# Analysis of Chaotic Response of Frenkel-Kontorova-Tomlinson Model

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## Abstract

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## 1. Introduction

## 2. The Governing Equations

## 3. The Convergence Criterion and the Lyapunov Exponent

## 4. Simulation and Results

## 5. Final Comments and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Phase diagram of Duffing equations for RELTOL and VNTOL equal to 1.0 × 10${}^{-10}$ (DTMIN = 1.0 × 10${}^{-15}$).

**Figure 4.**Zoom of Figure 3 between 900 and 1000 s.

**Figure 5.**Zoom of Figure 3 between 0 and 100 s.

**Figure 6.**Lyapunov exponents of the Duffing equations represented in Figure 3.

**Figure 7.**Phase diagram of Duffing equations for RELTOL and VNTOL equal to 1.0 × 10${}^{-15}$ (DTMIN = 1.0 × 10${}^{-20}$).

**Figure 8.**Phase diagram of Duffing equations for RELTOL and VNTOL equal to 1.0 × 10${}^{-6}$ (DTMIN = 1.0 × 10${}^{-15}$).

**Figure 9.**Phase diagram of Duffing equations for RELTOL and VNTOL equal to 1.0 × 10${}^{-20}$ (DTMIN = 1.0 × 10${}^{-30}$).

**Figure 10.**Phase diagram of FKT model with ${\kappa}_{2}=1.5$ for RELTOL and VNTOL equal to 1.0 × 10${}^{-6}$ (DTMIN = 1.0 × 10${}^{-15}$).

**Figure 11.**Phase diagram of FKT model with ${\kappa}_{2}=1.5$ for RELTOL and VNTOL equal to 1.0 × 10${}^{-5}$ (DTMIN = 1.0 × 10${}^{-15}$).

**Figure 12.**Phase diagram of FKT model with ${\kappa}_{2}=1.5$ for RELTOL and VNTOL equal to 1.0 × 10

^{−4}(DTMIN = 1.0 × 10${}^{-15}$).

**Figure 13.**Phase diagram of FKT model with ${\kappa}_{2}=1.5$ for RELTOL and VNTOL equal to 1.0 × 10${}^{-3}$ (DTMIN = 1.0 × 10${}^{-15}$).

**Figure 14.**Phase diagram of FKT model with the same parameters as Figure 13 in a range of 3950 and 4000 s.

**Figure 15.**Phase diagram of FKT model with the same parameters as Figure 13 in a range of 7950 and 8000 s.

**Figure 16.**Phase diagram of FKT model with the same parameters as Figure 12 in a range of 50 and 100 s.

**Figure 17.**Phase diagram of FKT model with the same parameters as Figure 9 but in a range of 50 and 100 s.

**Figure 18.**Phase diagram of FKT model with the same parameters as Figure 9 but using zero and infinite resistances of 10 − 20 and 10 + 20, respectively.

**Figure 19.**Phase diagram of FKT model with ${\kappa}_{2}=1.4$ for RELTOL and VNTOL equal to 1.0 × 10

^{−4}(DTMIN = 1.0 × 10${}^{-15}$).

**Figure 20.**Phase diagram of FKT model with ${\kappa}_{2}=1.4$ for RELTOL and VNTOL equal to 1.0 × 10${}^{-5}$ (DTMIN = 1.0 × 10${}^{-15}$).

**Figure 21.**Phase diagram of FKT model with ${\kappa}_{2}=1.4$ for RELTOL and VNTOL equal to 1.0 × 10${}^{-6}$ (DTMIN = 1.0 × 10${}^{-15}$).

**Figure 22.**Phase diagram of FKT model with ${\kappa}_{2}=\phantom{\rule{3.33333pt}{0ex}}1.4$ for RELTOL and VNTOL equal to 1.0 × 10${}^{-7}$ (DTMIN = 1.0 × 10${}^{-15}$).

**Figure 23.**Phase diagram of FKT model with ${\kappa}_{2}=\phantom{\rule{3.33333pt}{0ex}}1.4$ for RELTOL and VNTOL equal to 1.0 × 10${}^{-10}$ (DTMIN = 1.0 × 10${}^{-15}$).

**Figure 24.**Phase diagram of FKT model with ${\kappa}_{2}=\phantom{\rule{3.33333pt}{0ex}}1.4$ for RELTOL and VNTOL equal to 1.0 × 10${}^{-12}$ (DTMIN = 1.0 × 10${}^{-15}$).

