Dynamics of Locally Coupled Agents with Next Nearest Neighbor Interaction

J. Herbrych; A. G. Chazirakis; N. Christakis; J. J. P. Veerman

doi:10.1007/s12591-017-0377-3

Differential Equations and Dynamical Systems

pp 1–23 | Cite as

Dynamics of Locally Coupled Agents with Next Nearest Neighbor Interaction

Authors
Authors and affiliations

J. HerbrychEmail author
A. G. Chazirakis
N. Christakis
J. J. P. Veerman

Original Research

First Online: 06 July 2017

Abstract

We consider large but finite systems of identical agents on the line with up to next nearest neighbor asymmetric coupling. Each agent is modelled by a linear second order differential equation, linearly coupled to up to four of its neighbors. The only restriction we impose is that the equations are decentralized. In this generality we give the conditions for stability of these systems. For stable systems, we find the response to a change of course by the leader. This response is at least linear in the size of the flock. Depending on the system parameters, two types of solutions have been found: damped oscillations and reflectionless waves. The latter is a novel result and a feature of systems with at least next nearest neighbor interactions. Analytical predictions are tested in numerical simulations.

Keywords

Dynamical systems Chaotic Dynamics Optimization and control Multi-agent systems

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Notes

Acknowledgements

We acknowledge support by the European Union’s Seventh Framework Program FP7-REGPOT-2012-2013-1 under Grant Agreement No. 316165.

Appendix A: The Routh–Hurwitz Stability Criteria

The Routh–Hurwitz criterion is a standard strategy to derive a concise set of conditions that is equivalent to the condition that all roots of a given polynomial have negative real parts. In various systems similar to the ones discussed here, this criterion gives good results [9, 20]. In our current case the resulting equations are too complicated to give us much information and we only get one more necessary condition for stability that we can use, namely Corollary 7.1. Our discussion is based on Chapter 15, Sections 6, 8, and 13 of Ref. [29], where more details can be found.

Theorem 7.1

(Routh–Hurwitz) Assume that the determinants given below are nonzero. Given a real polynomial $R=x^4+a_3x^3+a_2x^2+a_1x+a_0$, all roots of R have negative real part if and only if all determinants of the upper-left submatrices (the leading principal minors) of:

$$\begin{aligned} A_{4}\equiv \begin{pmatrix} a_{3} &{}\quad a_{1} &{}\quad 0 &{}\quad 0 \\ 1 &{}\quad a_{2} &{}\quad a_{0} &{}\quad 0 \\ 0 &{}\quad a_{3} &{}\quad a_{1} &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad a_{2} &{}\quad a_{0} \end{pmatrix}, \end{aligned}$$

are positive. That is: $a_3>0$, $a_0>0$, $a_3a_2-a_1>0$, and $a_3a_2a_1-a_3^2a_0-a_1^2>0$.

An equivalent but less well-known set of conditions is given in the following:

Theorem 7.2

(Liénard–Chipart) Assume that the determinants in Theorem 7.1 are nonzero. Given a real polynomial $R=x^4+a_3x^3+a_2x^2+a_1x+a_0$, all roots of R have negative real part if and only if $a_3>0$, $a_2>0$, $a_0>0$, and $a_3a_2a_1-a_3^2a_0-a_1^2>0$.

The characteristic polynomial Q of Eq. (9) can be turned into a polynomial with real coefficients

$$\begin{aligned} R=QQ^{*}\equiv & {} \nu ^4-2\mathfrak {R}(\lambda _v)\nu ^3+ \left[ |\lambda _v|^2-2\mathfrak {R}(\lambda _x)\right] \nu ^2\\&+\, 2\left[ \mathfrak {R}(\lambda _x)\mathfrak {R}(\lambda _v)+ \mathfrak {I}(\lambda _x)\mathfrak {I}(\lambda _v)\right] \nu +|\lambda _x|^2, \end{aligned}$$

by taking its product with its complex conjugate. Clearly, all roots of Q have negative real part if and only if the same is true for R. Notice that in each of the two criteria, one of the equations is trivially satisfied, namely $a_0>0$ (where we are assuming nondegeneracy). Therefore, in the Routh–Hurwitz case three equations are obtained. The first two are:

$$\begin{aligned} \mathfrak {R}(\lambda _v)< & {} 0, \end{aligned}$$

(15)

$$\begin{aligned} -\mathfrak {R}(\lambda _v) \left[ |\lambda _v|^2-2\mathfrak {R}(\lambda _x)\right] -\left[ \mathfrak {R}(\lambda _x)\mathfrak {R}(\lambda _v)+\mathfrak {I}(\lambda _x)\mathfrak {I}(\lambda _v)\right]> & {} 0\,. \end{aligned}$$

(16)

The third inequality we do not utilize, since it is extremely complicated containing fifth order terms. We are left with the above two, which are now necessary conditions for all roots to have negative real part.

