# Locating the Sets of Exceptional Points in Dissipative Systems and the Self-Stability of Bicycles

## Abstract

**:**

## 1. Introduction

## 2. Complex Exceptional Points and the Self-Stability of Bicycles

#### 2.1. The TMS Bicycle Model

#### 2.2. Preliminaries on Lyapunov Stability and the Asymptotic Stability of Equilibria

#### 2.3. Asymptotic Stability of the TMS Bike and the Critical Froude Number for the Weaving Motion

#### 2.4. Minimizing the Spectral Abscissa of General TMS Bikes

#### 2.5. Self-Stable and Heavily-Damped TMS Bikes with ${\chi}_{H}=1$

#### 2.5.1. The Critical Froude Number and Its Minimum

#### 2.5.2. Exact Location of the Real Exceptional Point ${\mathrm{EP}}_{4}$

#### 2.5.3. Discriminant Surface and the EP-Set

#### 2.5.4. Location of the EP-Set and Stability Optimization

#### 2.5.5. Mechanism of Self-Stability and ${\mathrm{CEP}}_{2}$ as a Precursor to Bike Weaving

#### 2.5.6. How the Scaling Laws Found Match the Experimental TMS Bike Realization

## 3. Conclusions

## Funding

## Conflicts of Interest

## References

- Bender, C.M.; Berntson, B.K.; Parker, D.; Samuel, E. Observation of PT phase transition in a simple mechanical system. Am. J. Phys.
**2013**, 81, 173–179. [Google Scholar] [CrossRef] - Xu, X.-W.; Liu, Y.-X.; Sun, C.-P.; Li, Y. Mechanical PT symmetry in coupled optomechanical systems. Phys. Rev. A
**2015**, 92, 013852. [Google Scholar] [CrossRef] - Schindler, S.; Lin, Z.; Lee, J.M.; Ramezani, H.; Ellis, F.M.; Kottos, T. PT symmetric electronics. J. Phys. A Math. Theor.
**2012**, 45, 444029. [Google Scholar] [CrossRef] - Freitas, P. On some eigenvalue problems related to the wave equation with indefinite damping. J. Differ. Equ.
**1996**, 127, 320–335. [Google Scholar] [CrossRef] - Freitas, P.; Zuazua, E. Stability results for the wave equation with indefinite damping. J. Differ. Equ.
**1996**, 132, 338–353. [Google Scholar] [CrossRef] - Freitas, P.; Grinfeld, M.; Knight, P.A. Stability of finite-dimensional systems with indefinite damping. Adv. Math. Sci. Appl.
**1997**, 7, 437–448. [Google Scholar] - Freitas, P. Quadratic matrix polynomials with Hamiltonian spectrum and oscillatory damped systems. Z. Angew. Math. Phys.
**1999**, 50, 64–81. [Google Scholar] [CrossRef] - Kliem, W.; Pommer, C. Indefinite damping in mechanical systems and gyroscopic stabilization. Z. Angew. Math. Phys.
**2009**, 60, 785–795. [Google Scholar] [CrossRef] - Kirillov, O.N. PT symmetry, indefinite damping and dissipation-induced instabilities. Phys. Lett. A
**2012**, 376, 1244–1249. [Google Scholar] [CrossRef] - Kirillov, O.N. Stabilizing and destabilizing perturbations of PT symmetric indefinitely damped systems. Philos. Trans. R. Soc. A
**2013**, 371, 20120051. [Google Scholar] [CrossRef] [PubMed] - Kirillov, O.N. Exceptional and diabolical points in stability questions. Fortschr. Phys. Prog. Phys.
**2013**, 61, 205–224. [Google Scholar] [CrossRef] - Kirillov, O.N. Singular diffusionless limits of double-diffusive instabilities in magnetohydrodynamics. Proc. R. Soc. A
**2017**, 473, 20170344. [Google Scholar] [CrossRef] [PubMed][Green Version] - Jones, C.A. Multiple eigenvalues and mode classification in plane Poiseuille flow. Quart. J. Mech. Appl. Math.
**1988**, 41, 363–382. [Google Scholar] [CrossRef] - Dobson, I.; Zhang, J.; Greene, S.; Engdahl, H.; Sauer, P.W. Is strong modal resonance a precursor to power system oscillations? IEEE Trans. Circ. Syst. I
**2001**, 48, 340–349. [Google Scholar] [CrossRef][Green Version] - Friedland, S. Simultaneous similarity of matrices. Adv. Math.
**1983**, 50, 189–265. [Google Scholar] [CrossRef] - Christensen, E. On invertibility preserving linear mappings, simultaneous triangularization and Property L. Linear Algebra Appl.
**1999**, 301, 153–170. [Google Scholar] [CrossRef] - Kirillov, O.N.; Overton, M.L. Robust stability at the swallowtail singularity. Front. Phys.