**Figure 25.**Phase diagram of FKT model with ${\kappa}_{2}=\phantom{\rule{3.33333pt}{0ex}}1.4$ for RELTOL and VNTOL equal to 1.0 × 10${}^{-6}$ (DTMIN = 1.0 × 10${}^{-15}$) (50–100 s).

**Figure 26.**Phase diagram of FKT model with ${\kappa}_{2}=\phantom{\rule{3.33333pt}{0ex}}1.4$ for RELTOL and VNTOL equal to 1.0 × 10${}^{-6}$ (DTMIN = 1.0 × 10${}^{-15}$) (150–200 s).

**Figure 27.**Phase diagram of FKT model with ${\kappa}_{2}=\phantom{\rule{3.33333pt}{0ex}}1.4$ for RELTOL and VNTOL equal to 1.0 × 10${}^{-6}$ (DTMIN = 1.0 × 10${}^{-15}$) (250–300 s).

**Figure 28.**Phase diagram of FKT model with ${\kappa}_{2}=\phantom{\rule{3.33333pt}{0ex}}1.6$ for RELTOL and VNTOL equal to 1.0 × 10${}^{-10}$ (DTMIN = 1.0 × 10${}^{-15}$).

**Figure 29.**Phase diagram of FKT model with ${\kappa}_{2}=\phantom{\rule{3.33333pt}{0ex}}1.4$ for RELTOL and VNTOL equal to 1.0 × 10${}^{-10}$ (DTMIN = 1.0 × 10${}^{-15}$).

**Figure 30.**Phase diagram of FKT model with ${\kappa}_{2}=\phantom{\rule{3.33333pt}{0ex}}1.4$ for RELTOL and VNTOL equal to 1.0 × 10${}^{-5}$ (DTMIN = 1.0 × 10${}^{-15}$).

**Figure 31.**Lyapunov exponents of the FKT model with ${\kappa}_{2}=\phantom{\rule{3.33333pt}{0ex}}1.4$.

**Figure 32.**Phase diagram of FKT model using MATLAB with ${\kappa}_{2}=\phantom{\rule{3.33333pt}{0ex}}1.6$ for RELTOL and VNTOL equal to 1.0 × 10${}^{-5}$ (DTMIN = 1.0 × 10${}^{-15}$).

**Figure 33.**Phase diagram of FKT model using ngspice with ${\kappa}_{2}=\phantom{\rule{3.33333pt}{0ex}}1.6$ for RELTOL and VNTOL equal to 1.0 × 10${}^{-5}$ (DTMIN = 1.0 × 10${}^{-15}$).

**Figure 34.**Phase diagram of FKT model using MATLAB with ${\kappa}_{2}=\phantom{\rule{3.33333pt}{0ex}}1.4$ for RELTOL and VNTOL equal to 1.0 × 10${}^{-5}$ (DTMIN = 1.0 × 10${}^{-15}$).

**Figure 35.**Phase diagram of FKT model using ngspice with ${\kappa}_{2}=\phantom{\rule{3.33333pt}{0ex}}1.4$ for RELTOL and VNTOL equal to 1.0 × 10${}^{-5}$ (DTMIN = 1.0 × 10${}^{-15}$).

**Figure 36.**Lyapunov exponents of the FKT model with ${\kappa}_{2}=\phantom{\rule{3.33333pt}{0ex}}1.6$.

**Figure 37.**Lyapunov exponents of the FKT model with ${\kappa}_{2}=\phantom{\rule{3.33333pt}{0ex}}1.4$.

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**MDPI and ACS Style**

Ramírez, J.S.; Gallego, F.B.; Nicolás, J.A.M.; García, F.M.
Analysis of Chaotic Response of Frenkel-Kontorova-Tomlinson Model. *Symmetry* **2020**, *12*, 1413.
https://doi.org/10.3390/sym12091413

**AMA Style**

Ramírez JS, Gallego FB, Nicolás JAM, García FM.
Analysis of Chaotic Response of Frenkel-Kontorova-Tomlinson Model. *Symmetry*. 2020; 12(9):1413.
https://doi.org/10.3390/sym12091413

**Chicago/Turabian Style**

Ramírez, Joaquín Solano, Francisco Balibrea Gallego, José Andrés Moreno Nicolás, and Fulgencio Marín García.
2020. "Analysis of Chaotic Response of Frenkel-Kontorova-Tomlinson Model" *Symmetry* 12, no. 9: 1413.
https://doi.org/10.3390/sym12091413