Fast computation of steady-state response for high-degree-of-freedom nonlinear systems

Abstract

We discuss an integral equation approach that enables fast computation of the response of nonlinear multi-degree-of-freedom mechanical systems under periodic and quasi-periodic external excitation. The kernel of this integral equation is a Green’s function that we compute explicitly for general mechanical systems. We derive conditions under which the integral equation can be solved by a simple and fast Picard iteration even for non-smooth mechanical systems. The convergence of this iteration cannot be guaranteed for near-resonant forcing, for which we employ a Newton– Raphson iteration instead, obtaining robust convergence. We further show that this integral equation approach can be appended with standard continuation schemes to achieve an additional, significant performance increase over common approaches to computing steady-state response.

Appendices

Proof of Lemma 1

The general solution of (6) is given by the classic variation of constants formula

$$\begin{aligned} {\mathbf {w}}(t)=e^{\varvec{\Lambda }t}{\mathbf {w}}(0)+\int _{0}^{t}e^{\varvec{\Lambda }(t-s)}\varvec{\psi }(s)\,\mathrm{d}s. \end{aligned}$$

(65)

A T-periodic solution ${\mathbf {w}}_{0}(t)$ of (6) must satisfy (65), resulting in

$$\begin{aligned} {\mathbf {w}}_{0}(T)=e^{\varvec{\Lambda }T}{\mathbf {w}}_{0}(0)+\int _{0}^{T}e^{\varvec{\Lambda }(T-s)}\varvec{\psi }(s)\,\mathrm{d}s={\mathbf {w}}_{0}(0). \end{aligned}$$

(66)

Since the matrix $\left[ {\mathbf {I}}-e^{\varvec{\Lambda }T}\right]$ is invertible due to the non-resonance condition (7), this allows us to solve Eq. (66) for a unique initial condition ${\mathbf {w}}_0(0)$ as (cf. Burd [40], Chapter 2)

$$\begin{aligned} {\mathbf {w}}_{0}(0)=\left[ {\mathbf {I}}-e^{\varvec{\Lambda }T}\right] ^{-1}\int _{0}^{T}e^{\varvec{\Lambda }(T-s)}\varvec{\psi }(s)\,\mathrm{d}s. \end{aligned}$$

(67)

Substituting the unique initial condition from (67) into the general solution (65) provides us an explicit expression of the unique T-periodic solution ${\mathbf {w}}_{0}(t)$ to system (6) as

$$\begin{aligned} {\mathbf {w}}_{0}(t)&=e^{\varvec{\Lambda }t}\left[ {\mathbf {I}} -e^{\varvec{\Lambda }T}\right] ^{-1}\int _{0}^{T}e^{\varvec{\Lambda }\ (T-s)}\varvec{\psi }(s)\,\mathrm{d}s\nonumber \\&\quad +\,\int _{0}^{t}e^{\varvec{\Lambda }(t-s)}\varvec{\psi }(s)\,ds\nonumber \\&=e^{\varvec{\Lambda }T}\left[ {\mathbf {I}}-e^{\varvec{\Lambda }T} \right] ^{-1}\int _{0}^{T}e^{\varvec{\Lambda }(t-s)} \varvec{\psi }(s)\,\mathrm{d}s\nonumber \\&\quad +\,\int _{0}^{t}e^{\varvec{\Lambda }(t-s)}\varvec{\psi }(s)\,ds\nonumber \\&=e^{\varvec{\Lambda }T}\left[ {\mathbf {I}} -e^{\varvec{\Lambda }T}\right] ^{-1}\int _{0}^{t}e^{\varvec{\Lambda } (t-s)}\varvec{\psi }(s)\,\mathrm{ds}\nonumber \\&\quad +\,e^{\varvec{\Lambda }T} \left[ {\mathbf {I}}-e^{\varvec{\Lambda }T}\right] ^{-1}\int _{t}^{T} e^{\varvec{\Lambda }(t-s)}\varvec{\psi }(s)\,\mathrm{d}s\nonumber \\&\quad +\,\int _{0}^{t}e^{\varvec{\Lambda }(t-s)}\varvec{\psi }(s)\,ds\nonumber \\&=\int _{0}^{t}\left[ e^{\varvec{\Lambda }T}\left[ {\mathbf {I}} -e^{\varvec{\Lambda }T}\right] ^{-1}+{\mathbf {I}}\right] e^{\varvec{\Lambda }(t-s)}\varvec{\psi }(s)\,\mathrm{d}s\nonumber \\&\quad +\,\int _{t}^{T}e^{\varvec{\Lambda }T}\left[ {\mathbf {I}}-e^{\varvec{\Lambda }T}\right] ^{-1}e^{\varvec{\Lambda }(t-s)}\varvec{\psi }(s)\,\mathrm{d}s\nonumber \\&=\int _{0}^{T}\underbrace{\left[ e^{\varvec{\Lambda }T} \left[ {\mathbf {I}}-e^{\varvec{\Lambda }T}\right] ^{-1}+h(t-s) {\mathbf {I}}\right] e^{\varvec{\Lambda }(t-s)}}_{{\mathbf {G}}(t-s,T)} \varvec{\psi }(s)\,\mathrm{d}s\,, \end{aligned}$$

