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From: Ambassador Tkachenko

Proliferation of stability in phase and parameter spaces of nonlinear systems

In this work, we show how the composition of maps allows us to multiply, enlarge, and move stable domains in phase and parameter spaces of discrete nonlinear systems. Using Hénon maps with distinct parameters, we generate many identical copies of isoperiodic stable structures (ISSs) in the parameter space and attractors in phase space. The equivalence of the identical ISSs is checked by the largest Lyapunov exponent analysis, and the multiplied basins of attraction become riddled. Our proliferation procedure should be applicable to any two-dimensional nonlinear system.
In high-dimensional dynamical systems, the possibility of controlling the dynamics through parametric changes is of great interest. Adding a time dependent parameter, it is possible to control the dynamics of the paradigmatic Hénon map (HM) generating multiply Isoperiodic Stable Structures (ISSs) on the parameter space and consequently increasing the number of attractors on the phase space. Numerical simulations and analytical results explain the origin of new stable domains due to saddle-node bifurcations for specific parametric combinations. The distance between the multiplied ISSs can be controlled by the intensity of the time dependent parameter, and general rules for the occurrence of proliferation are treated in detail. We believe that the present study provides a significantly new insight into the use of alternating forces to control the dynamics of complex nonlinear systems modeled by two-dimensional maps and can be extended to various applications ranging from physics, biology, to engineering.
A huge number of physical systems in nature present regular or chaotic dynamics depending on parameters and initial conditions. We mention granular dynamics, coupled networks, brain dynamics, market crisis, chemical systems, laser physics, transport, extreme events, and weather forecast, among many others. In nonlinear dynamics, it is essential to know the correct parameter combination which leads to (or avoids) a specific dynamics. One crucial step for this was the discovery of structures in the parameter space of dynamical systems. Such structures, called Isoperiodic Stable Structures (ISSs), are Lyapunov stable islands in the parameter space and are supposed to be generic in dynamical systems. For parameters chosen inside the ISSs, the corresponding dynamics is stable and regular. ISSs were found in many systems, and we would like to mention some of them, namely, in theoretical1 and experimental2electronic circuits, continuous systems,3–9 maps3,10–15 laser models,16 cancer models,17 classical,18–20 and quantum Ratchet systems.21–23 For the description of nature processes, it is essential to discover generic properties for parameter combinations in nonlinear dynamical systems which can be applied to anyrealistic situation, independent of the specific physical system.
In this work, we investigate the non-trivial dynamics of the composition of two-dimensional discrete maps. We use the paradigmatic Hénon map (HM), whose relevant dynamics should be visible in any two-dimensional dissipative map. It is shown that composing HMs with distinct parameters, following a specific protocol, it is possible to generate multiple ISSs which can be split in the parameter space. Multioverlapping identical copies of the ISSs start to separate from each other with the increasing intensity of the perturbative parameter F, enlarging the available stable domain in phase and parameter spaces. The generated overlapping ISSs are enlarged ISSs, which are found to be the factorized composition of identical copies of the original ISSs. The proposed method is generic and can be applied to ordinary problems involving nonlinear behaviors. The results for du-, tri-, sextu-, and decuplications are described for the composition of Hénon maps with distinct parameters. Indications for the possible duplication of structures in parameter space were given for the composition of two quadratic coupled maps in the context of chaos suppression.24 The replication of a shrimp-like ISS was observed in a continuous oscillator,8 but its origin remained unknown. This work extends previous results for one-dimensional systems25 to the non-trivial two-dimensional case.
