Considering a system combining two generalized Boolean transformations, we found that depending on the parameters, we can generate generalized synchronization such that the two chaotic orbits have arbitrary proportional linear relations. We rigorously determined its synchronization conditions by the explicit computing conditional Lyapunov exponent using the ergodic property and stable property of the Cauchy distribution. We found that a phenomenon similar to chaotic synchronization occurs even when the synchronization conditions are not strictly satisfied, which exhibits some degree of structural stability of chaotic synchronization. Our model can be further extended to systems with more degrees of freedom and, in the future, can be applied to reservoir computing.

In a chaotic system coupled with generalized Boolean transformations with Cauchy ergodic invariant measures, xn+1=αxnβxn(0<α<1,β>0), we observed chaotic synchronization such as xn=kyn(k0,kR), for n, where k is an arbitrary real number. The key idea behind our model is that a sum of two Cauchy-distributed variables preserves the Cauchy-distributed property by the stable property of the Cauchy distribution. In particular, we analytically obtained the parameter conditions for conditional Lyapunov exponents and chaotic synchronization by using the ergodic property and explicit Cauchy density of the variables. Thus, this study is the first to explicitly obtain conditional Lyapunov exponents and synchronization conditions for generalized synchronization xn=kyn in mutually coupled map systems.

Fujisaka first observed the synchronization phenomenon of chaos in his research on chaos in 1983.1,2 He validated by numerical simulations that when two independent chaotic dynamical systems were combined, they acquired the same orbits. In 1990, Pecora and Carroll have proposed the conditional Lyapunov exponent that can be used to study chaotic synchronization.5 In 1995, Rulkov et al. observed generalized synchronization of chaos by numerical simulations.6 In particular, they had proved the existence of some generalized synchronization conditions x(t)=ϕ(y(t)) among drive variables x(t) and response variables y(t) by examining the attractors of both. However, the conditional Lyapunov exponents can only be calculated numerically, and most studies of chaotic synchronization have been conducted by numerical simulations.

In 2018, Shintani and Umeno have demonstrated that if the ergodic property holds for the chaotic dynamical system x(t+1)=f(x(t) and its ergodic distribution function is known, the conditional Lyapunov exponent for a type of the systems,

x(t+1)=f(x(t))+ξ(t),

can be obtained analytically.7,8 The key idea behind their analysis on chaotic synchronization is that a sum x(t)+ξ(t) is Cauchy-distributed with the scale parameter γ1+γ2 if drive variables ξ(t) and response variables x(t) are Cauchy-distributed with scale parameters γ1 and γ2, respectively.3 

Subsequently, Higa and Umeno analyzed the following two-degree-of-freedom coupled chaotic system:9 

{xn+1=12(xn1xn)+ε1yn,yn+1=12(yn1yn)+ε2xn.

This system is a bi-directional combination of two generalized Boole transformations,

xn+1=12(xn1xn),

which are ergodic and Cauchy-distributed chaotic dynamical systems. They have observed complete synchronization xn=yn and generalized synchronization xn=yn in the system and have shown their conditions by using the conditional Lyapunov exponent.

In this study, we analyzed the following more generalized model:

{xn+1=αxnβ1xn+ε1yn,yn+1=αynβ2yn+ε2xn.

Here, the equation xn+1=αxnβxn(0<α<1,β>0) is the general form of the generalized Boole transformations with Cauchy ergodic invariant measures. We have confirmed that arbitrary generalized synchronization xn=kyn(k0,kR) for n occurs in this model. Moreover, we have obtained the necessary and sufficient conditions for generalized synchronization.

Because this model is a simple model that can analyze chaotic synchronization, it can be treated as CML10 when considering a system that causes chaotic synchronization. The generation and detection of THz waves using mode synchronization of laser chaos is an example of engineering research handling chaotic synchronization of mutually coupled systems, such as this model.11 

This model is also expected to be applied as a reservoir dynamical system in reservoir computing. The model has the necessary properties for reservoir dynamical systems that two different initial states converge to the same state through time evolution.12 There is also a rule of thumb that reservoir computing has the highest information processing performance when the conditional Lyapunov exponent of the reservoir dynamics system is negative and close to 0. This model is suitable for reservoir dynamical systems because the conditional Lyapunov exponent is obtained analytically and the conditional Lyapunov exponents can be set to the best value.

We handle the following model:

{xn+1=αxnβ1xn+ε1yn,yn+1=αynβ2yn+ε2xn,0<α<1,β1,β2>0,ε1,ε20.
(1)

Here, ε1 and ε2 are coupling coefficients. This model combines the general form of the generalized Boole transformation, which is a chaotic dynamical system,13 

xn+1=αxnβxn:0<α<1,β>0.
(2)

Figure 1 is a conceptual diagram of this model.

FIG. 1.

Interaction between xn and yn.

FIG. 1.

Interaction between xn and yn.

Close modal

In model (1), divergence phenomenon and generalized synchronization are observed under certain parameter conditions. Divergence phenomenon is a phenomenon in which xn and yn diverge to positive or negative infinity, as detailed in Sec. IV. Generalized synchronization is a phenomenon in which two chaotic orbit satisfy the proportional linear relation, xn=kyn(k0,kR), following some steps. Notably, generalized synchronization refers to only chaotic synchronization in this paper.

This section presents several examples of divergence phenomenon and generalized synchronization phenomenon observed in model (1).

(a) Divergence phenomenon

In the case where α=0.5,β1=β2=1.0, and ε1=ε2=0.7, divergence phenomenon occurs. Figure 2 shows a graph in this case with the initial values x0=3 and y0=5, the horizontal axis n, and the vertical axis xn and yn.

FIG. 2.

In the case where α=0.5,β1=β2=1.0, and ε1=ε2=0.7, divergence phenomenon occurs.

FIG. 2.

In the case where α=0.5,β1=β2=1.0, and ε1=ε2=0.7, divergence phenomenon occurs.

Close modal

(b) Generalized synchronization xn=2yn

In the case where α=0.4,β1=4.0,β2=1.0,ε1=0.8, and ε2=0.2, generalized synchronization xn=2yn are observed. Figure 3 shows a graph in this case with the initial values x0=3 and y0=5, the horizontal axis n=[9900,10000], and the vertical axis xn and yn. We can see that xn and yn take chaotic orbits while satisfying xn=2yn.

FIG. 3.

In the case when α=0.4,β1=4.0,β2=1.0,ε1=0.8,ε2=0.2,x0=3, and y0=5, generalized synchronization xn=2yn occurs.

FIG. 3.

In the case when α=0.4,β1=4.0,β2=1.0,ε1=0.8,ε2=0.2,x0=3, and y0=5, generalized synchronization xn=2yn occurs.

Close modal

(c) Generalized synchronization xn=2yn, only the initial values differ from those in Example (b).

The parameters are the same as in Example (b); however, the initial value is changed to x0=3+0.0001 and y0=5. Figure 4 shows that the orbits have changed compared to Example (b); however, they are still generalized synchronization of xn=2yn. The result shows that even if the initial value is slightly different, the orbits dramatically change. Thus, the chaotic properties such as the sensitivity of the initial conditions are maintained even if chaotic synchronization occurs.

