**27**, 081101 (2017); https://doi-org.aurarialibrary.idm.oclc.org/10.1063/1.4994329

#### ABSTRACT

#### I. INTRODUCTION

^{1}and experimental

^{2}electronic circuits, continuous systems,

^{3–9}maps

^{3,10–15}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

*any*realistic situation, independent of the specific physical system.

*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 systems

^{25}to the non-trivial two-dimensional case.

#### II. RULES FROM THE ONE-DIMENSIONAL CASE

^{25}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 ago

^{26}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

*x*. For

_{n}*F*= 0, the above map suffers a PDB from period

*p*= 1 → 2 at

*a*

_{1→2}= 0.75 and a PDB from period

*p*= 2 → 4 at

*a*

_{2→4}= 1.25. It is clear that for

*F*≠ 0, no orbit with period

*p*= 1 exists anymore and the PDB at

*a*

_{1→2}becomes forbidden. In fact, it was shown

^{25}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

*a*

_{2→4}and subsequently along the whole PDB sequence

*a*

_{4→8},

*a*

_{8→16},…, leading to two independently shifted bifurcation diagrams.

*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,

*p*

_{F}=

*p*, where

*p*

_{F}is the orbital period for

*F*≠ 0. (b) When ω∉ℤ, the orbits of period

*p*become

*p*

_{F}-periodic, where

*p*

_{F}=

*kp*. These apparently simple rules generate complex behaviors. For example, suppose a PDB sequence

*p*→ 2

*p*→ 4

*p*→…,

*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.

#### III. TWO-DIMENSIONAL CASE

^{25}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 *x _{n}* and

*y*calculated at discrete times

_{n}*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)].

*k*= 2)

*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*→ 2

*p*→ 4

*p*→, …,

*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.

*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

*p*

_{F}= 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*=

*p*

_{F}= 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

*F*increases, until it reaches the grey region where it starts to disappear. In addition, the above mentioned large shrimp-like ISS with

*p*

_{F}= 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 laser

^{27}and, in this context, endorse the importance to control the intermediate dynamics to create and enlarge the ISSs.

*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.

*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

*p*

_{F}= 2 × 7 = 14, confirming rule (b). In this case, only one attractor with

*p*

_{F}= 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

*p*

_{F}= 14 (and PDB sequence 28 → 56…) remains, two smaller non-overlapping ISSs with period

*p*=

*p*

_{F}= 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).

*k*= 3), quadruplication (

*k*= 4), and more

*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.

#### IV. MULTIPLICATION OF ATTRACTORS AND RIDDLED BASINS

*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.

#### V. DUPLICATION OF OTHER STRUCTURES

^{12}However, 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}

*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.

*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*.

*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).

#### VI. ANALYTICAL RESULTS FOR *P*_{F} = 2

^{30,31}

W1(a,b)=(4a+1−2b+b2)=0,W1→2(a,b)=(4a−3+6b−3b2)3=0,W2→4(a,b)=(4a−5+6b−5b2)2×[5b4+4b3+(8a−2)b2+(16a+4)b+16a2+8a+5]=0. |

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 (*p*_{F} = 2), and Eq. (3) indicates the PDB from periods 1 to 2 for the composed map, which for the iterations of MHMs means *p*_{F} = 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 *W*_{1}(*a*, *b*) and *W*_{1→2}(*a*, *b*) become *coupled* in Eq. (2). In other words, the two independent conditions, *W*_{1}(*a*, *b*) = 0 for saddle-node bifurcation and *W*_{1→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.

*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 *p*_{F} = 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}

#### VII. CONCLUSIONS

*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

*p*

_{F}=

*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.

^{20}(also observed in the parameter space of the relativistic standard map

^{32}). 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.

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.