This paper is presented as follows. In Sec. II, we summarize the main properties observed in the one-dimensional case, and in Sec. III, the proliferation of shrimp-like ISSs in the parameter space of the two-dimensional Hénon map is presented. Section IV shows that multiple attractors are created in phase space with the corresponding riddled basin of attraction. In Sec. V, we generalize our procedure showing the duplication of other more complicated ISSs, and Sec. VIshows analytical results for the duplication of period 2 (shortly written per-2) stability boundaries in parameter space. This corresponds to the duplication of ISSs. Section VII summarizes our results.
Recently, it was shown25 that by controlling the dynamics of composed one-dimensional quadratic maps (QMs), multiple independent attractors and independently shifted bifurcation diagrams can be generated. The appearance of extra stable motion together with the prohibition of period doubling bifurcations (PDBs) is the mechanism which leads to shifted bifurcation diagrams. An analogous mechanism was revealed many years ago26 in the context of taming chaos in continuous systems under weak harmonic perturbation. The above mentioned mechanism can be briefly explained in a simple example. Consider the Modified Quadratic Map (MQM) xn+1=a−xn2+F (−1)n, with n = 0, 1, 2, …, N, and parameters (a, F), where F is the intensity of the external force with alternating signals +F,−F,+F,−F,…. Note that this is a composition of two QMs with alternating (k = 2 periodic) parameters. Besides period k of the external force, we have also period p of the variable xn. For F = 0, the above map suffers a PDB from period p = 1 → 2 at a1→2 = 0.75 and a PDB from period p = 2 → 4 at a2→4 = 1.25. It is clear that for F ≠ 0, no orbit with period p = 1 exists anymore and the PDB at a1→2 becomes forbidden. In fact, it was shown25 that this PDB is transformed in a saddle-node bifurcation and two orbits of period p = 2 exist with distinct stabilities. Consequently, for increasing values of a, these pairs of per-2 orbits suffer PDBs at distinct values of a2→4 and subsequently along the whole PDB sequence a4→8, a8→16,…, leading to two independently shifted bifurcation diagrams.
Other compositions of QMs can be used, and in general, it was shown for one-dimensional systems that the k-composition of QMs with distinct parameters induces dynamics which follows the rules: (a) generates k-attractors and k-independently shifted bifurcation diagrams when ω∈ℤ, where ω = p/k. In this case, pF = p, where pF is the orbital period for F ≠ 0. (b) When ω∉ℤ, the orbits of period p become pF -periodic, where pF = kp. These apparently simple rules generate complex behaviors. For example, suppose a PDB sequence p → 2p→ 4p →…,lp. For F 0, all PDBs with ω = lp/k < 1 become forbidden by the composition of the k QMs. This prohibition is responsible for the generation of new periodic orbits via a saddle-node bifurcation, and k-shifted bifurcation diagrams are created.
While our paper25 explains the basic mechanism for the appearance of multiple bifurcation diagrams in a family of one dimensional quadratic maps, the present work analyses the effects of such a mechanism in two dimensional systems with two parameters. Specifically, we are interested in the behaviour of the ISSs, whose relevance in the description of dynamical systems was explained in Sec. I. In general, the above rules should be extended to two-dimensional systems with two parameters. A complex behavior is expected for the ISSs as a function of the perturbation with many interesting new features, as will be discussed next. The two-dimensional modified Hénon map (MHM) is given by