FIG. 4.

In the case when α=0.4,β1=4.0,β2=1.0,ε1=0.8,ε2=0.2,x0=3+0.0001, and y0=5, generalized synchronization xn=2yn occurs.

FIG. 4.

In the case when α=0.4,β1=4.0,β2=1.0,ε1=0.8,ε2=0.2,x0=3+0.0001, and y0=5, generalized synchronization xn=2yn occurs.

Close modal

(d) Generalized synchronization 3xn=yn

In the case where α=0.5,β1=0.5,β2=4.5,ε1=0.1, and ε2=0.9, generalized synchronization 3xn=yn is observed. Figure 5 shows a graph in this case with the initial values x0=3 and y0=5, the horizontal axis n=[9900,10000], and the vertical axis xn and yn. We can observe that xn and yn acquire chaotic orbits while satisfying 3xn=yn.(e) Generalized synchronization 5xn=yn

FIG. 5.

In the case where α=0.5,β1=0.5,β2=4.5,ε1=0.1, and ε2=0.9, generalized synchronization 3xn=yn.

FIG. 5.

In the case where α=0.5,β1=0.5,β2=4.5,ε1=0.1, and ε2=0.9, generalized synchronization 3xn=yn.

Close modal

In the case where α=0.6,β1=1.0,β2=5.0,ε1=0.35, and ε2=0.35, generalized synchronization 5xn=yn is observed. Figure 6 shows a graph in this case with the initial values x0=3 and y0=5, the horizontal axis n=[9900,10000], and the vertical axis xn and yn. We can observe that xn and yn acquire chaotic orbits while satisfying 5xn=yn.

FIG. 6.

In the case where α=0.6,β1=1.0,β2=5.0,ε1=0.35, and ε2=0.35, generalized synchronization 5xn=yn occurs.

FIG. 6.

In the case where α=0.6,β1=1.0,β2=5.0,ε1=0.35, and ε2=0.35, generalized synchronization 5xn=yn occurs.

Close modal

(f) Generalized synchronization xn=108yn

In the case where α=0.5,β1=108,β2=108,ε1=0.4×108, and ε2=0.4×108, generalized synchronization xn=108yn are observed. Table I shows numerical results of xn,yn, and xnyn for the initial values x0=2 and y0=3. Figure 7 shows the error curve |xn108yn| in the same case as Table I. We can see that the relationship xn=108yn holds after about n=12

FIG. 7.

The error curve |xn108yn| in the case where α=0.5,β1=108,β2=108,ε1=0.4×108, and ε2=0.4×108.

FIG. 7.

The error curve |xn108yn| in the case where α=0.5,β1=108,β2=108,ε1=0.4×108, and ε2=0.4×108.

Close modal
TABLE I.

In the case where α = 0.5, β1 = 108, β2 = 10−8, ɛ1 = 0.4 × 108, and ɛ2 = 0.4 × 10−8, generalized synchronization xn = 108yn occurs.

nxnynxn/yn
1.414 213 562 373 1.732 050 807 569 0.816 496 580 928 
−1 428 645.108 8 0.866 025 403 668 −1 649 657.276 498 212 013 
33 926 763.5887 0.427 298 109 852 79 398 346.977 191 999 555 
34 055 303.2409 0.349 356 085 878 97 480 206.063 467 457 891 
31 001 892.1192 0.310 899 227 278 99 716 851.632 439 181 209 
27 936 911.9251 0.279 457 149 951 99 968 499.392 47 2937 703 
25 146 738.3811 0.251 476 186 892 99 996 499.437 427 580 357 
22 632 412.6896 0.226 325 007 205 99 999 611.041 868 224 740 
20 369 202.2146 0.203 692 110 177 99 999 956.782 261 341 810 
18 332 280.6050 0.183 322 814 853 99 999 995.198 015 257 716 
10 16 499 047.4417 0.164 990 475 298 99 999 999.466 444 358 230 
11 14 849 136.6718 0.148 491 366 806 99 999 999.940 715 804 696 
12 13 364 216.2738 0.133 642 162 747 99 999 999.993 412 822 485 
13 12 027 787.1641 0.120 277 871 642 99 999 999.999 268 099 666 
14 10 825 000.1336 0.108 250 001 336 99 999 999.999 918 684 363 
15 9 742 490.8824 0.097 424 908 824 99 999 999.999 990 954 995 
16 8 768 231.5298 0.087 682 315 298 99 999 999.999 999 001 622 
17 7 891 396.9720 0.078 913 969 720 99 999 999.999 999 880 791 
18 7 102 244.6028 0.071 022 446 028 99 999 999.999 999 985 099 
19 6 392 006.0625 0.063 920 060 625 100 000 000.000 000 014 901 
20 5 752 789.8117 0.057 527 898 117 100 000 000.000 000 000 000 
21 5 177 493.447 6 0.051 774 934 476 99 999 999.999 999 985 099 
22 4 659 724.7885 0.046 597 247 885 100 000 000.000 000 000 000 
23 4 193 730.8492 0.041 937 308 492 100 000 000.000 000 000 000 
24 3 774 333.9191 0.037 743 339 191 100 000 000.000 000 000 000 
25 3 396 874.0325 0.033 968 740 325 100 000 000.000 000 000 000 
26 3 057 157.1904 0.030 571 571 904 100 000 000.000 000 000 000 
27 2 751 408.7612 0.027 514 087 612 100 000 000.000 000 000 000 
28 2 476 231.5401 0.024 762 315 401 99 999 999.999 999 970 198 
29 2 228 568.0021 0.022 285 680 021 99 999 999.999 999 985 099 
30 2 005 666.3301 0.020 056 663 301 100 000 000.000 000 000 000 
nxnynxn/yn
1.414 213 562 373 1.732 050 807 569 0.816 496 580 928 
−1 428 645.108 8 0.866 025 403 668 −1 649 657.276 498 212 013 
33 926 763.5887 0.427 298 109 852 79 398 346.977 191 999 555 
34 055 303.2409 0.349 356 085 878 97 480 206.063 467 457 891 
31 001 892.1192 0.310 899 227 278 99 716 851.632 439 181 209 
27 936 911.9251 0.279 457 149 951 99 968 499.392 47 2937 703 
25 146 738.3811 0.251 476 186 892 99 996 499.437 427 580 357 
22 632 412.6896 0.226 325 007 205 99 999 611.041 868 224 740 
20 369 202.2146 0.203 692 110 177 99 999 956.782 261 341 810 
18 332 280.6050 0.183 322 814 853 99 999 995.198 015 257 716 
10 16 499 047.4417 0.164 990 475 298 99 999 999.466 444 358 230 
11 14 849 136.6718 0.148 491 366 806 99 999 999.940 715 804 696 
12 13 364 216.2738 0.133 642 162 747 99 999 999.993 412 822 485 
13 12 027 787.1641 0.120 277 871 642 99 999 999.999 268 099 666 
14 10 825 000.1336 0.108 250 001 336 99 999 999.999 918 684 363 
15 9 742 490.8824 0.097 424 908 824 99 999 999.999 990 954 995 
16 8 768 231.5298 0.087 682 315 298 99 999 999.999 999 001 622 
17 7 891 396.9720 0.078 913 969 720 99 999 999.999 999 880 791 
18 7 102 244.6028 0.071 022 446 028 99 999 999.999 999 985 099 
19 6 392 006.0625 0.063 920 060 625 100 000 000.000 000 014 901 
20 5 752 789.8117 0.057 527 898 117 100 000 000.000 000 000 000 
21 5 177 493.447 6 0.051 774 934 476 99 999 999.999 999 985 099 
22 4 659 724.7885 0.046 597 247 885 100 000 000.000 000 000 000 
23 4 193 730.8492 0.041 937 308 492 100 000 000.000 000 000 000 
24 3 774 333.9191 0.037 743 339 191 100 000 000.000 000 000 000 
25 3 396 874.0325 0.033 968 740 325 100 000 000.000 000 000 000 
26 3 057 157.1904 0.030 571 571 904 100 000 000.000 000 000 000 
27 2 751 408.7612 0.027 514 087 612 100 000 000.000 000 000 000 
28 2 476 231.5401 0.024 762 315 401 99 999 999.999 999 970 198 
29 2 228 568.0021 0.022 285 680 021 99 999 999.999 999 985 099 
30 2 005 666.3301 0.020 056 663 301 100 000 000.000 000 000 000 