**2013**, 1, 24. [Google Scholar] [CrossRef] - Borrell, B. Physics on two wheels. Nature
**2016**, 535, 338–341. [Google Scholar] [CrossRef] [PubMed] - Meijaard, J.P.; Papadopoulos, J.M.; Ruina, A.; Schwab, A.L. Linearized dynamics equations for the balance and steer of a bicycle: A benchmark and review. Proc. R. Soc. A
**2007**, 463, 1955–1982. [Google Scholar] [CrossRef] - Sharp, R.S. On the stability and control of the bicycle. Appl. Mech. Rev.
**2008**, 61, 060803. [Google Scholar] [CrossRef] - Meijaard, J.P.; Papadopoulos, J.M.; Ruina, A.; Schwab, A.L. Historical Review of Thoughts on Bicycle Self-Stability; Cornell University: Ithaca, NY, USA, 2011. [Google Scholar]
- Boyer, F.; Porez, M.; Mauny, J. Reduced dynamics of the non-holonomic Whipple bicycle. J. Nonlinear Sci.
**2018**, 28, 943–983. [Google Scholar] [CrossRef] - Borisov, A.V.; Mamaev, I.S.; Bizyaev, I.A. Dynamical systems with non-integrable constraints, vakonomic mechanics, sub-Riemannian geometry, and non-holonomic mechanics. Russ. Math. Surv.
**2017**, 72, 783–840. [Google Scholar] [CrossRef] - Levi, M.; Tabachnikov, S. On bicycle tire tracks geometry, hatchet planimeter, Menzin’s conjecture, and oscillation of unicycle tracks. Exp. Math.
**2009**, 18, 173–186. [Google Scholar] [CrossRef] - Levi, M. Schrödinger’s equation and “bike tracks” —A connection. J. Geom. Phys.
**2017**, 115, 124–130. [Google Scholar] [CrossRef] - Bor, G.; Levi, M.; Perline, R.; Tabachnikov, S. Tire tracks and integrable curve evolution. Int. Math. Res. Not.
**2018**, rny087. [Google Scholar] [CrossRef] - Kooijman, J.D.G.; Meijaard, J.P.; Papadopoulos, J.M.; Ruina, A.; Schwab, A.L. A bicycle can be self-stable without gyroscopic or caster effects. Science
**2011**, 332, 339–342. [Google Scholar] [CrossRef] [PubMed] - Hess, R.; Moore, J.K.; Hubbard, M. Modeling the manually controlled bicycle. IEEE Trans. Syst. Man Cybern. Part A Syst. Hum.
**2012**, 42, 545–557. [Google Scholar] [CrossRef] - Ricci, F.; Frosali, G. A symbolic method for the analysis of a nonlinear Two-Mass-Skate model. arXiv, 2016; arXiv:1611.07796. [Google Scholar]
- Collins, S.; Ruina, A.; Tedrake, R.; Wisse, M. Efficient bipedal robots based on passive-dynamic walkers. Science
**2005**, 307, 1082–1085. [Google Scholar] [CrossRef] [PubMed] - Borisov, A.V.; Kilin, A.A.; Mamaev, I.S. On the Hadamard-Hamel problem and the dynamics of wheeled vehicles. Reg. Chaot. Dyn.
**2015**, 20, 752–766. [Google Scholar] [CrossRef] - Lyapunov, A.M. The general problem of the stability of motion. Int. J. Control
**1992**, 55, 531–773. [Google Scholar] [CrossRef] - Kirillov, O.N. Nonconservative Stability Problems of Modern Physics; De Gruyter: Berlin, Germany, 2013. [Google Scholar]
- Barkwell, L.; Lancaster, P. Overdamped and gyroscopic vibrating systems. J. Appl. Mech.
**1992**, 59, 176–181. [Google Scholar] [CrossRef] - Veselic, K. Damped Oscillations of Linear Systems: A Mathematical Introduction; Springer: Berlin, Germany, 2011. [Google Scholar]
- Berry, M.V.; Shukla, P. Curl force dynamics: Symmetries, chaos and constants of motion. New J. Phys.
**2016**, 18, 063018. [Google Scholar] [CrossRef] - Kirillov, O.N. Classical results and modern approaches to nonconservative stability. Ch. 4. In Dynamic Stability and Bifurcation in Nonconservative Mechanics; Bigoni, D., Kirillov, O., Eds.; CISM International Centre for Mechanical Sciences 586; Springer: Berlin, Germany, 2018; pp. 129–190. [Google Scholar]
- Austin Sydes, G.L. Self-Stable Bicycles; BSc (Hons) Mathematics Final Year Project Report; Northumbria University: Newcastle upon Tyne, UK, 2018. [Google Scholar]
- Freitas, P.; Lancaster, P. On the optimal value of the spectral abscissa for a system of linear oscillators. SIAM J. Matrix Anal. Appl.
**1999**, 21, 195–208. [Google Scholar] [CrossRef] - Blondel, V.D.; Gurbuzbalaban, M.; Megretski, A.; Overton, M.L. Explicit solutions for root optimization of a polynomial family with one affine constraint. IEEE Trans. Autom. Control
**2012**, 57, 3078–3089. [Google Scholar] [CrossRef]