xn+1=a−xn2+b yn+g(F,n),yn+1=xn, (1)

with states xn and yn calculated at discrete times n = 0, 1, 2,…, N, parameters a, b, and F, and the function g(F, n) with period k which will define the protocol of proliferation. For b = 0, map (1) reduces to the MQM in Ref. 25. The MHM corresponds to the HM with a time dependent parameter aj′=[a+g(F,n)].

A. Duplication of shrimp-like ISSs (k = 2)
For the duplication, we use g(F,n)=F (−1)n so that the protocol is F, −F, F, −F,… and function g(F, n) is k = 2 periodic. Figure 1 shows the period of trajectories in the parameter space (a, b). Each color represents a given period (see the color bar). In Fig. 1(a), the case F = 0 is displayed and almost any ISS has a shrimp-like form with distinct periods. Inside each ISS, a sequence of PDBs p → 2p → 4p →, …, lp occurs. For a detailed description of the periods, the properties of the shrimp-like ISSs shown in Fig. 1(a), we refer the readers to the work.12 If parameters are chosen inside one ISS, the Hénon map will generate a stable orbit with the period corresponding to the color. For example, the largest ISS in the center of Fig. 1(a) has per-5, while the smaller ISS just above this per-5 ISS has per-6 (see the green box). When parameters are chosen outside the ISSs, a chaotic motion occurs and is presented in Fig. 1 by the black color. The grey color represents the case when the trajectory diverges and no bound motion is expected.
The results for F = 7 × 10−3 are shown in Fig. 1(b), which displays the same parameter interval as Fig. 1(a). For simplicity, we have chosen to count the periods including all states. In other words, we do not display the periods of the composed map. The first observation in Fig. 1(b) is that while ISSs with even periods keep their period, for all ISSs with odd periods, the period is duplicated, as expected by rule (b). For example, the large shrimp-like ISS mentioned above with p = 5 (ω = 5/2) now has pF = 10 (see the white arrow). There are obviously no per-1 orbits anymore, and orbits with odd periods p are prohibited since ω = p/k = p/2 is rational. The second observation is that all ISSs with even periods p start to duplicate since ω = p/2 is an integer and satisfies rule (a). This is better observed in the ISSs with periods p = pF = 6 (ω = 6/2), which separate from each other. See the black arrow indicating both ISSs. From the resolution of Fig. 1(b), some of the duplications from other ISSs cannot be seen. The separation between the duplicated ISSs in the parameter space increases with F, as can be checked in Fig. 1(c) for F = 2 × 10−2 and in Fig. 1(d) for F = 4 × 10−2. An interesting aspect is that the per-6 ISS from the right move further to the right as Fincreases, until it reaches the grey region where it starts to disappear. In addition, the above mentioned large shrimp-like ISS with pF = 10 (white arrow) is transformed into three (not a triplication in this case, see the explanation below) interconnected shrimps observed in Fig. 1(d). Such interconnected shrimps were observed to be relevant in a tunnel diode and a fiber-ring laser27 and, in this context, endorse the importance to control the intermediate dynamics to create and enlarge the ISSs.
1. Magnification of ISSs with integer ω = p/k
To better understand the duplication of ISSs, Fig. 2 presents the details of the effect of increasing values of F in two distinct shrimp-like ISSs with PDBs p=pF=8→16→32… (ω=4→8→16…) in Figs. 2(a) and 2(b) and p=pF=10→20→40… (ω=5→10→20…) in Figs. 2(e) and 2(f). For F ≠ 0, in both cases, we obtain k = 2 identical ISSs which are copies of the original ones and k = 2 attractors in phase space (see Sec. IV). To turn the statement of identical copies more convincing, we plot the largest Lyapunov exponent (LE) for each case [see Figs. 2(c), 2(d), 2(g), and 2(h)]. Grey, yellow, to red are used for increasing positive LE and blue to cyan for increasing negative LE. The figures nicely show that the internal structures of the ISSs, which contain information about the local stability, are unaltered by the duplication. Thus, identical copies refer to the shape of the ISSs and the corresponding stability for parameters chosen inside the ISSs.
2. Magnification of ISSs with rational ω = p/k