Similarly, the Liénard–Chipart stability criterion also gives two necessary conditions for all roots to have negative real part:

$$\begin{aligned} \mathfrak {R}(\lambda _v)< & {} 0, \end{aligned}$$

(17)

$$\begin{aligned} 2\mathfrak {R}(\lambda _x)-|\lambda _v|^2< & {} 0. \end{aligned}$$

(18)

The third inequality is the same as before and will not be utilized. Since the second inequality of the Liénard–Chipart conditions seems less complicated than the corresponding one of the Routh–Hurwitz conditions, we will continue with the former.

Substituting the expressions for the $\lambda $’s (Lemma 3.1) we get:

$$\begin{aligned}&g_v \, \sum _{j=0}^{2} \alpha _{v,j}\cos (j\phi )< 0, \\&2g_x \,\sum _{j=0}^{2} \alpha _{x,j}\cos (j\phi )- g_v^2\,\left| \sum _{j=0}^{2} \rho _{v,j} e^{\imath \phi j}\right| ^2 < 0. \end{aligned}$$

These are complicated relations, and therefore we will use the equivalent relations averaged over $\phi $. The first of these inequalities was already used in Lemma 4.1. After some calculations we can work out the average over $\phi $ of the second relation. This gives the final necessary condition for all roots to have negative real part. $\square $

Corollary 7.1

The following is a necessary condition for the collection $\mathcal{S}^*$ to not be unstable:

$$\begin{aligned} 2g_{x}-g_{v}^2\sum _{j=-2}^{2} \rho _{v,j}^2\le 0. \end{aligned}$$

Appendix B: Analysis of Type II Trajectories

Proposition 7.1

Let $K_0>0$ fixed. Suppose that $\mathcal{S}^*$ is stable and that $0<c_-<c_+$. Suppose further that there are $\alpha \in (0,1)$ and $q>0$ such that $Na_m m^{1+q}$ and $Nb_m m^{1+q}$ are bounded, and $(2-q)\alpha \le 1$. Then

$$\begin{aligned} \lim _{N\rightarrow \infty } \sup _{t\in [0,K_0N]}\,|z_N(t)-{\overline{z}}_N(t)|=0. \end{aligned}$$

where ${\overline{z}}_N(t)$ is given by

$$\begin{aligned} {\overline{z}}_N(t)= \left\{ \begin{array}{ccl} -t &{} \text { if } &{}\quad t\in \left[ 0,\frac{N}{c_+}\right] \\ \frac{-N}{c_+} + \frac{c_-}{c_+ - c_-}\left( t-\frac{N}{c_+}\right) &{} \text { if } &{}\quad t\in \left( \frac{N}{c_+},\frac{N}{c_-}\right) \\ 0 &{} \text { if } &{}\quad t\in \left[ \frac{N}{c_-}, \infty \right) \end{array} \right. \end{aligned}$$

The signal velocities are as in Theorem 5.1.

Sketch of Proof

We consider the equations of motion for the acceleration $\xi _k$ of agent k. These are given by the second derivative with respect to time of Definition 2.1. In those equations, the only expression that depends on time is the initial condition of leader. So nothing changes, except that now $\xi _0(t)=\delta (t)$ (for $>0$), where $\delta $ is the Dirac function. We replace the Dirac function by a smooth pulse p(t) that enables us to satisfy the decay constraint on the decay of $a_m$ and $b_m$ but with the condition that $\int \,p(s)\,ds=1$. So now we obtain:

$$\begin{aligned} \xi _0(t) = p(t) \end{aligned}$$

(19)

Theorem 5.1 now implies that in $\mathcal{S}^*$ we have

$$\begin{aligned} \xi _k(t)= f_+(k-c_+t)+f_-(k-c_-t) \end{aligned}$$

(20)

By the periodic boundary conditions conjectures, we see that away from the boundaries the behavior of $\mathcal{S}$ and $\mathcal{S}^*$ is the same. So we have the above relation from $t=0$ until the signal runs into the boundary at N.