(68)

where ${\mathbf {G}}(t,T)$ is a diagonal matrix with the entries given by (9). Using the linear modal transformation ${\mathbf {z}}=\mathbf {Vw}$, we find the unique T-periodic solution to system (4) in the form

$$\begin{aligned} {\mathbf {z}}(t)={\mathbf {V}}\int _{0}^{T}{\mathbf {G}}(t-s,T)\varvec{\psi }(s)\,\mathrm{d}s. \end{aligned}$$

Proof of Theorem 1

If ${\mathbf {z}}(t)$ is a T-periodic solution of (3), then it satisfies the linear inhomogeneous differential equation

$$\begin{aligned} {\mathbf {B}}\dot{{\mathbf {z}}}={\mathbf {A}}{\mathbf {z}}+{\mathbf {F}}(t)-{\mathbf {R}}({\mathbf {z}}(t)), \end{aligned}$$

where we view ${\mathbf {F}}(t)-{\mathbf {R}}({\mathbf {z}}(t))$ as a $T-$periodic forcing term. Thus, according to Lemma 1, we have

$$\begin{aligned} {\mathbf {z}}(t)={\mathbf {V}}\int _{0}^{T}{\mathbf {G}}(t-s,T){\mathbf {V}}^{-1}\left[ {\mathbf {F}}(s)-{\mathbf {R}}({\mathbf {z}}(s))\right] \,\mathrm{d}s, \end{aligned}$$

as claimed in statement (i).

Now, let ${\mathbf {z}}(t)$ be a continuous, $T-$periodic solution to (11). After introducing the notation $\varvec{\chi }(t)={\mathbf {V}}^{-1}\left[ {\mathbf {F}}(t)-{\mathbf {R}}({\mathbf {z}}(t))\right]$, we have

$$\begin{aligned} {\mathbf {z}}(t)&={\mathbf {V}}\int _{0}^{T}{\mathbf {G}}(t-s,T)\varvec{\chi }(s)\,\mathrm{d}s,\nonumber \\&={\mathbf {V}}e^{\varvec{\Lambda }t}\left( {\mathbf {I}} -e^{\varvec{\Lambda }T}\right) ^{-1}\int _{0}^{T}e^{\varvec{\Lambda } (T-s)}\varvec{\chi }(s)\,\mathrm{d}s\nonumber \\&\quad +\,{\mathbf {V}}\int _{0}^{t}e^{\varvec{\Lambda }(t-s)}\varvec{\chi }(s)\,\mathrm{d}s\,, \end{aligned}$$

(69)

where (69) is a direct consequence of (68). By the continuity of ${\mathbf {z}}(t)$, $\varvec{\chi }(t)$ is also at least $C^{0}$ (${\mathbf {F}}$ is at least $C^{0}$ and ${\mathbf {R}}$ is Lipschitz). Thus, for any $t\in [0,T]$, the right-hand side of (69) can be differentiated with respect to t according to the Leibniz rule, to obtain

$$\begin{aligned} \frac{d{\mathbf {z}}(t)}{dt}&={\mathbf {V}}\varvec{\Lambda }e^{\varvec{\Lambda }t} \left( {\mathbf {I}}-e^{\varvec{\Lambda }T}\right) ^{-1}\int _{0}^{T} e^{\varvec{\Lambda }(T-s)}\varvec{\chi }(s)\,ds\\&\quad +\,{\mathbf {V}}\varvec{\Lambda }\int _{0}^{t}e^{\varvec{\Lambda }(t-s)}\varvec{\chi }(s)\,ds\,+{\mathbf {V}}\varvec{\Lambda }\varvec{\chi }(t)\\&={\mathbf {V}}\varvec{\Lambda }{\mathbf {V}}^{-1}{\mathbf {z}}(t)\\&\quad +\,{\mathbf {V}}\varvec{\Lambda }\varvec{\chi }(t)\qquad \text {[using (69)]}\\&={\mathbf {V}}\varvec{\Lambda }{\mathbf {V}}^{-1}{\mathbf {z}}(t)+{\mathbf {V}}\varvec{\Lambda }{\mathbf {V}}^{-1}\left[ {\mathbf {F}}(t)-{\mathbf {R}}({\mathbf {z}}(t))\right] , \end{aligned}$$

which implies

$$\begin{aligned} {\mathbf {B}}\frac{d{\mathbf {z}}(t)}{dt}={\mathbf {A}}{\mathbf {z}}(t)+{\mathbf {F}}(t)-{\mathbf {R}}({\mathbf {z}}(t)), \end{aligned}$$

as claimed in statement (ii).

