Nonlinear Network Modes in Cyclic Systems with Applications to Connected Vehicles

Abstract

In this paper, we propose a novel technique to decompose networked systems with cyclic structure into nonlinear modes and apply these ideas to a system of connected vehicles. We perform linear and nonlinear transformations that exploit the network structure and lead to nonlinear modal equations that are decoupled. Each mode can be obtained by solving a small set of algebraic equations without deriving the coefficients for any other mode. By focusing on the mode that is loosing stability, bifurcation analysis can be carried out. The techniques developed are applied to evaluate the impact of connected cruise control on the nonlinear dynamics of a connected vehicle system.

Appendices

Appendix 1: Simplification of Cubic Terms

Before applying the cubic near-identity transformation (52), we need to express all the cubic terms of (50) using multiples of two circulant matrices. However, the cubic term ${\hat{\mathfrak {L}}}^{(\beta )}\Big ({\hat{\mathbf {S}}}_b\big (\hat{\varvec{\Psi }}({\hat{\mathbf {u}}}){\hat{\mathbf {u}}}\big )\Big )\,\hat{\mathbf {K}}^{(\beta )}_{b}\,{\hat{\mathbf {u}}}$ does not appear to have such structure. By spelling out the components corresponding to the dynamics of mode $k$ (in other words the $k+1$-st row pair), we obtain

$$\begin{aligned} \begin{aligned}&\bigg [\sum _{b=0,1}\sum _{\beta =1,2}{\hat{\mathfrak {L}}}^{(\beta )}\big ({\hat{\mathbf {S}}}_{b}(\hat{\varvec{\Psi }}({\hat{\mathbf {u}}}){\hat{\mathbf {u}}})\big )\hat{\mathbf {K}}^{(\beta )}_{b}{\hat{\mathbf {u}}}\bigg ]_{k}\\&=\sum ^{N-1}_{j=0}\,\sum ^{N-1}_{\ell =0}\,\sum _{b=0,1}\,\sum _{\beta =1,2}\,\sum _{\delta =1,2} \begin{bmatrix} \mathrm {e}^{\mathrm {i}\frac{2\,\pi \,b\,(k-\ell )}{N}}\Big [\mathrm {\partial }^{(\beta )}_{b}\,\kappa ^{(1\,\delta )}_{\ell 0}\Big ]^{*}\Big (\overline{\mathrm {\varvec{\psi }}}^{(\beta 1)}_{m(\ell )\,j}\cdot \mathbf {u}_{f_{m(\ell )\,j}}\,u^{(1)}_{j}+\overline{\mathrm {\varvec{\psi }}}^{(\beta 2)}_{m(\ell )\,j}\cdot \mathbf {u}_{f_{m(\ell )\,j}}\,u^{(2)}_{j}\Big )\,u^{(\delta )}_{\ell }\\ \mathrm {e}^{\mathrm {i}\frac{2\,\pi \,b\,(k-\ell )}{N}}\Big [\mathrm {\partial }^{(\beta )}_{b}\,\kappa ^{(2\,\delta )}_{\ell 0}\Big ]^{*}\Big (\overline{\mathrm {\varvec{\psi }}}^{(\beta 1)}_{m(\ell )\,j}\cdot \mathbf {u}_{f_{m(\ell )\,j}}\,u^{(1)}_{j}+\overline{\mathrm {\varvec{\psi }}}^{(\beta 2)}_{m(\ell )\,j}\cdot \mathbf {u}_{f_{m(\ell )\,j}}\,u^{(2)}_{j}\Big )\,u^{(\delta )}_{\ell } \end{bmatrix}, \end{aligned} \end{aligned}$$

(103)

where $m(\ell )$ is given as

$$\begin{aligned} m(\ell )=f_{k\ell }={\left\{ \begin{array}{ll} k-\ell \quad &{}\text {if} \quad k\,\ge \,\ell ,\\ N+k-\ell \quad &{}\text {if} \quad k\,<\,\ell . \end{array}\right. } \end{aligned}$$

(104)