(g) Generalized synchronization 108xn=yn

In the case where α=0.4,β1=106,β2=1010,ε1=0.5×108, and ε2=0.5×108, generalized synchronization 108xn=yn are observed. Table II shows numerical results xn,yn, and ynxn for the initial values x0=2 and y0=3. Figure 8 shows the error curve |108xnyn| in the same case as Table II. We can see that the relationship 108xn=yn holds after about n=14.

FIG. 8.

The error curve |108xnyn| in the case where α=0.4,β1=106,β2=1010,ε1=0.5×108, and ε2=0.5×108.

FIG. 8.

The error curve |108xnyn| in the case where α=0.4,β1=106,β2=1010,ε1=0.5×108, and ε2=0.5×108.

Close modal
TABLE II.

In the case where α = 0.4, β1 = 10−6, β2 = 1010, ɛ1 = 0.5 × 10−8, and ɛ2 = 0.5 × 108, generalized synchronization 108xn = yn occurs.

nxnynyn/xn
1.414 213 562 373 1.732 050 807 568 1.224 744 871 39 
0.565 684 726 503 −5 702 792 013.084 8 −1 008 121 9707.559 936 523 
−28.287 687 942 592 −2 252 832 567.155 3 79 640 038.865 220 382 810 
−22.579 237 977 462 −2 315 517 419.552 8 102 550 733.636 986 926 198 
−20.609 282 244 461 −2 055 168 862.375 5 99 720 545.237 714 245 915 
−18.519 557 161 140 −1 852 531 652.307 5 100 031 098.810 216 769 576 
−16.670 481 071 996 −1 666 990 513.582 0 99 996 545.173 626 497 388 
−15.003 144 936 722 −1 500 320 253.033 8 100 000 383.876 952 707 767 
−13.502 859 173 205 −1 350 285 341.384 4 99 999 957.347 097 903 490 
−12.152 570 302 145 −1 215 257 087.808 2 100 000 004.739 212 244 749 
10 −10.937 313 477 612 −1 093 731 342.001 8 99 999 999.473 420 888 186 
11 −9.843 582 009 624 −984 358 201.538 3 100 000 000.058 508 798 480 
12 −8.859 223 709 952 −885 922 370.937 6 99 999 999.993 499 025 702 
13 −7.973 301 225 792 −797 330 122.585 0 100 000 000.000 722 333 789 
14 −7.175 970 977 823 −717 597 097.781 7 99999999.999919742346 
15 −6.458 373 740 684 −645 837 374.068 5 100 000 000.000 008 910 894 
16 −5.812 536 211 778 −581 253 621.177 8 99 999 999.999999001622 
17 −5.231 282 418 558 −523 128 241.855 8 100 000 000.000 000 104 308 
18 −4.708 153 985 545 −470 815 398.554 5 99 999 999.999 999 985 099 
19 −4.237 338 374 593 −423 733 837.459 3 99 999 999.999 999 985 099 
20 −3.813 604 301 136 −381 360 430.113 6 100 000 000.000 000 014 901 
21 −3.432 243 608 804 −343 224 360.880 4 100 000 000.000 000 014 901 
22 −3.089 018 956 569 −308 901 895.656 9 99 999 999.999 999 985 099 
23 −2.780 116 737 184 −278 011 673.718 4 100 000 000.000 000 000 000 
24 −2.502104703769 −250 210 470.376 9 100 000 000.000 000 000 000 
25 −2.251 893 833 728 −225 189 383.372 8 100 000 000.000 000 014 901 
26 −2.026 704 006 285 −202 670 400.628 5 100 000 000.000 000 014 901 
27 −1.824 033 112 244 −182 403 311.224 4 100 000 000.000 000 000 000 
28 −1.641 629 252 784 −164 162 925.278 4 100 000 000.000 000 000 000 
29 −1.477465718355 −147 746 571.835 5 100 000 000.000 000 000 000 
30 −1.329 718 469 685 −132 971 846.968 5 100 000 000.000 000 000 000 
nxnynyn/xn
1.414 213 562 373 1.732 050 807 568 1.224 744 871 39 
0.565 684 726 503 −5 702 792 013.084 8 −1 008 121 9707.559 936 523 
−28.287 687 942 592 −2 252 832 567.155 3 79 640 038.865 220 382 810 
−22.579 237 977 462 −2 315 517 419.552 8 102 550 733.636 986 926 198 
−20.609 282 244 461 −2 055 168 862.375 5 99 720 545.237 714 245 915 
−18.519 557 161 140 −1 852 531 652.307 5 100 031 098.810 216 769 576 
−16.670 481 071 996 −1 666 990 513.582 0 99 996 545.173 626 497 388 
−15.003 144 936 722 −1 500 320 253.033 8 100 000 383.876 952 707 767 
−13.502 859 173 205 −1 350 285 341.384 4 99 999 957.347 097 903 490 
−12.152 570 302 145 −1 215 257 087.808 2 100 000 004.739 212 244 749 
10 −10.937 313 477 612 −1 093 731 342.001 8 99 999 999.473 420 888 186 
11 −9.843 582 009 624 −984 358 201.538 3 100 000 000.058 508 798 480 
12 −8.859 223 709 952 −885 922 370.937 6 99 999 999.993 499 025 702 
13 −7.973 301 225 792 −797 330 122.585 0 100 000 000.000 722 333 789 
14 −7.175 970 977 823 −717 597 097.781 7 99999999.999919742346 
15 −6.458 373 740 684 −645 837 374.068 5 100 000 000.000 008 910 894 
16 −5.812 536 211 778 −581 253 621.177 8 99 999 999.999999001622 
17 −5.231 282 418 558 −523 128 241.855 8 100 000 000.000 000 104 308 
18 −4.708 153 985 545 −470 815 398.554 5 99 999 999.999 999 985 099 
19 −4.237 338 374 593 −423 733 837.459 3 99 999 999.999 999 985 099 
20 −3.813 604 301 136 −381 360 430.113 6 100 000 000.000 000 014 901 
21 −3.432 243 608 804 −343 224 360.880 4 100 000 000.000 000 014 901 
22 −3.089 018 956 569 −308 901 895.656 9 99 999 999.999 999 985 099 
23 −2.780 116 737 184 −278 011 673.718 4 100 000 000.000 000 000 000 
24 −2.502104703769 −250 210 470.376 9 100 000 000.000 000 000 000 
25 −2.251 893 833 728 −225 189 383.372 8 100 000 000.000 000 014 901 
26 −2.026 704 006 285 −202 670 400.628 5 100 000 000.000 000 014 901 
27 −1.824 033 112 244 −182 403 311.224 4 100 000 000.000 000 000 000 
28 −1.641 629 252 784 −164 162 925.278 4 100 000 000.000 000 000 000 
29 −1.477465718355 −147 746 571.835 5 100 000 000.000 000 000 000 
30 −1.329 718 469 685 −132 971 846.968 5 100 000 000.000 000 000 000 