Proof of Lemma 2

By the linearity of (4), one can verify that the sum of periodic solutions given by Lemma 1 for each periodic forcing summand in (12) is the unique, bounded solution of (4). In case of a forcing written as a Fourier series, we can carry out the integration appearing in Lemma 1 for each summand in this bounded solution explicitly in diagonalized coordinates ${\mathbf {z}}={\mathbf {V}}{\mathbf {w}}$. With the notation $\varvec{\psi _{\varvec{\kappa }}}={\mathbf {V}}^{-1}{\mathbf {F}}_{\varvec{\kappa }}$, we then obtain for the jth degree of freedom:

$$\begin{aligned} w_{j}(t)&=\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}\int _{0}^{T_{\varvec{\kappa }}}G_{j}(t-s,T_{\varvec{\kappa }})\psi _{\varvec{\kappa },j}e^{i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle s}\,\mathrm{d}s\\&=\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}\int _{0}^{T_{\varvec{\kappa }}}e^{\lambda _{j}t} \left( \frac{e^{\lambda _{j}T_{\varvec{\kappa }}}}{1-e^{\lambda _{j}T_{\varvec{\kappa }}}} +h(t)\right) \psi _{\varvec{\kappa },j}e^{i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle s}\,\mathrm{d}s\\&=\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}\int _{0}^{t}e^{\lambda _{j} (t-s)}\left( \frac{1}{1-e^{\lambda _{j}T_{\varvec{\kappa }}}}\right) \psi _{\varvec{\kappa },j}e^{i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle s}\,\mathrm{d}s\\&\quad +\,\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}\int _{t-s} ^{T_{\varvec{\kappa }}}e^{\lambda _{j}(t-s)} \left( \frac{e^{\lambda _{j}T_{\varvec{\kappa }}}}{1-e^{\lambda _{j}T_{\varvec{\kappa }}}}\right) \psi _{\varvec{\kappa },j}e^{i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle s}\,\mathrm{d}s\\&=\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}} e^{\lambda _{j}t}\left( \frac{1}{1-e^{\lambda _{j}T_{\varvec{\kappa }}}} \right) \psi _{\varvec{\kappa },j}\int _{0}^{t}e^{-\lambda _{j}s} e^{i\left\langle \varvec{\kappa },\varvec{\Omega } \right\rangle s}\,\mathrm{d}s\\&\quad +\,\sum _{\varvec{\kappa } \in {\mathbb {Z}}^{k}}e^{\lambda _{j}t} \left( \frac{e^{\lambda _{j}T_{\varvec{\kappa }}}}{1-e^{\lambda _{j}T_{\varvec{\kappa }}}}\right) \psi _{\varvec{\kappa },j}\int _{t}^{T_{\varvec{\kappa }}} e^{-\lambda _{j}s}e^{i\left\langle \varvec{\kappa }, \varvec{\Omega }\right\rangle s}\,\mathrm{d}s\\&=\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}e^{\lambda _{j}t} \left( \frac{1}{1-e^{\lambda _{j}T_{\varvec{\kappa }}}} \right) \psi _{\varvec{\kappa },j}\frac{1}{i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\lambda _{j}}\\&\quad \left. e^{(i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\lambda _{j})s}\right| _{s=0}^{s=t} \\&\quad +\,\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}e^{\lambda _{j}t} \left( \frac{e^{\lambda _{j}T_{\varvec{\kappa }}}}{1-e^{\lambda _{j}T_{\varvec{\kappa }}}}\right) \psi _{\varvec{\kappa },j}\frac{1}{i\left\langle \varvec{\kappa },\ \varvec{\Omega }\right\rangle -\lambda _{j}}\\&\quad \left. e^{(i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\lambda _{j})s}\right| _{s=t}^{s=T_{\varvec{\kappa }}}\\&=\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}e^{\lambda _{j}t} \left( \frac{1}{1-e^{\lambda _{j}T_{\varvec{\kappa }}}}\right) \psi _{\varvec{\kappa },j}\frac{1}{i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\lambda _{j}}\\&\quad \left[ e^{(i\left\langle \varvec{\kappa }, \varvec{\Omega }\right\rangle -\lambda _{j})t}-1+\,e^{\lambda _{j}T_{\varvec{\kappa }}} (e^{(i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\lambda _{j})T_{\varvec{\kappa }}} \right. \\&\quad \left. -\,\,e^{(i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\lambda _{j})t})\right] \\&=\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}e^{\lambda _{j}t} \psi _{\varvec{\kappa },j}\frac{1}{i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\lambda _{j}}\frac{1}{1-e^{\lambda _{j}T_{\varvec{\kappa }}}}\\&\quad \left[ (1-e^{\lambda _{j}T_{\varvec{\kappa }}})(e^{(i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\lambda _{j})t})-1+e^{i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle T_{\varvec{\kappa }}}\right] \\&=\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}\frac{1}{i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\lambda _{j}}\psi _{\varvec{\kappa },j}e^{i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle t}\,. \end{aligned}$$