If we consider a change of indexes $\sigma =f_{k\ell }$, we get

$$\begin{aligned} \begin{aligned}&\bigg [\sum _{b=0,1}{\hat{\mathfrak {L}}}^{(\beta )}\big ({\hat{\mathbf {S}}}_{b}(\hat{\varvec{\Psi }}({\hat{\mathbf {u}}}){\hat{\mathbf {u}}})\big )\hat{\mathbf {K}}^{(\beta )}_{b}{\hat{\mathbf {u}}}\bigg ]_{k}\\&=\sum ^{N-1}_{j=0}\,\sum ^{N-1}_{\sigma =0}\,\sum _{b=0,1}\,\sum _{\beta =1,2}\,\sum _{\delta =1,2} \begin{bmatrix} \mathrm {e}^{\mathrm {i}\frac{2\,\pi \,b\,\sigma }{N}}\Big [\mathrm {\partial }^{(\beta )}_{b}\,\kappa ^{(1\,\delta )}_{f_{k\sigma }0}\Big ]^{*}\Big (\overline{\mathrm {\varvec{\psi }}}^{(\beta 1)}_{\sigma \,j}\cdot \mathbf {u}_{f_{\sigma \,j}}\,u^{(1)}_{j}+\overline{\mathrm {\varvec{\psi }}}^{(\beta 2)}_{\sigma \,j}\cdot \mathbf {u}_{f_{\sigma \,j}}\,u^{(2)}_{j}\Big )\,u^{(\delta )}_{f_{k\sigma }}\\ \mathrm {e}^{\mathrm {i}\frac{2\,\pi \,b\,\sigma }{N}}\Big [\mathrm {\partial }^{(\beta )}_{b}\,\kappa ^{(2\,\delta )}_{f_{k\sigma }0}\Big ]^{*}\Big (\overline{\mathrm {\varvec{\psi }}}^{(\beta 1)}_{\sigma \,j}\cdot \mathbf {u}_{f_{\sigma \,j}}\,u^{(1)}_{j}+\overline{\mathrm {\varvec{\psi }}}^{(\beta 2)}_{\sigma \,j}\cdot \mathbf {u}_{f_{\sigma \,j}}\,u^{(2)}_{j}\Big )\,u^{(\delta )}_{f_{k\sigma }} \end{bmatrix}. \end{aligned} \end{aligned}$$

(105)

Similarly, the $k$-th modal component of ${\hat{\mathfrak {L}}}^{(\beta )}\big ({\hat{\mathbf {S}}}_{b}({\hat{\mathbf {u}}})\big )\hat{\mathbf {K}}^{(\beta )}_{b}\hat{\mathrm {\varvec{\Psi }}}({\hat{\mathbf {u}}}){\hat{\mathbf {u}}}$ can be expanded as

$$\begin{aligned} \begin{aligned}&\bigg [\sum _{b=0,1}\sum _{\beta =1,2}{\hat{\mathfrak {L}}}^{(\beta )}\big ({\hat{\mathbf {S}}}_{b}({\hat{\mathbf {u}}})\big )\hat{\mathbf {K}}^{(\beta )}_{b}\hat{\mathrm {\varvec{\Psi }}}({\hat{\mathbf {u}}}){\hat{\mathbf {u}}}\bigg ]_{k}\\&=\sum ^{N-1}_{j=0}\,\sum ^{N-1}_{\ell =0}\,\sum _{b=0,1}\,\sum _{\delta =1,2}\,\sum _{\beta =1,2} \begin{bmatrix} \mathrm {e}^{\mathrm {i}\frac{2\,\pi \,b\,(k-\ell )}{N}}\Big [\mathrm {\partial }^{(\beta )}_{b}\,\kappa ^{(1\,\delta )}_{\ell 0}\Big ]^{*}\Big (\overline{\mathrm {\varvec{\psi }}}^{(\delta 1)}_{\ell \,j}\cdot \mathbf {u}_{f_{\ell \,j}}\,u^{(1)}_{j}+\overline{\mathrm {\varvec{\psi }}}^{(\delta 2)}_{\ell \,j}\cdot \mathbf {u}_{f_{\ell \,j}}\,u^{(2)}_{j}\Big )\,u^{(\beta )}_{f_{k\ell }}\\ \mathrm {e}^{\mathrm {i}\frac{2\,\pi \,b\,(k-\ell )}{N}}\Big [\mathrm {\partial }^{(\beta )}_{b}\,\kappa ^{(2\,\delta )}_{\ell 0}\Big ]^{*}\Big (\overline{\mathrm {\varvec{\psi }}}^{(\delta 1)}_{\ell \,j}\cdot \mathbf {u}_{f_{\ell \,j}}\,u^{(1)}_{j}+\overline{\mathrm {\varvec{\psi }}}^{(\delta 2)}_{\ell \,j}\cdot \mathbf {u}_{f_{\ell \,j}}\,u^{(2)}_{j}\Big )\,u^{(\beta )}_{f_{k\ell }} \end{bmatrix}. \end{aligned} \end{aligned}$$

(106)