Figure 3 presents the magnification for shrimp-like ISS with PDB p=7→14→26→… (ω=7/2→7→14…). Here, the effect of increasing values of F on the ISS is more complicated since the lowest period of the PDB sequence follows rule (b), while all subsequent periods follow rule (a). Since ω = 7/2 for p = 7, we obtain pF = 2 × 7 = 14, confirming rule (b). In this case, only one attractor with pF = 14 is found and no duplication of the ISS with this period occurs. This breaks the ISS apart as observed in Figs. 3(b)–3(d). While one main shrimp-like ISS with period pF = 14 (and PDB sequence 28 → 56…) remains, two smaller non-overlapping ISSs with period p = pF = 14 move apart (see white arrows). However, here, the internal structure of the ISS is changed qualitatively when compared to the F = 0 case, as can be checked in the largest LE analysis in Figs. 3(e)–3(h).
B. Triplication (k = 3), quadruplication (k = 4), and more
Next, it is shown that the above behavior can be extended to multiple ISSs in the parameter space. For the triplicated case (k = 3), the external force must have per-3, as can be obtained by using the protocol −F, 0, F, −F, 0, F,… perturbing the HM. In this case, the ISSs with periods of multiples of 3 are triplicated. In this section, we focus on integer values of ω. The results are shown in Fig. 4(a), which displays a magnification of the parameter space in Fig. 1(a) and for F = 1.2 × 10−3. This is a triplication of the per-6 shrimp (ω = 6/3). In fact, it creates k = 3 per-6 stable periodic orbits which separate more and more for increasing values of F. In Fig. 4(b), the case of the quadruplication of the per-8 shrimp [see the magenta box in Fig. 1(a)] is shown using +F,−F/2,F/2,−F,…,+F,−F/2 with F = 2.0 × 10−3. Figure 4(c) displays the sextuplication of the per-12 shrimp using +F,−F/2,F/4,−F/4,F/2,−F,+F,−F/2,… with F = 1.2× 10−3. To show that a proliferation of the ISS is possible, we present in Fig. 4(d) the case of the decuplications of two per-10 ISSs so that twenty ISSs are observed.
It remains to show that the multiplication of ISSs in the parameter space is a consequence of the multiplication of attractors in phase space. To exemplify this, we show the duplication and triplication of shrimp-like ISSs in Fig. 1. Figure 5(a) shows the basin of attraction inside the per-6 ISS in Fig. 1(a) for F = 0. Figure 5(b) shows the basin of attraction for the duplication for which the identical copies of the ISSs still overlap. We clearly observe that it generates another basin of attraction related to the duplicated per-6 orbit. Figure 5(c) shows the basin of attraction for the case of three attractors, each one with per-6.
Figures 5(d) and 5(e) display the duplication and triplication of one per-6 orbital point [see inside the red box in Fig. 5(a)] as a function of F, respectively. It is very interesting to observe that k small parametric changes in the HM generate kriddled basin of attractions.28
The ISSs discussed in Sec. III A have the well known shrimp-like form.12However, other ISSs exist, which may be more complicated or not. For example, simpler ISSs than the shrimp-like ISSs are cuspidal and non-cuspidal, as shown in Figs. 1(a) and 1(b), respectively, in Ref. 19. More complicated and higher order ISSs were described in the very last paper from Lorenz.29
The purpose of the present section is to show that our multiplication procedure is also valid for such ISSs. The first example is shown in Fig. 6 for the duplication of a per-18 ISS. By performing a visual analysis, the ISS in Fig. 6(a) for F = 0 could be wrongly interpreted as a composition of shrimp-like ISSs which are overlapped. But this is not the case, as shown by Lorenz and checked here analyzing the LE in Fig. 6(b). Compared to the shrimp-like ISSs, now we have two superstable regimes (cyan lines) inside the ISSs. This also suggests that the dynamics inside the ISS in Fig. 6(a) is different, regarding stability, from the dynamics inside shrimp-like ISS.
As the values of F increase, the duplication of the ISS is visible and nice complex pictures are generated. While the inner structure of the LE inside the ISSs in Fig. 6(d) is still identical to the original one in Fig. 6(b) (compare cyan lines), it changes in Fig. 6(f). We observed in general that the higher-order ISSs are more sensitive to F. In other words, the duplication generates identical copies of the higher-order ISSs, but they change very fast with increasing values of F.
Figure 7 shows the example of the duplication of a per-40 higher-order ISS. Again, the duplications are visible, but the copies are only identical for very small values of F. Both examples above are related to integer values of ω = p/k, namely, 18/2 = 9 and 40/2 = 20, respectively. Both cases obey rule (a).
For low periods, it is possible to give an analytical demonstration of the duplication. Years ago, the boundaries between the appearance of per-1 and the PDBs 1 → 2 and 2 → 4 were determined analytically in the parameter space of the Hénon map. These boundaries are given by the relations30,31