We show some relevant theorems to explain various phenomena presented in Sec. III.

The necessary and sufficient conditions for the model (1) to diverge are as follows:

ε1ε2<(1α2),(1α)2<ε1ε2.
(3)

This property is independent of the initial values x0 and y0.

Proof of Theorem 1.
First, we show that the divergence conditions of model (1) is equal to that of the following dynamical system:
{xn+1=αxn+ε1yn,yn+1=αyn+ε2xn.
(4)
Putting un=1xn and vn=1yn, the model (1) can be expressed as
{un+1=unvn(αβ1un2)vn+ε1un,vn+1=unvn(αβ2vn2)un+ε2vn.
(5)
Model (1) diverges to infinity if and only if Eq. (5) has a fixed point (un,vn)=(0,0) and the point is stable. The fixed point (u,v) satisfies the following equation:
{u=uv(αβ1u2)v+ε1u,v=uv(αβ2v2)u+ε2v.
(6)

We assume that u=v=O(1Nk) for k>0 and sufficiently large N>0.

Then,
uv(αβ1u2)v+ε1u=O(1Nk)O(1Nk)O(αβ1O(1Nk))O(1Nk)+ε1O(1Nk)=O(1Nk).
If N, both sides of Eq. (6) approach zero in the same order O(1Nk). Therefore, (u,v)=(0,0) is a fixed point of Eq. (5).
Stability of the fixed point depend on Jacobi matrix of Eq. (5), and it is calculated as
J=(α(α+ε1)2ε1(α+ε1)2ε2(α+ε2)2α(α+ε2)2).
(7)
The fixed point, (u,v)=(0,0), is stable if and only if the absolute values of the Jacobi matrix eigenvalues are all less than one.

In Eq. (7), we can observe that the Jacobi matrix is not dependent on β1 and β2. Thus, we proved that the divergence conditions of model (1) is the same as that of the dynamical system (4).

Finally, we derive the divergence conditions from the general term of Eq. (4). Equation (4) leads to the following equation:
xn+22αxn+1+(α2ε1ε2)xn=0.
(8)
The characteristic equation of the recurrence relation can be expressed as
λ22αλ+α2ε1ε2=0.
The solution to the characteristic equation is λ=α±ε1ε2, and we put a=α+ε1ε2 and b=αε1ε2. Then, the general term of Eq. (8) can be expressed as
xn=pan+qbn,
where p and q are constant. Considering ab, the necessary and sufficient conditions for xn to diverge can be expressed as
|a|>1 or |b|>1.
(9)
For the case ε1ε2>0, we have
|a|=α+ε1ε2>1,ε1ε2>(1α)2.
The relation |b|=|αε1ε2|>1 is calculated as
ε1ε2>(1+α)2.
In the case of ε1ε2<0, |a|=|b| and |a|=|α+ε1ε2i|>1, we have
α2+(ε1ε2)2>1,ε1ε2<(1α2),
where i is the imaginary unit. From the aforementioned results, Eq. (9), the necessary and sufficient conditions for the model (1) divergence can be expressed as
ε1ε2<(1α2),(1α)2<ε1ε2.

Actually, in the example in Fig. 2, the divergence condition (3) is satisfied, as ε1ε2=0.49,(1α)2=0.25, and a divergence phenomenon occurs.

In model (1), generalized synchronization xnβ1=ynβ2(for n) occurs, and its necessary and sufficient conditions are as follows:

{εβ2β1ε1=β1β2ε2>0,α(1α)<ε<1α.
(10)

In model (1), generalized synchronization xnβ1=ynβ2(for n) also occurs, and its necessary and sufficient conditions are as follows:

{εβ2β1ε1=β1β2ε2>0,α(1α)<ε<1α.
(11)

This property is independent of the initial values, x0 and y0.

1. Preparing for proof of Theorem 2

By transforming xnβ1αxn and ynβ2αyn in model (1), we can represent model (1) in a form that uses the basic form of the generalized Boole transformation,

{xn+1=α(xn1xn)+β2β1ε1yn,yn+1=α(yn1yn)+β1β2ε2xn.

This shows that instead of showing Theorem 2, we can show the simpler corollary as follows:

Corollary 4.1
We consider the following dynamical system:
{xn+1=α(xn1xn)+ε1yn,yn+1=α(yn1yn)+ε2xn.
(12)
The necessary and sufficient conditions for complete synchronization xn=yn and generalized synchronization xn=yn of the dynamical system (12) are as follows:
Complete synchronization xn=yn (for n) conditions:
{εε1=ε2>0,α(1α)<ε<1α.
(13)
Generalized synchronization xn=yn (for n) conditions:
{εε1=ε2>0,α(1α)<ε<1α.
(14)
Here, complete synchronization means that two dynamical systems acquire the same chaotic orbits.

To prove Corollary 4.1, we show some lemma. We note that the divergence condition of dynamical system (12) is the same as that of model (1) from Theorem 1, because the dynamical system (12) is equal to the model (1) with β1=β2=α.

Lemma 4.2

The necessary and sufficient conditions for generalized synchronization xn=yn of the dynamical system (12) is equal to that of complete synchronization in which signs of ε1 and ε2 are reversed.

Proof.
Transforming the dynamical system (12) to allow it to be an expression for xn and yn, we obtain the following equation:
{xn+1=α(xn1xn)+(ε1)(yn),yn+1=α(yn1(yn))+(ε2)xn.
(15)
This equation can be viewed as the dynamical system (12) whose coupling coefficients are ε1 and ε2. Thus, the generalized synchronization xn=yn conditions of the dynamical system (12) is equal to the complete synchronization conditions in which signs of ε1 and ε2 are reversed.