If we re-label the $\delta$ and $\beta$, we obtain

$$\begin{aligned} \begin{aligned}&\bigg [\sum _{b=0,1}\sum _{\beta =1,2}{\hat{\mathfrak {L}}}^{(\beta )}\big ({\hat{\mathbf {S}}}_{b}({\hat{\mathbf {u}}})\big )\hat{\mathbf {K}}^{(\beta )}_{b}\hat{\mathrm {\varvec{\Psi }}}({\hat{\mathbf {u}}}){\hat{\mathbf {u}}}\bigg ]_{k}\\&=\sum ^{N-1}_{j=0}\,\sum ^{N-1}_{\ell =0}\,\sum _{b=0,1}\,\sum _{\beta =1,2}\,\sum _{\delta =1,2} \begin{bmatrix} \mathrm {e}^{\mathrm {i}\frac{2\,\pi \,b\,(k-\ell )}{N}}\Big [\mathrm {\partial }^{(\delta )}_{b}\,\kappa ^{(1\,\beta )}_{\ell 0}\Big ]^{*}\Big (\overline{\mathrm {\varvec{\psi }}}^{(\beta 1)}_{\ell \,j}\cdot \mathbf {u}_{f_{\ell \,j}}\,u^{(1)}_{j}+\overline{\mathrm {\varvec{\psi }}}^{(\beta 2)}_{\ell \,j}\cdot \mathbf {u}_{f_{\ell \,j}}\,u^{(2)}_{j}\Big )\,u^{(\delta )}_{f_{k\ell }}\\ \mathrm {e}^{\mathrm {i}\frac{2\,\pi \,b\,(k-\ell )}{N}}\Big [\mathrm {\partial }^{(\delta )}_{b}\,\kappa ^{(2\,\beta )}_{\ell 0}\Big ]^{*}\Big (\overline{\mathrm {\varvec{\psi }}}^{(\beta 1)}_{\ell \,j}\cdot \mathbf {u}_{f_{\ell \,j}}\,u^{(1)}_{j}+\overline{\mathrm {\varvec{\psi }}}^{(\beta 2)}_{\ell \,j}\cdot \mathbf {u}_{f_{\ell \,j}}\,u^{(2)}_{j}\Big )\,u^{(\delta )}_{f_{k\ell }} \end{bmatrix}. \end{aligned} \end{aligned}$$

(107)

Considering (105) and (107) for the same values of $j,\,\beta ,\,$ and $\delta$, and for $\sigma =\ell$, we can show that

$$\begin{aligned} \sum _{b=0,1}\mathrm {e}^{\mathrm {i}\frac{2\,\pi \,b\,\sigma }{N}}\Big [\mathrm {\partial }^{(\beta )}_{b}\,\kappa ^{(1\,\delta )}_{f_{k\sigma }0}\Big ]^{*}=\sum _{b=0,1}\mathrm {e}^{\mathrm {i}\frac{2\,\pi \,b\,(k-\ell )}{N}}\Big [\mathrm {\partial }^{(\delta )}_{b}\,\kappa ^{(1\,\beta )}_{\ell 0}\Big ]^{*}, \end{aligned}$$

(108)

and hence (105) and (107) are equal. This means that (50) and (51) are identical.

Appendix 2: Cubic Near-Identity Coefficients (Left Hand Side)