**Figure 1.**The two-mass-skate (TMS) bicycle model [27].

**Figure 2.**(

**a**) The growth rates for the benchmark TMS bicycle (7); (

**b**) the growth rates of the optimized TMS bicycle with ${\zeta}_{B}=-0.4$, ${\zeta}_{H}=-0.2$, ${\chi}_{B}=1.19$, ${\chi}_{H}=1.02$, $\mu =20.84626701$ and ${\lambda}_{s}=0.8514403685$ showing that the spectral abscissa attains its minimal value ${a}_{min}=-1$ at ${\mathrm{Fr}}_{\mathrm{EP}}=2.337214017$ at the real exceptional point of order four, ${\mathrm{EP}}_{4}$.

**Figure 3.**(

**a**) The discriminant surface of the characteristic polynomial of the TMS bike with ${\chi}_{H}=1$, ${\zeta}_{H}=-0.2$ and ${\zeta}_{B}=-0.4$ showing the swallowtail singularity at ${\mathrm{EP}}_{4}$. The cross-section of the domain of asymptotic stability and the discriminant surface at (

**b**) $\mathrm{Fr}={\mathrm{Fr}}_{{\mathrm{EP}}_{4}}=\frac{3\sqrt{110\sqrt{2}-120}}{8}$, (

**c**) $\mathrm{Fr}={\mathrm{Fr}}_{{\mathrm{EP}}_{4}}-0.1$ and (

**d**) $\mathrm{Fr}={\mathrm{Fr}}_{{\mathrm{EP}}_{4}}+0.5$.

**Figure 4.**${\chi}_{H}=1$, ${\zeta}_{H}=-0.2$ and ${\zeta}_{B}=-0.4$. (

**a**) For ${\chi}_{B}=\sqrt{2}$, the boundary between the domains of weaving and asymptotic stability in the $(\mathrm{Fr},{\lambda}_{s})$-plane shown together with the domain of heavy damping that has a cuspidal point corresponding to a negative real eigenvalue ${\omega}_{0}=-\sqrt[4]{\frac{25}{2}}$ with the Jordan block of order four (${\mathrm{EP}}_{4}$). The ${\mathrm{EP}}_{4}$ belongs to a curve (23) that corresponds to (dashed part) conjugate pairs of double complex eigenvalues with the Jordan block of order two (complex ${\mathrm{EP}}_{2}$) and (solid part) to couples of double real negative eigenvalues with the Jordan block of order two ($2{\mathrm{EP}}_{2}$). (