Setting $k=0$ in the last equation and comparing with Eq. (19) gives

$$\begin{aligned} p(t) = f_+(-c_+t)+f_-(-c_-t). \end{aligned}$$

(21)

The second part of Eq. (13) then gives:

$$\begin{aligned} f_+'(-c_+t)+f_-'(-c_-t)=0. \end{aligned}$$

Integrate with respect to t to get

$$\begin{aligned} \frac{-1}{c_+}f_+(-c_+t)-\frac{1}{c_-}f_-(-c_-t)=0 \quad \Longrightarrow \quad f_-(s)=-\frac{c_-}{c_+}f_+\left( \frac{c_+}{c_-}\,s\right) . \end{aligned}$$

Substitute this into Eq. (21):

$$\begin{aligned} p(t)=\frac{c_+-c_-}{c_+}f_+(-c_+t)\quad \Longrightarrow \quad f_+(s)=\frac{c_+}{c_+-c_-}p\left( \frac{s}{c_+}\right) . \end{aligned}$$

Now use both of the last equations to eliminate $f_-$ and $f_+$ from Eq. (20):

$$\begin{aligned} \xi _k(t)=\frac{c_+}{c_+-c_-}\, p\left( t-\frac{k}{c_+}\right) - \frac{c_-}{c_+-c_-}\, p\left( t-\frac{k}{c_-}\right) . \end{aligned}$$

Now set $k=N$ and integrate twice with respect to time and add a Galilean transformation so that for small positive t we get $z_N(t)=-t$. With some rewriting this gives the final result. (As before, to actually prove the remainder indeed tends to zero, one needs the assumption on the decay of the $a_m$ and $b_m$. This part of the argument is given in [20]).