According to Lemma 4.2, we only need to show the necessary and sufficient conditions for complete synchronization in Eq. (13).

Lemma 4.3
In the dynamical system (12), the necessary condition for complete synchronization xn=yn is as follows:
ε1=ε2.
(16)
Proof.
We consider the case where xn and yn are perturbed from some identical orbit hn. Let us denote
xn=hn+ξx,n,yn=hn+ξy,n.
Here ξx,n and ξy,n satisfy |ξx,n||hn|,|ξy,n||hn|.
Putting
δxy(n)xnyn=ξx,nξy,n,
the following equation holds for δxy(n):
δxy(n+1)=αδxy(n)+ε1ynε2xn+α(1hn+ξy,n1hn+ξx,n)=αδxy(n)+ε1ynε2xn+αhn(11+ξy,nhn11+ξx,nhn)αδxy(n)+ε1ynε2xn+αhn{(1ξy,nhn)(1ξx,nhn)}=α(1+1hn2)δxy(n)+ε1ynε2xn.
We note that we created an approximation using |ξx,n||hn| and |ξy,n||hn| in the calculation process. A necessary condition for complete synchronization is δxy(n+1)=δxy(n)=0; therefore, ε1ynε2xn=0 is necessary. Thus, the necessary condition for complete synchronization can be expressed as
ε1=ε2.

Additionally, from the aforementioned equation, the following equation holds in the perturbed state:

δxy(n+1)=(αε1+αhn2)δxy(n).
(17)
Lemma 4.4
If the dynamical system (12) has a limiting distribution, it is the Cauchy distribution, and the scale parameters of its probability density function γx and γy satisfy the following equation:
{γx=α(γx+1γx)+|ε1|γy,γy=α(γy+1γy)+|ε2|γx.
(18)
Proof.

The basic form of the generalized Boole transformations f(x)=α(x1x) has been proven to form a Cauchy distribution.13 Furthermore, from the properties of the Cauchy distribution, when the random variables X1 and X2 are independent and both follow the Cauchy distribution, their sum also follows Cauchy distribution because Cauchy distribution is stable distribution.3,14

Based on this and the reproducibility of the Cauchy distribution, the distribution of the dynamical system (12) is the Cauchy distribution.

In addition, from  Appendix A, the iteration relation γ=α(γ+1γ) holds for the scale parameter γ of the iterated variable by the generalized Boole transformations distribution. Moreover, if we add together Cauchy distribution of two independent scale parameters, γ and γ, we obtain that of scale parameter γ+γ. Furthermore, when the Cauchy distribution of the scale parameter γ is multiplied by ε, its scale parameter is multiplied by |ε| (because scale parameters need to be positive). Based on this and the fact that scale parameters are invariant in the same dynamical system, the scale parameters γx and γy of the limiting distribution of the dynamical system (12) satisfy the following equation:
{γx=α(γx+1γx)+|ε1|γy,γy=α(γy+1γy)+|ε2|γx.

2. Calculation of the conditional Lyapunov exponent

First, we explain the conditional Lyapunov exponent.5 The conditional Lyapunov exponent λ determines whether two chaotic dynamical systems are synchronized or not, and λ<0 is a necessary and sufficient condition for synchronization. Notably, it does not necessarily mean that the orbits are chaotic at that time. In addition, the dynamical system (12) is ergodic and follows a known distribution, and the conditional Lyapunov exponent of such a dynamical system can be determined analytically.7 

For a conditional Lyapunov exponent to exist in the dynamical system (12), it is necessary that the scale parameter of the distribution γx,γy>0 exists and that γx=γy. From Lemma 4.3, the necessary condition for complete synchronization is ε1=ε2, and when this is true, γx=γy is true. Hereinafter, we denote

εε1=ε2.
(19)

Furthermore, from Eq. (18) in Lemma 4.4, the necessary and sufficient condition for the existence of the scale parameters can be expressed as

0<|ε|<1α.
(20)

In this case, the scale parameters can be expressed as

γx=γy=α1α|ε|.
(21)

We note that Eq. (20) is equivalent to the non-divergence condition in Theorem 1 except for the boundary. Hereinafter, we denote γγx=γy=α1α|ε|.

Second, we calculate the conditional Lyapunov exponent assuming that its existing conditions in Eqs. (19) and (20) hold. Because the scale parameters of xn,yn are equal and the difference δxy(n) satisfies Eq. (17) in the perturbed state, the conditional Lyapunov exponent λ is computed by the scale parameters γ and hn,

λ=Pγ(hn)log|αε+αhn2|dhn.
(22)

We note that Pγ(x)=γπ(x2+γ2) is the probability density function of the Cauchy distribution; therefore, the following equation holds:

Pγ(x)dx=1.

To calculate the conditional Lyapunov exponent (22), we consider λ separately in the following cases:

  • αε>0.

  • αε=0.

  • αε<0.

(i) In the case where αε>0

λ=Pγ(hn)log|αε+αhn2|dhn=log(αε)+Pγ(hn)log|1+ααε1hn2|dhn.

Putting β=αεα and Hn=βhn, we obtain the following equation:

Pγ(hn)=γπ(hn2+γ2)=β(βγ)π(Hn2+(βγ)2)=βPβγ(Hn),1+ααε1hn2=1+1Hn2,dhn=dHnβ.

Hence,

λ=log(αε)+βPβγ(Hn)log|1+1Hn2|dHnβ=log(αε)+Pβγ(Hn)log|Hn2+1|dHn40Pβγ(Hn)logHndHn.

Additionally, from  Appendixes B and  C, the following equations hold:

0Pγ(x)logxdx=12logγ,
(23)
Pγ(x)log|x2+1|dx=2log(γ+1).
(24)

From the aforementioned equation, β=αεα and γ=α1α|ε|; λ is calculated as

λ=log(αε)+2log(βγ+1)2logβγ=2log(αε+1α|ε|)=log{1ε|ε|+2(αε)(1α|ε|)}.

(ii) In the case where αε=0, λ is calculated as

λ=Pγ(hn)log|αε+αhn2|dhn=logα2Pγ(hn)log|hn|dhn=logα40Pγ(hn)loghndhn.

From Eq. (23) and γ=α1α|ε|, λ is calculated as

λ=logα2logγ=log(12α).

(iii) In the case where αε<0, λ is calculated as

λ=Pγ(hn)log|αε+αhn2|dhn=log(εα)+Pγ(hn)log|1αεα1hn2|dhn.

Putting β=εαα and Hn=βhn, by the same calculation as in the case (i),

λ=log(εα)+Pβγ(Hn)log|11Hn2|dHn=log(εα)+Pβγ(Hn)log|Hn21|dHn40Pβγ(Hn)logHndHn.

From Eq. (23) and

Pγ(x)log|x21|dx=log(γ2+1),
(25)

which is derived from  Appendix D, λ is calculated as

λ=log(εα)+log{(βγ)2+1}2log(βγ),=log(12α).