Explicit Green’s function for mechanical systems: Proof of Lemma 3

The first-order ODE formulation for (26) is given by

$$\begin{aligned} \frac{d}{dt}\left( \begin{array}{c} y_{j}\\ \dot{y}_{j} \end{array}\right)&=\underbrace{\left( \begin{array}{ll} 0 &{} 1\\ -\omega _{0,j}^{2} &{} -2\zeta _{j}\omega _{0,j} \end{array}\right) }_{{\mathbf {A}}_{j}}\left( \begin{array}{l} y_{j}\\ \dot{y}_{j} \end{array}\right) +\left( \begin{array}{c} 0\\ \varphi _{j}(t) \end{array}\right) ,\nonumber \\&\quad j=1,\ldots ,n. \end{aligned}$$

(70)

By the classic variation of constants formula for first-order systems of ordinary differential equations, the general solution of (70) is of the form

$$\begin{aligned}&\left( \begin{array}{c} y_{j}(t)\\ \dot{y}_{j}(t) \end{array}\right) ={\mathbf {N}}^{(j)}(t)\left( \begin{array}{c} y_{j}(0)\\ \dot{y}_{j}(0) \end{array}\right) \nonumber \\&\quad +\int _{0}^{t}{\mathbf {N}}^{(j)}(t-s)\left( \begin{array}{c} 0\\ \varphi _{j}(s) \end{array}\right) \mathrm{d}s,\qquad j=1,\ldots ,n, \end{aligned}$$

(71)

with ${\mathbf {N}}(t)=e^{{\mathbf {A}}_{j}t}$ denoting the fundamental matrix solution for the jth mode with ${\mathbf {N}}(0)={\mathbf {I}}.$ Thus, the homogeneous (unforced) version of (26), the explicit solution can be obtained as

$$\begin{aligned} \left( \begin{array}{c} y_{j}(t)\\ \dot{y}_{j}(t) \end{array}\right) ={\mathbf {N}}^{(j)}(T)\left( \begin{array}{c} y_{j}(0)\\ \dot{y}_{j}(0) \end{array}\right) ,\qquad j=1,\ldots ,n. \end{aligned}$$

(72)

Since ${\mathbf {F}}(t)$ is uniformly bounded for all times and all ${\mathbf {A}}_{j}$ matrices are hyperbolic ($\zeta _{j}>0$ for $j=1,\ldots ,n$), then a unique uniformly bounded solution exists for the 2n-dimensional system of linear ordinary differential equations (ODEs) (70) (see, e.g., Burd [40]). The initial condition $\left( y_{j}(0),\dot{y}_{j}(0)\right)$ for the unique T-periodic solution of (71) is obtained by imposing periodicity, i.e., $y_{j}(0)=y_{j}(T)$ for $j=1,\dots ,n$ and is given by

$$\begin{aligned} \left( \begin{array}{c} y_{j}(0)\\ y_{j}(0) \end{array}\right) =\frac{1}{1-\text {Trace}\left( {\mathbf {N}}^{(j)}(T)\right) +\text {det}\left( {\mathbf {N}}^{(j)}(T)\right) }\int _{0}^{T} \left( \begin{array}{c} \left( 1-N_{22}^{(j)}(T)\right) N_{12}^{(j)}(T-s)+N_{12}^{(j)}(T)N_{22}^{(j)}(T-s)\\ \left( 1-N_{11}^{(j)}(T)\right) N_{22}^{(j)}(T-s)+N_{21}^{(j)}(T)N_{12}^{(j)}(T-s) \end{array}\right) \quad \varphi _{j}(s)\,\mathrm{d}s.\nonumber \\ \end{aligned}$$

(73)