**b**) The same in the $(\mathrm{Fr},{\chi}_{B})$-plane at ${\lambda}_{s}=arctan\left(\right)open="("\; close=")">\frac{15}{4}-\frac{75}{32}\sqrt{2}$ rad. The domain of heavy damping degenerates into a singular point: the swallowtail singularity.

**Figure 5.**${\chi}_{H}=1$, ${\zeta}_{H}=-0.2$ and ${\zeta}_{B}=-0.4$. (

**a**) For ${\chi}_{B}=\sqrt{2}-0.1$, the boundary between the domains of weaving and asymptotic stability in the $(\mathrm{Fr},{\lambda}_{s})$-plane shown together with the domain of heavy damping that has a cusp corresponding to a negative real eigenvalue with the Jordan block of order three (${\mathrm{EP}}_{3}$). The ${\mathrm{EP}}_{3}$ belongs to the cuspidal edge of the swallowtail surface bounding the domain of heavy damping. (

**b**) The same in the $(\mathrm{Fr},{\chi}_{B})$-plane at ${\lambda}_{s}=arctan\left(\right)open="("\; close=")">\frac{15}{4}-\frac{75}{32}\sqrt{2}$ rad. Notice the cuspidal ${\mathrm{EP}}_{3}$-points and the self intersection at the $2{\mathrm{EP}}_{2}$ point on the boundary of the domain of heavy damping.

**Figure 6.**${\chi}_{H}=1$, ${\zeta}_{H}=-0.2$, ${\zeta}_{B}=-0.4$. Stabilization of the TMS bike as $\mathrm{Fr}$ is increasing from 0–5 for (

**a,b**) ${\lambda}_{s}=arctan\left(\right)open="("\; close=")">\frac{15}{4}-\frac{75}{32}\sqrt{2}$ rad, (

**c,d**) ${\lambda}_{s}=arctan\left(\right)open="("\; close=")">\frac{15}{4}-\frac{75}{32}\sqrt{2}$ rad and (

**e,f**) ${\lambda}_{s}=arctan\left(\right)open="("\; close=")">\frac{15}{4}-\frac{75}{32}\sqrt{2}$ rad. The eigenvalue curves are shown for (black) ${\chi}_{B}=\sqrt{2}$, (blue) ${\chi}_{B}=\sqrt{2}-0.01$ and (red) ${\chi}_{B}=\sqrt{2}+0.01$ in the upper and middle rows and for (black) ${\chi}_{B}=\sqrt{2}$, (blue) ${\chi}_{B}=\sqrt{2}-0.1$ and (red) ${\chi}_{B}=\sqrt{2}+0.1$ in the lower row. Notice the existence at ${\chi}_{B}=\sqrt{2}$ of (

**a,b**) a real exceptional point ${\mathrm{EP}}_{4}$, (

**c,d**) a couple of real exceptional points ${\mathrm{EP}}_{2}$ and (

**e,f**) a couple of complex exceptional points ${\mathrm{EP}}_{2}$ and repelling of eigenvalue curves near $\mathrm{EP}$s when ${\chi}_{B}\ne \sqrt{2}$.

**Figure 7.**Experimental realization of a self-stable TMS bicycle design found by trial and error in [18,27] with ${\chi}_{B}=1.526$, ${\chi}_{H}=0.921$, ${\zeta}_{B}=-1.158$ and ${\zeta}_{H}=-0.526$ approximately fitting the scaling law (20). Indeed, $\sqrt{\frac{{\zeta}_{B}}{{\zeta}_{H}}}=1.484$ is close to ${\chi}_{B}=1.526$.