Here, αε<0 leads to ε>0; therefore, γ can be expressed as γ=α1αε.

Therefore, the conditional Lyapunov exponent of the dynamical system (12) when Eqs. (19) and (20) hold is as follows:

λ={log{1ε|ε|+2(αε)(1α|ε|)}(αε>0)log(12α)(αε0).
(26)

We note that in Eq. (26), if ε=0, i.e., the coupling is eliminated, the conditional Lyapunov exponent is equal to the Lyapunov exponent of the generalized Boole transformations, λ=log(1+2α(1α)).13 It shows that Eq. (26) is an extension of Ref. 9.

3. Proof of Corollary 4.1

Proof of Corollary 4.1.

From Lemma 4.2, we only need to show the necessary and sufficient conditions for complete synchronization (13).

A necessary and sufficient condition for complete synchronization of the dynamical system (12) is that the conditional Lyapunov exponent is negative and the orbit at synchronization is chaotic.

From Eq. (26), the condition λ<0 can be expressed as
α(1α)<ε.
(27)
Combining Eq. (27) and the λ existence conditions, Eqs. (19) and (20), the necessary and sufficient condition for synchronization xn=yn of the dynamical system (12) can be expressed as
{εε1=ε2,α(1α)<ε<1α.
(28)
Second, we need to find the condition that the orbits at synchronization are chaotic, i.e., the condition that the synchronization is complete synchronization when (28) holds. From xn=yn and ε1=ε2, the dynamical system (12) can be expressed as
{sn+1=α(sn1sn)+ε1sn,sn+1=α(sn1sn)+ε2sn.
Here, snxn=yn. Thus,
sn+1=(α+ε)snαsn.
(29)
Equation (29) has the same form as the generalized Boole transformations xn+1=αxnβxn and the Lyapunov exponent of the transformation considers a positive value when 0<α<1. Thus, the condition for the generalized Boolean transformations to consider a chaotic orbit is 0<α<1. Therefore, Eq. (29) has a chaotic orbit when
α<ε<1α.
(30)
Therefore, the necessary and sufficient condition for complete synchronization of the dynamical system (12) can be expressed as
{εε1=ε2,α(1α)<ε<1α.

Actually, in the example in Figs. 3–6 and Tables I and II, the generalized synchronization condition (10) or (11) is satisfied, and a generalized synchronization phenomenon occurs.

The condition in the second line of Eqs. (10) and (11), α(1α)<ε<1α, is illustrated by the colored area in Fig. 9, without including boundaries.

FIG. 9.

Range of α(1α)<ε<1α.

FIG. 9.

Range of α(1α)<ε<1α.

Close modal

As shown in Theorem 2, model (1) has the important property that arbitrary generalized synchronization can occur if the parameters can be chosen freely.

In model (1), when the parameters satisfy condition (10), generalized synchronization xnβ1=ynβ2 occurs. However, this does not mean that chaotic synchronization does not occur when this condition is not strictly satisfied. In addition, the phenomenon similar to generalized synchronization occurs even when there is an error deviation in the parameters from the condition of generalized synchronization.

Example

(a) Orbits close to complete synchronization

In the case where α=0.5,β1=β2=1.0,ε1=0.3, and ε2=0.3, these parameters satisfy condition (10); therefore, generalized synchronization occurs. For example, as shown in Fig. 10, even when there is an error in ε2, such as ε2=0.3001, the orbits are close to complete synchronization.

FIG. 10.

In the case where α=0.5,β1=β2=1.0,ε1=0.3,ε2=0.3001,x0=2, and y0=3, orbits close to complete synchronization xn=yn are observed.

FIG. 10.

In the case where α=0.5,β1=β2=1.0,ε1=0.3,ε2=0.3001,x0=2, and y0=3, orbits close to complete synchronization xn=yn are observed.

Close modal

(b) Orbits close to complete synchronization, only the initial values differ from those in Example (a)

The same phenomenon occurs for the same parameters as in Example (a) but with different initial values. This means that the stability of the generalized synchronization does not depend on the initial values. Figure 11 shows an example.

FIG. 11.

In the case where α=0.5,β1=β2=1.0,ε1=0.3,ε2=0.3001,x0=5, and y0=7, orbits close to complete synchronization xn=yn are observed.

FIG. 11.

In the case where α=0.5,β1=β2=1.0,ε1=0.3,ε2=0.3001,x0=5, and y0=7, orbits close to complete synchronization xn=yn are observed.

Close modal

(c) Orbits close to generalized synchronization 2xn=yn

In the case where α=0.4,β1=0.2,β2=0.8,ε1=0.2, and ε2=0.8, these parameters satisfy condition (11); therefore, generalized synchronization 2xn=yn occurs. For example, as shown in Fig. 12, even when there is an error in the parameters, such as ε1=0.200001, the orbits are close to generalized synchronization 2xn=yn.

FIG. 12.

In the case where α=0.4,β1=0.2,β2=0.8,ε1=0.200001,ε2=0.8,x0=3, and y0=5, orbits close to generalized synchronization 2xn=yn are observed.

FIG. 12.

In the case where α=0.4,β1=0.2,β2=0.8,ε1=0.200001,ε2=0.8,x0=3, and y0=5, orbits close to generalized synchronization 2xn=yn are observed.

Close modal

(d) Orbits close to complete synchronization, in the case where α has an error.

Examples (a)–(c) are the cases where an error occurs in ε1 or ε2; however, the same phenomenon can be observed when an error occurs in α. For example, if we express model (1) as

{xn+1=αxxnβ1xn+ε1yn,yn+1=αyynβ2yn+ε2xn,

even when αx=0.5,αy=0.499999,β1=β2=1.0, and ε1=ε2=0.3, the orbits are close to complete synchronization as shown in Fig. 13.

FIG. 13.

In the case where αx=0.5,αy=0.499999,β1=β2=1.0,ε1=ε2=0.3,x0=3, and y0=5, orbits close to complete synchronization xn=yn are observed.

FIG. 13.

In the case where αx=0.5,αy=0.499999,β1=β2=1.0,ε1=ε2=0.3,x0=3, and y0=5, orbits close to complete synchronization xn=yn are observed.

Close modal

The model investigated in this study not only causes chaotic synchronization but also determines the conditional Lyapunov exponent and the synchronization conditions.

Let us consider an applicability of the analysis presented here for model (1) in Eq. (1). A more general model is given by

{xn+1=f(xn)+ε1yn,yn+1=g(yn)+ε2xn,ε1,ε20.
(31)

Here, the mappings f and g are assumed to have mixing properties and thus are ergodic with respect to some invariant measures. Even in this case, we can investigate chaotic synchronization condition numerically because we can estimate conditional Lyapunov exponents by ergodic theorems with empirical measures. However, analytical obtaining conditional Lyapunov exponents to give explicit chaotic synchronization condition presented in this paper needs to have explicit probability distributions for xn and yn. Since the model in Eq. (31) is a coupled chaotic equation, some sort of mechanism for obtaining explicit limiting distributions is necessary for the analysis. In this respect, a key ingredient of the mechanism to enable us to analyze that chaotic synchronization conditions explicitly are the stable property of the ergodic invariant measure such as stating symbolically,

Stable Law=Stable Law+Stable Law.