Finally, the unique periodic response $\left( y_{j}(t),\dot{y}_{j}(t)\right)$ is obtained by substituting the initial condition (73) into the Duhamel’s integral formula (71) as

$$\begin{aligned} \left( \begin{array}{c} y_{j}(t)\\ \dot{y}_{j}(t) \end{array}\right)&=\frac{{\mathbf {N}}^{(j)}(t)}{1-\text {Trace}\left( {\mathbf {N}}^{(j)}(T)\right) +\text {det}\left( {\mathbf {N}}^{(j)}(T)\right) }\int _{0}^{T}\left( \begin{array}{c} \left( 1-N_{22}^{(j)}(T)\right) N_{12}^{(j)}(T-s)+N_{12}^{(j)}(T)N_{22}^{(j)}(T-s)\\ \left( 1-N_{11}^{(j)}(T)\right) N_{22}^{(j)}(T-s)+N_{21}^{(j)}(T)N_{12}^{(j)}(T-s) \end{array}\right) \qquad \varphi _{j}(s)\,\mathrm{d}s\nonumber \\&\quad +\int _{0}^{t}{\mathbf {N}}^{(j)}(t-s)\left( \begin{array}{c} 0\\ \varphi _{j}(s) \end{array}\right) \mathrm{d}s,\qquad j=1,\ldots ,n\,. \end{aligned}$$

(74)

With the notation introduced in (28), i.e.,

$$\begin{aligned}&\alpha _{j}:=\text {Re}(\lambda _{2j}),\quad \omega _{j}:=|\text {Im}(\lambda _{2j})|,\quad \beta _{j}:=\alpha _{j}+\omega _{j},\\&\gamma _{j}:=\alpha _{j}-\omega _{j,}\quad j=1,\dots ,n, \end{aligned}$$

the specific expressions for the fundamental matrix of solutions of (70) in the underdamped, the critically damped and overdamped cases are given by

$$\begin{aligned} {\mathbf {N}}^{(j)}(t)={\left\{ \begin{array}{ll} \frac{e^{\alpha _{j}t}}{\omega _{j}}\left( \begin{array}{cc} \omega _{j}\cos \omega _{j}t-\alpha _{j}\sin \omega _{j}t &{} \sin \omega _{j}t\\ -\alpha _{j}^{2}\sin \omega _{j}t-\omega _{j}^{2}\sin \omega _{j}t &{} \omega _{j}\cos \omega _{j}t+\alpha _{j}\sin \omega _{j}t \end{array}\right) , &{} \zeta _{j}<1\\ \left( \begin{array}{cc} e^{\alpha _{j}t}-\alpha _{j}te^{\alpha _{j}t} &{} te^{\alpha _{j}t}\\ -\alpha _{j}^{2}te^{\alpha _{j}t} &{} e^{\alpha _{j}t}+\alpha _{j}te^{\alpha _{j}t} \end{array}\right) \,, &{} \zeta _{j}=1\\ \frac{1}{\beta _{j}-\gamma _{j}}\left( \begin{array}{cc} \beta _{j}e^{\gamma _{j}t}-\alpha _{j}e^{\beta _{j}t} &{} e^{\beta _{j}t}-e^{\gamma _{j}t}\\ \gamma _{j}\beta _{j}\left( e^{\gamma _{j}t}-e^{\beta _{j}t}\right) &{} \beta _{j}e^{\beta _{j}t}-\gamma _{j}e^{\gamma _{j}t} \end{array}\right) \,,&\zeta _{j}>1 \end{array}\right. },\quad j=1,\dots ,n\,. \end{aligned}$$

(75)

Furthermore, we have

$$\begin{aligned} {\mathrm {Trace}}\,{\mathbf {N}}^{(j)}(t)&={\left\{ \begin{array}{ll} 2e^{\alpha _{j}t}\cos \omega _{j}t\,, &{} \zeta _{j}<1\\ 2e^{\beta _{j}t}\,, &{} \zeta _{j}=1\\ e^{\beta _{j}t}+e^{\gamma _{j}t}\,, &{} \zeta _{j}>1 \end{array}\right. },\quad j=1,\dots ,n\,.\nonumber \\ \det {\mathbf {N}}^{(j)}(t)&={\left\{ \begin{array}{ll} e^{2\alpha _{j}t}\,, &{} \zeta _{j}<1\\ e^{2\beta _{j}t}\,, &{} \zeta _{j}=1\\ e^{\left( \beta _{j}+\gamma _{j}\right) t}\,, &{} \zeta _{j}>1 \end{array}\right. },\quad j=1,\dots ,n\,. \end{aligned}$$

(76)

Thus, we can explicitly compute the particular periodic solution given in (74) using (75) as

$$\begin{aligned}&{\mathbf {y}}(t)=\int _{0}^{T}{\mathbf {L}}(t-s,T)\varvec{\varphi }(s) \,\mathrm{d}s,\quad {\mathbf {L}}(t-s,T)\\&\quad ={\mathrm {diag}}\left( L_{1}(t-s,T), \ldots ,L_{n}(t-s,T)\right) \in {\mathbb {R}}^{n\times n},\\&\dot{{\mathbf {y}}}(t)=\int _{0}^{T}{\mathbf {J}}(t-s,T) \varvec{\varphi }(s)\,\mathrm{d}s,\quad {\mathbf {J}}(t-s,T)\\&\quad ={\mathrm {diag}}\left( J_{1}(t-s,T),\ldots ,J_{n}(t-s,T)\right) \in {\mathbb {R}}^{n\times n}, \end{aligned}$$