The coefficient vector ${\tilde{\mathbf {b}}}_{k j\ell }$ in (57) is given by

$$\begin{aligned} {\tilde{\mathbf {b}}}_{k j\ell }\,=\, \begin{bmatrix} \gamma _{kj}^{(111)}\,\phi _{ j\ell }^{(111)}+\gamma _{kj}^{(121)}\,\phi _{ j\ell }^{(211)}\\ \gamma _{kj}^{(111)}\,\phi _{ j\ell }^{(112)}+\gamma _{kj}^{(121)}\,\phi _{ j\ell }^{(212)}\\ \gamma _{kj}^{(112)}\,\phi _{ j\ell }^{(111)}+\gamma _{kj}^{(122)}\,\phi _{ j\ell }^{(211)}\\ \gamma _{kj}^{(112)}\,\phi _{ j\ell }^{(112)}+\gamma _{kj}^{(122)}\,\phi _{ j\ell }^{(212)}\\ \gamma _{kj}^{(111)}\,\phi _{ j\ell }^{(121)}+\gamma _{kj}^{(121)}\,\phi _{ j\ell }^{(221)}\\ \gamma _{kj}^{(111)}\,\phi _{ j\ell }^{(122)}+\gamma _{kj}^{(121)}\,\phi _{ j\ell }^{(222)}\\ \gamma _{kj}^{(112)}\,\phi _{ j\ell }^{(121)}+\gamma _{kj}^{(122)}\,\phi _{ j\ell }^{(221)}\\ \gamma _{kj}^{(112)}\,\phi _{ j\ell }^{(122)}+\gamma _{kj}^{(122)}\,\phi _{ j\ell }^{(222)}\\ \gamma _{kj}^{(211)}\,\phi _{ j\ell }^{(111)}+\gamma _{kj}^{(221)}\,\phi _{ j\ell }^{(211)}\\ \gamma _{kj}^{(211)}\,\phi _{ j\ell }^{(112)}+\gamma _{kj}^{(221)}\,\phi _{ j\ell }^{(212)}\\ \gamma _{kj}^{(212)}\,\phi _{ j\ell }^{(111)}+\gamma _{kj}^{(222)}\,\phi _{ j\ell }^{(211)}\\ \gamma _{kj}^{(212)}\,\phi _{ j\ell }^{(112)}+\gamma _{kj}^{(222)}\,\phi _{ j\ell }^{(212)}\\ \gamma _{kj}^{(211)}\,\phi _{ j\ell }^{(121)}+\gamma _{kj}^{(221)}\,\phi _{ j\ell }^{(221)}\\ \gamma _{kj}^{(211)}\,\phi _{ j\ell }^{(122)}+\gamma _{kj}^{(221)}\,\phi _{ j\ell }^{(222)}\\ \gamma _{kj}^{(212)}\,\phi _{ j\ell }^{(121)}+\gamma _{kj}^{(222)}\,\phi _{ j\ell }^{(221)}\\ \gamma _{kj}^{(212)}\,\phi _{ j\ell }^{(122)}+\gamma _{kj}^{(222)}\,\phi _{ j\ell }^{(222)} \end{bmatrix}, \end{aligned}$$

(109)

where the constants $\gamma _{kj}^{(\cdots )}$ and $\phi _{ j\ell }^{(\cdots )}$ are defined in (53).

Appendix 3: Cubic Near-Identity Coefficients (Right Hand Side)

The coefficient vector ${\tilde{\mathbf {c}}}_{k j\ell }$ in (57) is given by

(110)

Appendix 4: Remaining Quadratic Terms for Mode 0

In Eq. (77), $\frac{1}{2}{\hat{\mathbf {R}}}({\hat{\mathbf {u}}}){\hat{\mathbf {u}}}$ represents the quadratic terms that cannot be eliminated by the near-identity transformation (38) when modes 1 and $N-1$ undergo a Hopf bifurcation. The only two nonzero entries in matrix ${\hat{\mathbf {R}}}({\hat{\mathbf {u}}})$ are given by

$$\begin{aligned}{}[{\hat{\mathbf {R}}}({\hat{\mathbf {u}}})]_{(1,2)}= \begin{bmatrix} 0&0\\ 4\,q_{1}\,\sin ^{2}\Big (\frac{\pi }{N}\Big )\,z^{(1)}_{N-1}&0 \end{bmatrix},\quad [{\hat{\mathbf {R}}}({\hat{\mathbf {u}}})]_{(1,N)}= \begin{bmatrix} 0&0\\ 4\,q_{1}\,\sin ^{2}\Big (\frac{\pi }{N}\Big )\,z^{(1)}_1&0 \end{bmatrix}, \end{aligned}$$

(111)

where subscripts denote the location of the block in ${\hat{\mathbf {R}}}({\hat{\mathbf {u}}})$, and $q_{1}$ gives the value of $q$ at the bifurcation point; cf. (10). Indeed, these quadratic terms correspond to the modes $1$ and $N-1$, and they only appear in the dynamics of the $k=0$ mode. In the special case, when $\beta =0$ and even $N$ the term

$$\begin{aligned}{}[{\hat{\mathbf {R}}}({\hat{\mathbf {u}}})]_{(1,\frac{N}{2}+1)}= \begin{bmatrix} 0&0\\ 4\,q_{1}\,z^{(1)}_{\frac{N}{2}}&0 \end{bmatrix}, \end{aligned}$$

(112)

also remains in the equation of the $k=0$ mode due to mode $N/2$.