Furthermore, Cauchy distribution, which the model (1) has as ergodic invariant measure, is an exceptional explicit example of stable probability distribution among all the stable distributions satisfying

Cauchy Law=Cauchy Law+Cauchy Law.

Thus, if f and g have Cauchy law as ergodic probability measures, then it is possible to obtain the probability distributions even for mutually coupled variables xn and yn by solving a certain set of coupled self-consistent equations of the scaling parameter of Cauchy law as Eq. (18). Therefore, we can extend model (1) to more general model as long as two chaotic mappings f and g have Cauchy stable laws. Here we note that model (1) can be considered as the simplest model having the above key property because a generalized Boole transformation f(x)=αxβx is the simplest chaotic map having Cauchy distributions. Concerning delayed coupled equations such as

{xn+1=f(xn)+ε1ynd,yn+1=g(yn)+ε2xnd,ε1,ε20,d>0,
(32)

a stable property such as Cauchy law keeps to be a key ingredient and this shows a generalization possibility of the analysis presented here although its analysis should be more complex because of higher dimensionality of coupled self-consistent equations of the scale parameters. Another possibility for generalization of the model is to investigate the occurrence of multistability such as coexisting attractors and cluster of hyperchaotic attractors as discovered for higher dimensional coupled systems other than the present two-dimensional models,4 which should be of the future study.

We proved that divergence and generalized synchronization occur in the model (1), and we analytically obtained their necessary and sufficient conditions. In addition, we found that even when generalized synchronization is occurring, a slight difference in the initial values results in a completely different chaotic orbit. In the case of generalized synchronization, even a slight difference in the initial value resulted in a completely different chaotic orbit.

This model is the first model that can generate not only complete synchronization but also generalized synchronization whose conditions are explicitly provided. Furthermore, we found that the chaotic synchronization observed in this model has a certain structural stability. Thus, this model is expected to have engineering applications not only because the chaotic synchronization conditions are obtained analytically but also because some parameter errors are allowed in practical engineering situations.

The authors have no conflicts to disclose.

Takumi Kano: Conceptualization (equal); Investigation (equal); Methodology (equal); Software (equal); Visualization (equal); Writing – original draft (equal). Ken Umeno: Conceptualization (equal); Investigation (equal); Methodology (equal); Supervision (lead); Validation (equal); Writing – review & editing (equal).

Takumi Kano: Conceptualization (equal); Investigation (equal); Methodology (equal); Software (equal); Visualization (equal); Writing – original draft (equal). Ken Umeno: Conceptualization (equal); Investigation (equal); Methodology (equal); Supervision (lead); Validation (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

We show that the scale parameter of the Cauchy distribution of the iterated variable by the generalized Boole transformations (f(x)=α(x1x)) satisfies the following equation:

γ=α(γ+1γ),

where γ is the original scale parameter of variable x.

First, a probability density function of Cauchy distribution is expressed as ργ(x)=γπ(x2+γ2). Let X be a random variable in f(x)=α(x1x) and let Y satisfy Y=f(X). There are two X that satisfy Y=f(X). We denote them as X1 and X2, and let p(y) be a probability density function of Y distribution. From the probability conservation law,

p(Y)|dY|=ργ(X1)|dX1|+ργ(X2)|dX2|,p(Y)=ργ(X1)|dYdX|X=X1+ργ(X2)|dYdX|X=X2.
(A1)

Furthermore, X1 and X2 are the solutions to Y=f(X), X2YαX1=0; therefore, we obtain the following equation:

X1+X2=Yα,X1X2=1,X12+X22=(Yα)2+2.

In addition, dYdX=α(1+1X2). Therefore, p(y) can be expressed as

p(Y)=X12α(1+X12)γπ(γ2+X12)+X22α(1+X22)γπ(γ2+X22)=γαπX12(1+X22)(γ2+X22)+X22(1+X12)(γ2+X12)(1+X12)(1+X22)(γ2+X12)(γ2+X22)=γαπX12X22(X12+X22+2γ2+2)+γ2(X12+X22)(X12X22+X12+X22+1){X12X22+γ2(X12+X22)+γ4}=γαπ{(Yα)2+4}(1+γ2){(Yα)2+4}{(1+γ2)2+γ2(Yα)2}=α(γ+1γ)π[Y2+{α(γ+1γ)}2].

Furthermore, because the scale parameter remains the same before and after the transformation for the limiting distribution, the aforementioned equation provides γ=α(γ+1γ) for the limiting stable distribution with the scale parameter γ of the generalized Boole transformations. Thus,

γ=α1α.

We calculate 0γlogxπ(x2+γ2)dx by complex integral through the path shown in Fig. 14.

FIG. 14.

Path of the complex integral in the calculation of 0γlogxπ(x2+γ2)dx.

FIG. 14.

Path of the complex integral in the calculation of 0γlogxπ(x2+γ2)dx.

Close modal

Putting f(z)Pγ(z)logz=γlogzπ(z2+γ2), f is regular in the path, and

Cf(z)dz=CRf(z)dz+Rεf(x)dx+Cεf(z)dz+εRf(x)dx.
(B1)

0γlogxπ(x2+γ2)dx, the integral that we are calculating, is the limit considered in εRf(x)dx as R,ε0.

First, from the residue theorem,

Cf(z)dz=2πiRes(iγ,f)=log(iγ)=logγ+iπ2.

Second,

|CRf(z)dz|=γπ|0πlog(Reiθ)(Reiθ)2+γ2Rieiθdθ|γπ0π|O(logR)O(R)|dθ0(R).

Therefore CRf(z)dz0 for R. Furthermore,

|Cεf(z)dz|=γπ|0πlog(εeiθ)(εeiθ)2+γ2εieiθdθ|γπ0πε{log|ε|+|θ|}(εeiθ)2+γ2dθ0(ε0).

Therefore, Cεf(z)dz0 for ε0. Furthermore,

Rεf(x)dx+εRf(x)dx=εRγ(logx+iπ)π(x2+γ2)dx+εRγlogxπ(x2+γ2)dx,=2εRγlogxπ(x2+γ2)dx+iγεR1x2+γ2dx,20γlogxπ(x2+γ2)dx+iγ01x2+γ2dx(R,ε0).

01x2+γ2dx=π2γ, therefore for R,ε0,

Rεf(x)dx+εRf(x)dx20γlogxπ(x2+γ2)dx+iπ2.

Finally, for R,ε0, from Eq. (B1),

logγ+iπ2=20γlogxπ(x2+γ2)dx+iπ2,0γlogxπ(x2+γ2)dx=12logγ.

We calculate Pγ(x)log|x2+1|dx=γlog|x2+1|π(x2+γ2)dx by complex integral. First,

γlog|x2+1|π(x2+γ2)dx=γlog|x+i|π(x2+γ2)dx+γlog|xi|π(x2+γ2)dx.