with the diagonal elements of the Green’s function matrices ${\mathbf {L}},{\mathbf {J}}$ defined in (30) and (32), i.e.,

$$\begin{aligned}&L_{j}(t,T)={\left\{ \begin{array}{ll} \frac{e^{\alpha _{j}t}}{\omega _{j}}\left[ \frac{e^{\alpha _{j}T}\left[ \sin \omega _{j}(T+t)-e^{\alpha _{j}T}\sin \omega _{j}t\right] }{1+e^{2\alpha _{j}T}-2e^{\alpha _{j}T}\cos \omega _{j}T}+h(t)\sin \omega _{j}t\right] , &{} \zeta _{j}<1\\ \frac{e^{\alpha _{j}(T+t)}\left[ \left( 1-e^{\alpha _{j}T}\right) t+T\right] }{\left( 1-e^{\alpha _{j}T}\right) ^{2}}+h(t)te^{\alpha _{j}t}\,, &{} \zeta _{j}=1\\ \frac{1}{(\beta _{j}-\gamma _{j})}\left[ \frac{e^{\beta _{j}(T+t)}\left( 1-e^{\gamma _{j}T}\right) -e^{\gamma _{j}(T+t)}\left( 1-e^{\beta _{j}T}\right) }{1-e^{\gamma _{j}T}-e^{\beta _{j}T}+e^{\left( \gamma _{j}+\beta _{j}\right) T}}+h(t)\left( e^{\beta _{j}t}-e^{\gamma _{j}t}\right) \right] , &{} \zeta _{j}>1 \end{array}\right. },\\&\begin{aligned}J_{j}(t,T)&={\left\{ \begin{array}{ll} \begin{array}{c} \frac{e^{\alpha _{j}t}}{\omega _{j}}\left[ \frac{e^{\alpha _{j}T}\left[ \omega _{j}\left( \cos \omega _{j}(T+t)-e^{\alpha _{j}T}\cos \omega _{j}t\right) +\alpha _{j}\left( \sin \omega _{j}(T+t)-e^{\alpha _{j}T}\sin \omega _{j}t\right) \right] }{1+e^{2\alpha _{j}T}-2e^{\alpha _{j}T}\cos \omega _{j}T}+\right. \\ \left. h(t)\left( \frac{1}{\omega _{j}}\cos \omega _{j}t+\alpha _{j}\sin \omega _{j}t\right) \right] \end{array}\quad , &{} \zeta _{j}<1\\ \frac{e^{\alpha _{j}(T+t)}\left[ \left( 1-e^{\alpha _{j}T}\right) \left( 1+\alpha _{j}t\right) +\alpha _{j}T\right] }{\left( 1-e^{\alpha _{j}T}\right) ^{2}}+h(t)\left( e^{\alpha _{j}t}+\alpha _{j}te^{\alpha _{j}t}\right) , &{} \zeta _{j}=1, \quad j=1,\dots ,n.\\ \frac{1}{(\beta _{j}-\gamma _{j})}\left[ \frac{\beta _{j}e^{\beta _{j}(T+t)}\left( 1-e^{\gamma _{j}T}\right) -\gamma _{j}e_{j}^{\gamma _{j}(T+t)}\left( 1-e^{\beta _{j}T}\right) }{1-e^{\gamma _{j}T}-e^{\beta _{j}T}+e^{\left( \gamma _{j}+\beta _{j}\right) T}}+h(t)\left( \beta _{j}e^{\beta _{j}t}-\gamma _{j}e^{\gamma _{j}t}\right) \right] , &{} \zeta _{j}>1 \end{array}\right. }\end{aligned} \end{aligned}$$

Finally, the linear periodic response ${\mathbf {x}}_{P}(t)$ in the original system coordinates can then obtained by the linear transformation ${\mathbf {x}}_{P}(t)={\mathbf {U}}{\mathbf {y}}(t)$ as

$$\begin{aligned} {\mathbf {x}}_{P}(t)={\mathbf {U}}\int _{0}^{T}{\mathbf {L}}(t-s,T){\mathbf {U}}^{\top }{\mathbf {f}}(s)\,\mathrm{d}s. \end{aligned}$$

Derivative of Green’s function with respect to T

The derivative with respect to the time period T of the first-order periodic Green’s function ${\mathbf {G}}$ given in (9) is simply given by

$$\begin{aligned} \frac{\partial G_{j}}{\partial T}(t,T)= \lambda _je^{\lambda _{j}t}\frac{e^{\lambda _{j}T}}{(1-e^{\lambda _{j}T})^2},\quad j=1,\dots ,2n\,. \end{aligned}$$

(77)

