diagrams in Fig. 13a,b look identical, because they represent the same attractor.Figure 11: (a), (b) bifuraction tree and graph of two largest Lyapunov exponents for the bifurcation sequenceof the synchronous attractor on the path AB: · R < d < · R , P ac = 1 . MPa. Compare to Fig. 7.
Route CD: · R < d < . · R , P ac = 1 . MPa. (a), (b) bifurcation tree and two largestLyapunov exponents, corresponding to the synchronous attractor. (c), (d) bifurcation tree and two largestLyapunov exponents, corresponding to the asynchronous attractor.
Figure 13: (a), (b), bifuraction trees corresponding to different attractors and (c) graph of two largest Lya-punov exponents for both the attractors on the route MN: . · R < d < · R , P ac = 1 . MPa. Lyapunovexponents λ and λ correspond to the attractor associated with plot (a), λ and λ – to the attractor associatedwith plot (b). A variety of types of multistability can be observed on a single path. d/R ≈ . . The Lyapunov exponents λ and λ leap atthis point and begin to overlap with λ , λ to the left of it, see the left side of Fig. 13c. Itrepresents the shift from the disappearing stable limit cycle (the very left side of the bifurcationtree Fig. 13b) to the chaotic attractor corresponding to Fig. 13a. For all the lower values of d these attractor (and their spectra) coincide and we don‘t draw all the exponents further left.Thus the two attractors discussed here coexist in the interval . < d/R < . . A lot ofdifferent kinds of multistability can be observed in this interval with these two attractors. InFig. 13c one can see the following types of coexisting attractors: chaotic with periodic, chaoticwith quasiperiodic, hyperchaotic with periodic, hyperchaotic and quasiperiodic, hyperchaoticand chaotic.Note that the the structure of the border between quasiperiodic and hyperchaotic regimesaround the route MN in Fig. 10 looks very similar to the same border around the route AB inFig. 2a. The structure of the graph of the Lyapunov exponents λ , λ in Fig. 13c also lookssimilar to those in Fig. 7b and in Fig. 8b. This leads us to the conclusion that it is highelylikely that the onset of the hyperchaotic attractor corresponding to Fig. 13b happens by thesame scenario that was described in Sec. 4.2. In this work we have studied the nonlinear dynamics of two encapsulated interacting gas bub-bles in a liquid. We have showed that the oscillations of bubbles can be periodic, quasyperiodic,chaotic and hyperchaotic. Moreover, we have observed multistability phenomenon in a wideregion of the control parameters, which makes bubbles’ dynamics even more complicated. Webelieve that both quasyperiodic and hyperchaotic oscillations along with multistability phe-nomenon are reported for the first time.Concernring the onset of chaotic dynamics, we have studied typical roots to chaos andhyperchaos in system (1). We have demonstrated that simple chaotic attractors (which onlyone positive Lyapunov exponent) occur either via the Feigenbaum’s cascade of period-doublingbifurcations or by the Afraimovich–Shilnikov scenario of the destruction of invariant tori. Onthe other hand, for the onset of hyperchaotic oscillations we propose a new scenario which isbased on the appearance of a discrete Shilnikov attractor containing a saddle-focus periodicorbit with its two-dimensional unstable manifold. Orbits on this attractor can pass arbitraryclose to this saddle-focus orbit, where two-dimensional areas are expanded. As a result, twoLyapunov exponents become positive i.e. a hyperchaotic attractor is born. As we know theproposed scenario gives one of few known explanations of the emergency of hyperchaotic behav-ior. Moreover, we believe that this scenario may be typical for other multidimensional systemsdemonstrating transition to hyperchaos via the destruction of a torus.We have also studied multistability phenomenon in system (1). We have showed that vari-ous types of attractors, both synchronous and asynchronous, regular (periodic or quasiperiodic)and chaotic, and even hyperchaotic can co-exist at the same values of the control parameters.In particular, synchronous periodic regimes can coexist with asynchronous periodic, quasiperi-odic, asynchronous chaotic or hyperchaotic states, and even coexistence of several sycnhronousregimes is possible (for example, two different synchronous periodic limit cycles). We havedemonstrated that in multistable states hyperchaotic regimes can coexist with regular andchaotic (both synchronous and asynchonous) ocillations, as well as with asynchronous qusiperi-odic regimes.As far as applications are concerned, it is known [3, 6] that chaotic oscillations of bubblescan be beneficial for blood flow visualization. Thus, we believe that the regions of the controlparameters where either one chaotic or hyperchaotic attractor exists or both of these attractorsco-exist, may be recommended for this type of applications. On the other hand, the regions19f the control parameters where different types of attractors exist (e.g. periodic and quasipar-iodic) should be avoided in applications, since in this case the dynamics of bubbles becomesvirtually unpredictable due to the fact that small perturbations in the initial conditions orcontrol parameters may lead to a substantial change in bubbles’ acoustic response.
Acknowledgments
This paper (except Section 3) was supported by RSF grant 17-71-10241. The results in Section3 were supported by RFBR grant 18-31-20052. A.K also thanks Basic Research Program atNRU HSE in 2019 for support of scientific researches.