If we convert xx in γlog|xi|π(x2+γ2)dx,

γlog|xi|π(x2+γ2)dx=γlog|xi|π{(x)2+γ2}(dx)=γlog|x+i|π(x2+γ2)dx.

Therefore, γlog|x2+1|π(x2+γ2)dx=2γlog|x+i|π(x2+γ2)dx. Accordingly, we only need to calculate γlog|x+i|π(x2+γ2)dx.

We calculate it through the path shown in Fig. 15.

FIG. 15.

Path of complex integral in the calculation of γlog|x+i|π(x2+γ2)dx.

FIG. 15.

Path of complex integral in the calculation of γlog|x+i|π(x2+γ2)dx.

Close modal

Putting f(z)γlog|z+i|π(z2+γ2), f is regular in the path, and

Cf(z)dz=CRf(z)dz+RRf(x)dx.
(C1)

γlog|x+i|π(x2+γ2)dx, the integral that we are calculating, is the limit considered in RRf(x)dx as R.

From the residue theorem,

Cf(z)dz=2πiRes(iγ,f)=log(γ+1).

Furthermore,

|CRf(z)dz|=γπ|0πlog|Reiθ+i|(Reiθ)2+γ2Rieiθdθ|γπ0π|O(logR)O(R)|dθ0(R).

Therefore, CRf(z)dz0 for R. Thus, for R, from Eq. (C1),

γlog|x+i|π(x2+γ2)dx=log(γ+1).

As a result,

γlog|x2+1|π(x2+γ2)dx=2log(γ+1).

We calculate γlog|x21|π(x2+γ2)dx by complex integral through the path shown in Fig. 16.

FIG. 16.

Path of the complex integral in the calculation of γlog|x21|π(x2+γ2)dx.

FIG. 16.

Path of the complex integral in the calculation of γlog|x21|π(x2+γ2)dx.

Close modal

Putting f(z)Pγ(z)log|z21|=γlog|z21|π(z2+γ2), f is regular in the path, and

Cf(z)dz=CRf(z)dz+R1ε1f(x)dx+Cε1f(z)dz+1+ε11ε2f(x)dx+Cε2f(z)dz+1+ε2Rf(x)dx.
(D1)

γlog|x21|π(x2+γ2)dx, the integral that we are calculating is the limit considered in R1ε1f(x)dx+1+ε11ε2f(x)dx+1+ε2Rf(x)dx as R,ε10,ε20.

First from the residue theorem,

Cf(z)dz=2πiRes(iγ,f)=log(γ2+1).

Second,

|CRf(z)dz|=γπ|0πlog|Reiθ+1|+log|Reiθ1|(Reiθ)2+γ2Rieiθdθ|γπ0π|O(logR)O(R)|dθ0(R).

Therefore CRf(z)dz0 for R. Furthermore,

|Cε1f(z)dz|=γπ|0πlog{ε1eiθ(2+ε1eiθ)}(1+ε1eiθ)2+γ2ε1ieiθdθ|γπ0πε1{log|ε1|+θ+log|2+ε1eiθ|+arg(2+ε1eiθ)}(1+ε1eiθ)2+γ2dθ0(ε10).

Therefore, Cε1f(z)dz0 for ε10. Furthermore,

|Cε2f(z)dz|=γπ|0πlog{ε2eiθ(2+ε2eiθ)}(1+ε2eiθ)2+γ2ε2ieiθdθ|γπ0πε2{log|ε2|+θ+log|2+ε2eiθ|+arg(2+ε2eiθ)}(1+ε2eiθ)2+γ2dθ0(ε20).

Therefore, Cε2f(z)dz0 for ε20.

Finally, for R,ε10,ε20, from Eq. (D1),

γlog|x21|π(x2+γ2)dx=log(γ2+1).
1.
H.
Fujisaka
and
T.
Yamada
, “
Stability theory of synchronized motion in coupled-oscillator systems
,”
Prog. Theor. Phys.
69
(
1
),
32
(
1983
).
2.
T.
Yamada
and
H.
Fujisaka
, “
Stability theory of synchronized motion in coupled-oscillator systems. II: The mapping approach
,”
Prog. Theor. Phys.
70
(
5
),
1240
(
1983
).
3.
B. V.
Gnedenko
and
A. N.
Kolmogorov
,
Limit Distributions for Sums of Independent Random Variables
(
Addison Wesley
,
Reading, MA
,
1954
).
4.
H.
Natiq
,
S.
Banerjee
,
M. R. K.
Ariffin
, and
M. R. M.
Said
, “
Can hyperchaotic maps with high complexity produce multistability?
,”
Chaos
29
,
011103
(
2019
).
5.
L. M.
Pecora
and
T. L.
Carroll
, “
Synchronization in chaotic systems
,”
Phys. Rev. Lett.
64
,
821
(
1990
).
6.
N. F.
Rulkov
,
M. M.
Sushchik
,
L. S.
Tsimring
, and
H. D. I.
Abarbanel
, “
Generalized synchronization of chaos in directionally coupled chaotic systems
,”
Phys. Rev. E
51
,
980
(
1995
).
7.
M.
Shintani
and
K.
Umeno
, “
Conditional Lyapunov exponent criteria in terms of ergodic theory
,”
Prog. Theor. Exper. Phys.
2018
(
1
), 013A01.
8.
K.
Umeno
, “On the proposal of coupled solvable chaotic systems and their singular behavior—Toward a theory of solvable chaotic fields,” in Proceedings of the Japan Society for Industrial and Applied Mathematics (JSIAM, 2017) (in Japanese).
9.
S.
Higa
and
K.
Umeno
, “Synchronization phenomena of coupled solvable chaotic systems” (unpublished).
10.
K.
Kaneko
, “
Transition from torus to chaos accompanied by frequency lockings with symmetry breaking-in connection with the coupled-logistic map
,”
Prog. Theor. Phys.
69
(
5
),
1427–1442
(
1983
).
11.
F.
Kuwashima
,
M.
Jarrahi
,
S.
Cakmakyapan
,
O.
Morikawa
,
T.
Shirao
,
K.
Iwao
,
K.
Kurihara
,
H.
Kitahara
,
T.
Furuya
,
K.
Wada
,
M.
Nakajima
, and
M.
Tani
, “
Evaluation of high-stability optical beats in laser chaos by plasmonic photomixing
,”
Opt. Express
28
,
24833
(
2020
).
12.
M.
Inubushi
and
K.
Yoshimura
, “
On the characteristics and structures of dynamical systems suitable for reservoir computing
,”
J. Inst. Electron. Inform. Commun. Engrs.
102
(
2
),
114–120
(
2019
) (in Japanese).
13.
K.
Umeno
and
K.
Okubo
, “
Exact Lyapunov exponents of the generalized Boole transformations
,”
Prog. Theor. Exper. Phys.
2016
(
2
),
021A01
(
2016
).
14.
K.
Umeno
, “
Superposition of chaotic processes with convergence to Lévy’s stable law
,”
Phys. Rev. E
58
,
2644
2647
(
1998
).