We also provide the derivative of the Green’s function ${\mathbf {L}}$ with respect to T to ease the computation of the Jacobian of the zero function in during numerical continuation. This is obtained by simply differentiating (30) with respect to T. We use a symbolic toolbox for this procedure:

$$\begin{aligned} \frac{\mathrm{d}L_{j}}{\mathrm{d}T}(t,T)={\left\{ \begin{array}{ll} \begin{array}{c} \frac{e^{\alpha _{j}(t+T)}}{\omega _{j}\left( 1+e^{2\alpha _{j}T}-2e^{\alpha _{j}T}\cos \omega _{j}T\right) ^{2}}\left[ \omega _{j}\cos \omega _{j}(T+t)+\alpha _{j}\sin \omega _{j}(T+t)\right. -\\ \left. 2e^{\alpha _{j}T}\left( \omega _{j}\cos \omega _{j}t+\alpha _{j}\sin \omega _{j}t\right) +e^{2\alpha _{j}T}\left( \alpha _{j}\sin \omega _{j}(t-T)+\omega _{j}\cos \omega _{j}(t-T)\right) \right] \end{array}, &{} \zeta _{j}<1\\ \frac{\alpha _{j}e^{\alpha _{j}(t+T)}}{(e^{\alpha _{j}T}-1)^{2}}\left[ t+T-2te^{\alpha _{j}T}+1-\frac{2e^{\alpha _{j}T}(T-t(e^{\alpha _{j}T}-1))}{(e^{\alpha _{j}T}-1)}\right] &{} \zeta _{j}=1, \quad j=1,\dots ,n. \\ \begin{array}{c} \frac{1}{(\beta _{j}-\gamma _{j})}\left[ \frac{\left( \beta _{j}-\left( \gamma _{j}+\beta _{j}\right) e^{\gamma _{j}T}\right) e^{\beta _{j}(T+t)}+\left( \left( \gamma _{j}+\beta _{j}\right) e^{\beta _{j}T}-\gamma _{j}\right) e^{\gamma _{j}(T+t)}}{1-e^{\gamma _{j}T}-e^{\beta _{j}T}+e^{\left( \gamma _{j}+\beta _{j}\right) T}}+\right. \\ \left. \frac{\left( e^{\beta _{j}(T+t)}\left( 1-e^{\gamma _{j}T}\right) -e^{\gamma _{j}(T+t)}\left( 1-e^{\beta _{j}T}\right) \right) \left( \gamma _{j}e^{\gamma _{j}T}+\beta _{j}e^{\beta _{j}T}-\left( \gamma _{j}+\beta _{j}\right) e^{\left( \gamma _{j}+\beta _{j}\right) T}\right) }{\left( 1-e^{\gamma _{j}T}-e^{\beta _{j}T}+e^{\left( \gamma _{j}+\beta _{j}\right) T}\right) ^{2}}\right] \end{array}\,,&\zeta _{j}>1 \end{array}\right. } \end{aligned}$$

Proof of Remark 2

We derive an estimate for the sup norm of the integral of the operator norm of the Green’s function, i.e., for $\int _{0}^{T}\left\| G_{j}(t-s,T)\right\| \,\mathrm{d}s$ defined in equation (9). For $t>s$, we start by noting that

$$\begin{aligned} \left| G_{j}(t-s,T)\right|&=\left| e^{\lambda _{j}(t-s)}\left( \frac{1}{1-e^{\lambda _{j}T}}\right) \right| \\&\le \left| e^{\lambda _{j}(t-s)}\right| \frac{1}{\left| 1-e^{\lambda _{j}T}\right| }\\&\le \frac{\max (\left| e^{\lambda _{j}t}\right| ,1)}{\left| 1-e^{\lambda _{j}T}\right| },\quad 0\le s\le t<T. \end{aligned}$$

For the case $T>s>t$, we obtain

$$\begin{aligned}&\left| G_{j}(t-s,T)\right| \\&\quad =\left| e^{\lambda _{j}(t-s)} \left( \frac{e^{\lambda _{j}T}}{1-e^{\lambda _{j}T}}\right) \right| \\&\quad \le \max \left( \left| \left( \frac{e^{\lambda _{j}T}}{1-e^{\lambda _{j}T}}\right) \right| ,\left| e^{\lambda _{j}(t-T)}\left( \frac{e^{\lambda _{j}T}}{1-e^{\lambda _{j}T}}\right) \right| \right) \\&\quad \le \frac{\max (\left| e^{\lambda _{j}t}\right| ,1)}{\left| 1-e^{\lambda _{j}T}\right| },\quad 0\le s\le t<T. \end{aligned}$$

The upper bounds on the Green’s function in the two intervals are equal and we therefore obtain