References

1.
Lewis, F.L., Zhang, H., Hengster-Movric, K., Das, A.: Cooperative Control of Multi-Agent Systems: Optimal and Adaptive Design Approaches. Springer Verlag, Berlin (2014)CrossRefMATHGoogle Scholar
2.
Backhaus, S., Chertkov, M.: Getting a grip on the electrical grid. Phys. Today 66, 42–48 (2013)CrossRefGoogle Scholar
3.
Motter, A.E., Myers, S.A., Anghel, M., Nishikawa, T.: Spontaneous synchrony in power-grid networks. Nat. Phys. 9, 191–197 (2013)CrossRefGoogle Scholar
4.
Bamieh, B., Paganini, F., Dahleh, M.: Distributed control of spatially invariant systems. IEEE Trans. Autom. Control 47, 1091–1107 (2002)MathSciNetCrossRefMATHGoogle Scholar
5.
Cruz, D., McClintock, J., Perteet, B., Orqueda, O., Cao, Y., Fierro, R.: Decentralized cooperative control: a multivehicle platform for research in networked embedded systems. IEEE Control. Syst. Mag. 27, 58–78 (2007)MathSciNetCrossRefGoogle Scholar
6.
Defoort, M., Floquet, F., Kokosy, A., Perruquetti, W.: Sliding-mode formation control for cooperative autonomous mobile robots. IEEE Trans. Ind. Electron. 55, 3944–3953 (2008)CrossRefGoogle Scholar
7.
Lin, F., Fardad, M., Jovanovic, M.R.: Optimal control of vehicular formations with nearest neighbor interactions. IEEE Trans. Autom. Control 57, 2203–2218 (2012)MathSciNetCrossRefGoogle Scholar
8.
Barooah, P., Mehta, P.G.P., Hespanha, J.J.P.: Mistuning-based control design to improve closed-loop stability margin of vehicular platoons. IEEE Trans. Autom. Control 54, 2100–2113 (2013)MathSciNetCrossRefGoogle Scholar
9.
Herman, I., Martinec, D., Veerman, J.J.P.: Transients of platoons with asymmetric and different laplacians. Syst. Control Lett. 91, 28–35 (2016)MathSciNetCrossRefMATHGoogle Scholar
10.
Reynolds, C.W.: Flocks, herds, and schools:a distributed behavioral model. Comput. Graph. 21, 25–34 (1987)CrossRefGoogle Scholar
11.
Strogatz, S.H., Stewart, I.: Coupled oscillators and biological synchronization. Sci. Am. 269, 102–109 (1993)CrossRefGoogle Scholar
12.
Couzin, I.D., Krause, J., Franks, N.R., Levin, S.A.: Effective leadership and decision-making in animal groups on the move. Nature 433, 513–516 (2005)CrossRefGoogle Scholar
13.
Attanasi, A., Cavagna, A., Del Castello, L., Giardina, I., Grigera, T.S., Jelic, A., Melillo, S., Parisi, L., Pohl, O., Shen, E., Viale, M.: Information transfer and behavioral inertia in starling flocks. Nat. Phys. 10, 691–696 (2014)CrossRefGoogle Scholar
14.
Pearce, D.J.G., Miller, A.M., Rowlands, G., Turner, M.S.: Role of projection in the control of bird flocks. Proc. Natl. Acad. Sci. 29, 10422–10426 (2014)CrossRefGoogle Scholar
15.
Ren, W., Beard, R.W., Atkins, E.M.: A survey of consensus problems in multi-agent coordination. In: Proceedings of the 2005 American Control Conference, vol. 3, pp. 1859–1864 (2005)Google Scholar
16.
Hegselmann, R., Krause, U.: Opinion dynamics and bounded confidence: models, analysis and simulation. J. Artif. Soc. Soc. Simul. 5(3) (2002)Google Scholar
17.
Ashcroft, N.W., Mermin, N.D.: Solid State Physics. Saunders College, Philadelphia (1976)MATHGoogle Scholar
18.
Haberman, R.: Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, 4th edn. Prentice Hall, New Jersey (2004)Google Scholar
19.
Golub, G.H., Van Loan, C.F.: Matrix Computations, 4th edn. Johns Hopkins University Press, Baltimore (2013)MATHGoogle Scholar
20.
Cantos, C.E., Hammond, D.K., Veerman, J.J.P.: Signal velocity in oscillator networks. Eur. Phys. J. Spec. Top. 225, 1115–1126 (2016)CrossRefGoogle Scholar
21.
Cantos, C.E., Hammond, D.K., Veerman, J.J.P.: Transients in the synchronization of oscillator arrays. Eur. Phys. J. Spec. Top. 225, 1199–1209 (2016)CrossRefGoogle Scholar
22.
Veerman, J.J.P.: Symmetry and stability of homogeneous flocks (a position paper). In: Proc. 1st Intern. Conf on Pervasive and Embedded Computing and Communication Systems, Algarve, vol. 21, p. 6 (2010)Google Scholar
23.
Trefethen, L.N., Trefethen, A.E., Reddy, S.C., Driscoll, T.A.: Hydrodynamic stability without eigenvalues. Science 261, 578–584 (1993)MathSciNetCrossRefMATHGoogle Scholar
24.
Trefethen, L.N.: Pseudospectra of linear operators. SIAM Rev. 39, 383–406 (1997)MathSciNetCrossRefMATHGoogle Scholar
25.
Kra, I., Simanca, S.R.: On circulant matrices. Not. Am. Math. Soc. 59, 368–377 (2012)CrossRefMATHGoogle Scholar
26.
Boost c++ libraries. http://www.boost.org. Accessed 6 Sept 2016
27.
Schöing, B.: The Boost C++ Libraries. XML Press, Laguna Hills (2011)Google Scholar
28.
Giardina I. Personal communicationGoogle Scholar
29.
Gantmacher, F.R.: The Theory of Matrices, vol. 2. American Mathematical Society, Providence (2000)MATHGoogle Scholar

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Authors and Affiliations

J. Herbrych
- 1
- 2
- 3
Email author
A. G. Chazirakis
- 4
N. Christakis
- 1
J. J. P. Veerman
- 1
- 5

1.Department of Physics, Crete Center for Quantum Complexity and NanotechnologyUniversity of CreteHeraklionGreece
2.Department of Physics and AstronomyUniversity of TennesseeKnoxvilleUSA
3.Materials Science and Technology DivisionOak Ridge National LaboratoryOak RidgeUSA
4.Department of Mathematics and Applied MathematicsUniversity of CreteHeraklionGreece
5.Fariborz Maseeh Department of Mathematics and StatisticsPortland State UniversityPortlandUSA

Dynamics of Locally Coupled Agents with Next Nearest Neighbor Interaction

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