Dimensional | Meaning | Dimensionless | Meaning |
---|---|---|---|

v | Velocity of the bike | ||

g | Gravity acceleration | $\mathrm{Fr}$ | Froude number |

w | Wheel base | ||

${\lambda}_{s}$ | Steer axis tilt (rad) | ${\lambda}_{s}$ | Steer axis tilt (rad) |

${m}_{H}$ | Front fork and handlebar | ||

assembly (FHA) mass | $\mu $ | Mass ratio (${m}_{H}/{m}_{B}$) | |

${m}_{B}$ | Rear frame assembly (RFA) mass | ||

${x}_{H}$ (≥0) | Horizontal coordinate of the | ${\chi}_{H}$ (≥0) | Horizontal coordinate of the |

FHA centre of mass | FHA centre of mass | ||

${z}_{H}$ (≤0) | Vertical coordinate of the | ${\zeta}_{H}$ (≤0) | Vertical coordinate of the |

FHA centre of mass | FHA centre of mass | ||

${x}_{B}$ (≥0) | Horizontal coordinate of the | ${\chi}_{B}$ (≥0) | Horizontal coordinate of the |

RFA centre of mass | RFA centre of mass | ||

${z}_{B}$ (≤0) | Vertical coordinate of the | ${\zeta}_{B}$ (≤0) | Vertical coordinate of the |

RFA centre of mass | RFA centre of mass | ||

t | Time | $\tau $ | Time |

Bike | ${\mathit{\chi}}_{\mathit{H}}$ | ${\mathit{\chi}}_{\mathit{B}}$ | ${\mathit{\zeta}}_{\mathit{H}}$ | ${\mathit{\zeta}}_{\mathit{B}}$ | ${\mathit{\omega}}_{0}$ | ${\mathit{\lambda}}_{\mathit{s}}$ (rad) | ${\mathbf{Fr}}_{\mathit{c}}$ | ${\mathbf{Fr}}_{\mathbf{EP}}$ |
---|---|---|---|---|---|---|---|---|

${\mathrm{EP}}_{4}$ | 1 | $\sqrt{2}$ | $-0.2$ | $-0.4$ | $-\frac{\sqrt{5}}{\sqrt[4]{2}}$ | $arctan\left(\right)open="("\; close=")">\frac{15}{4}-\frac{75}{32}\sqrt{2}$ | $\frac{\sqrt{30\sqrt{2}+120}}{8}$ | $\frac{3\sqrt{110\sqrt{2}-120}}{8}$ |

${2\mathrm{EP}}_{2}$ | 1 | $\sqrt{2}$ | $-0.2$ | $-0.4$ | $-\frac{\sqrt{5}}{\sqrt[4]{2}}$ | $arctan\left(\right)open="("\; close=")">\frac{15}{4}-\frac{75}{32}\sqrt{2}$ | ≈1.482682090 | ≈2.257421384 |

${\mathrm{CEP}}_{2}$ | 1 | $\sqrt{2}$ | $-0.2$ | $-0.4$ | $-\frac{\sqrt{5}}{\sqrt[4]{2}}$ | $arctan\left(\right)open="("\; close=")">\frac{15}{4}-\frac{75}{32}\sqrt{2}$ | ≈3.934331969 | ≈4.103508160 |

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**MDPI and ACS Style**

Kirillov, O.N.
Locating the Sets of Exceptional Points in Dissipative Systems and the Self-Stability of Bicycles. *Entropy* **2018**, *20*, 502.
https://doi.org/10.3390/e20070502

**AMA Style**

Kirillov ON.
Locating the Sets of Exceptional Points in Dissipative Systems and the Self-Stability of Bicycles. *Entropy*. 2018; 20(7):502.
https://doi.org/10.3390/e20070502

**Chicago/Turabian Style**

Kirillov, Oleg N.
2018. "Locating the Sets of Exceptional Points in Dissipative Systems and the Self-Stability of Bicycles" *Entropy* 20, no. 7: 502.
https://doi.org/10.3390/e20070502