Compared to the original work, the last equation includes a polynomial with complex solutions for (a, b). This polynomial must be taken into account in case F ≠ 0. Applying the same procedure for the duplication in MHM, we obtain

WpF=2(a,b,F)=W1(a,b)W1→2(a,b)+256F4−[288b4−1152b3+(1536a+1728)b2−(3072a+1152)b+512a2+1536a+288]F2=0, (2)
WpF=2→4(a,b,F)=W2→4(a,b)+256F4−[160b4−1152b3+(1536a+1472)b2−(3072a+1152)b+512a2+1536a+160]F2=0. (3)

Equation (2) gives the appearance (saddle-node bifurcation) of per-1 orbits for the map composed of two iterations of the MHM (pF = 2), and Eq. (3) indicates the PDB from periods 1 to 2 for the composed map, which for the iterations of MHMs means pF = 2 → 4. The solution of Eq. (3) is indicated with blue arrows in Fig. 1(a) for F = 0 and in Fig. 1(b) for F = 7 × 10−3. It is interesting to observe that for F ≠ 0, the boundaries W1(a, b) and W1→2(a, b) become coupled in Eq. (2). In other words, the two independent conditions, W1(a, b) = 0 for saddle-node bifurcation and W1→2(a, b) = 0 for PDB, are transformed in the one saddle-node bifurcation condition WpF=2(a,b,F)=0. Therefore, the PDB 1 → 2 from F = 0 becomes forbidden.

To explain this better, we show an example using b = 0.3 and F = 0.01. The solutions for the first boundaries and F = 0 are

W1(a,0.3):a=−0.1225, W1→2(a,0.3):a=0.3675,

while the solutions for F = 0.01 become

WpF=2(a,0.3,0.01):a=−0.1224 and a=0.4377.

This shows that the appearance of pF = 2 is shifted to the left (−0.1225 → −0.1224) and the PDB 1 → 2 at a = 0.3675 becomes forbidden (since ω = 1/2), transforming it into a saddle-node bifurcation at a = 0.4377. Thus, for F = 0.01, we have two saddle-node bifurcating points. The other boundaries are given by

W2→4(a,0.3):a=0.9125  and  a=−0.4225±i 0.4550,WpF=2→4(a,0.3,0.01):a=0.8983  and  a=0.9267,

which show that the complex solution in the F = 0 case becomes real, and we end up with two PDBs 2 → 4, one in a = 0.8983 and the other one in a = 0.9267. This explains the origin of the duplication of the PDB sequence and the ISSs which contain them. For more simple examples of the origin of shifted bifurcation diagrams via prohibition of PDBs, we refer the reader to the one-dimensional case.25

In this work, we show that the parametric control in composed maps can be used to enlarge stable domains in phase and parameter spaces of two-dimensional discrete nonlinear dynamical systems. Since the stable domains in parameter space are generic, our results are expected to be applicable to a large number of systems. We present analytical and numerical results for the specific case of the composition of Hénon maps with distinct parameters. Using the composition of k Hénon maps with distinct parameters, we have observed the following properties: (1) when the ratio ω = p/k is an integer, where p is the period of the stable orbit, k-identical attractors in phase space and k-identical ISSs in parameter space are generated. The identical copies are split apart as a function of the parameter F. The equivalence between identical stable attractors and identical ISSs was checked by the largest LE analysis. Besides that, the additional basin of attraction regarding the identical copies of the ISSs is riddled. (2) When the ratio ω is not an integer, the number of attractors in phase space and ISSs in parameter space remain unaltered. The new orbital period is pF = kp, and the multiplied ISS is broken apart. (3) The sign of the parameters from the intermediate dynamics must change by each iteration; otherwise, no multiplication is observed.
The multiple compositions of maps lead to the appearance of multiple attractors in phase space and multiple shifted ISSs in the parameter space. Consequently, a considerable enlargement of the stable domains in phase and parameter spaces occurs. This is crucial for the survival of the desired dynamics under noise and temperature effects, which usually destroy the ISSs starting from their borders20(also observed in the parameter space of the relativistic standard map32). Future contributions intend to verify the enlargement of stable domains for practical applications subjected to thermal effects. The multiple compositions of Hénon maps may be related to the general Jung’s decomposition,33 which shows that any planar, invertible quadratic map can be reduced to a composition of Henon-like maps. However, it is not the purpose of the present work to show such a relation.

  1. Democratic People's Republic of Korea UN Office of Disarmament Affairs

    Anastasia, forgive my ignorance but I literally did not understand a word of this article. It is very difficult to comprehend without a strong prior knowledge base so I think it would be helpful if you could try and write a simplified short summary so that is easier to follow.

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