Appendix 5: Quadratic Near-Identity Coefficients for the Connected Vehicle Example

Equations (83) and (88) contain four nonzero quadratic near-identity coefficients that can be obtained by solving (57) for the car following model:

$$\begin{aligned} \begin{aligned} \psi ^{(111)}_{21}\,=&\,\frac{q_{1}\,(\mathrm {e}^{\mathrm {i}\frac{2\,\pi }{N}}-1)^2}{2\,\Delta }\,\bigg (-\big (2\,\kappa ^{(21)\;*}_{10}-\kappa ^{(21)\;*}_{20}\big )\,\big (4\,\kappa ^{(21)\;*}_{10}-\kappa ^{(21)\;*}_{20}\big )\\&-2\,\kappa ^{(21)\;*}_{10}\,\big (\kappa ^{(22)\;*}_{10}-\kappa ^{(22)\;*}_{20}\big )\kappa ^{(22)\;*}_{20}+\,\kappa ^{(21)\;*}_{20}\,\kappa ^{(22)\;*}_{10}\,\big (5\,\kappa ^{(22)\;*}_{10}-3\,\kappa ^{(22)\;*}_{20}\big )\\&-2\,\big (\kappa ^{(22)\;*}_{10}\big )^2\,\big (\kappa ^{(22)\;*}_{10}-\kappa ^{(22)\;*}_{20}\big )\,\big (2\,\kappa ^{(22)\;*}_{10}-\kappa ^{(22)\;*}_{20}\big )\bigg ),\\ \psi ^{(112)}_{21}\,+\psi ^{(121)}_{21}\,=&\,\frac{-q_{1}\,(\mathrm {e}^{\mathrm {i}\frac{2\,\pi }{N}}-1)^2}{\Delta }\,\bigg (4\,\kappa ^{(21)\;*}_{10}\,\kappa ^{(22)\;*}_{10}+\,\kappa ^{(21)\;*}_{20}\,\big (\kappa ^{(22)\;*}_{10}-\kappa ^{(22)\;*}_{20}\big )\\&-2\,\kappa ^{(22)\;*}_{10}\,\big (\kappa ^{(22)\;*}_{10}-\kappa ^{(22)\;*}_{20}\big )\,\big (2\kappa ^{(22)\;*}_{10}-\kappa ^{(22)\;*}_{20}\big ) \bigg ),\\ \psi ^{(122)}_{21}\,=&\,\frac{q_{1}\,(\mathrm {e}^{\mathrm {i}\frac{2\,\pi }{N}}-1)^2}{\Delta }\,\bigg (\big (4\,\kappa ^{(21)\;*}_{10}-\,\kappa ^{(21)\;*}_{20}\big )- \big (\kappa ^{(22)\;*}_{10}-\kappa ^{(22)\;*}_{20}\big )\,\big (2\kappa ^{(22)\;*}_{10}-\kappa ^{(22)\;*}_{20}\big ) \bigg ), \end{aligned} \end{aligned}$$

(113)

where the factor $\Delta$ is

$$\begin{aligned} \begin{aligned} \Delta \,=&\,\kappa ^{(21)\;*}_{20}\big (4\,\kappa ^{(21)\;*}_{10}-\kappa ^{(21)\;*}_{20}\big )^2-16\,\kappa ^{(22)\;*}_{10}\,\big (\kappa ^{(22)\;*}_{10}-\kappa ^{(22)\;*}_{20}\big )\,\big (\kappa ^{(21)\;*}_{10}\big )^2\\&+4\,\big (2\,(\kappa ^{(22)\;*}_{10})^2-(\kappa ^{(22)\;*}_{20})^2\big )\,\kappa ^{(21)\;*}_{10}\,\kappa ^{(21)\;*}_{20}-\kappa ^{(22)\;*}_{10}\,\big (5\kappa ^{(22)\;*}_{10}-3\kappa ^{(22)\;*}_{20}\big )\,\big (\kappa ^{(22)\;*}_{20}\big )^2\\&+2\,\kappa ^{(22)\;*}_{10}\,\big (\kappa ^{(22)\;*}_{10}-\kappa ^{(22)\;*}_{20}\big )\,\big (2\,\kappa ^{(22)\;*}_{10}-\kappa ^{(22)\;*}_{20}\big )\,\big (-2\,\kappa ^{(21)\;*}_{10}\,\kappa ^{(22)\;*}_{20}+\,\kappa ^{(21)\;*}_{20}\,\kappa ^{(22)\;*}_{10}\big ), \end{aligned} \end{aligned}$$

(114)

and $q_{1}$ gives the value of $q$ at the bifurcation point; cf. (10). We remark that the coefficients $\psi ^{(112)}_{21}$ and $\psi ^{(121)}_{21}$ do not need to be evaluated individually since (83, 88) only include their sum.