Proof of Theorem 5

In the following, we derive conditions under which the mapping ${\varvec{\mathcal {H}}}$ defined in equation (37) is a contraction mapping. We rewrite (37) as

$$\begin{aligned}&{\mathbf {z}}(t)=\varUpsilon _{P}({\mathbf {F}}(t)-{\mathbf {R}} ({\mathbf {z}}(t)))\\&\quad :=\int _{0}^{T}{\mathbf {V}}{\mathbf {G}} (t-s,T){\mathbf {V}}^{-1}\left[ {\mathbf {F}}(s) -{\mathbf {R}}({\mathbf {z}}(s))\right] \,\mathrm{d}s, \end{aligned}$$

where $\varUpsilon _{P}$ is a linear map representing the convolution operation with the Green’s function. Specifically, we define the space of n-dimensional periodic T-periodic functions as

$$\begin{aligned} {\mathcal {P}}_{n}:=\{{\mathbf {p}}:{\mathbb {R}}\rightarrow {\mathbb {R}}^{n},{\mathbf {p}}\in C^{0},{\mathbf {p}}(t)={\mathbf {p}}(t+T)\forall t\in {\mathbb {R}}\}. \end{aligned}$$

(78)

Under the non-resonance condition (7), the linear map

$$\begin{aligned} \varUpsilon _{P}:{\mathcal {P}}_{2n}\rightarrow {\mathcal {P}}_{2n},\,\varUpsilon _{P}{\mathbf {p}}={\mathbf {V}}\int _{0}^{T}{\mathbf {G}}(t-s,T){\mathbf {V}}^{-1}{\mathbf {p}}(s)\,\mathrm{d}s \end{aligned}$$

is well defined, i.e., $\varUpsilon _{P}$ maps T-periodic functions into T-periodic functions. Indeed, for any ${\mathbf {p}}\in {\mathcal {P}}_{2n}$, let ${\mathbf {q}}=\varUpsilon _{P}{\mathbf {p}}$. We have

$$\begin{aligned} {\mathbf {q}}(t)&={\mathbf {V}}\int _{0}^{T}{\mathbf {G}}(t-s,T){\mathbf {V}}^{-1} {\mathbf {p}}(s)\,\mathrm{d}s\\&={\mathbf {V}} \int _{0}^{T}{\mathbf {G}}(t+T-(s+T),T){\mathbf {V}}^{-1}{\mathbf {p}}(s)\,\mathrm{d}s\\&={\mathbf {V}}\int _{0}^{T}{\mathbf {G}}(t+T-(s+T),T) {\mathbf {V}}^{-1}{\mathbf {p}}(s+T)\,\mathrm{d}s\\&={\mathbf {V}}\int _{T}^{2T}{\mathbf {G}}(t+T-\sigma ,T){\mathbf {V}}^{-1}{\mathbf {p}}(\sigma )\,\mathrm{d}\sigma \\&={\mathbf {q}}(t+T)\,, \end{aligned}$$

i.e., ${\mathbf {q}}\in {\mathcal {P}}_{2n}$.

Since the space (41) consists of periodic functions, we know that it is well defined in the space $C_{\delta }^{{\mathbf {z}}_{0}}[0,T]$. Therefore, by the Banach fixed point theorem, the integral equation (37) has a unique solution if the mapping ${\varvec{\mathcal {H}}}$ is a contraction of the complete metric space $C_{\delta }^{{\mathbf {z}}_{0}}[0,T]$ into itself for an appropriate choice of the radius $\delta >0$ and the initial guess ${\mathbf {z}}_{0}$.

To find a condition under which this holds, we first note that for $\left\| \mathbf {z-}{\mathbf {z}}_{0}\right\| _{0}\le \delta ,$ Eq. (37) gives

$$\begin{aligned} \left| {{\varvec{\mathcal {H}}}}({\mathbf {z}}(t))\right|&=\left| \int _{0}^{T}{\mathbf {V}}{\mathbf {G}}(t-s,T){\mathbf {V}}^{-1}[{\mathbf {F}}(s)-{\mathbf {R}}({\mathbf {z}}_{0}(s)) \right. \\&\quad \left. +{\mathbf {R}}({\mathbf {z}}_{0}(s))-{\mathbf {R}}({\mathbf {z}}(s))]\,ds\right| \\&=\left\| \int _{0}^{T}{\mathbf {V}}{\mathbf {G}}(t-s,T){\mathbf {V}}^{-1}\left[ {\mathbf {F}}(s)-{\mathbf {R}}({\mathbf {z}}_{0}(s))\right] \,ds\right\| _{0}\\&\quad +\left\| \int _{0}^{T}{\mathbf {V}}{\mathbf {G}}(t-s,T){\mathbf {V}}^{-1}\left[ {\mathbf {R}}({\mathbf {z}}_{0}(s))-{\mathbf {R}}({\mathbf {z}}(s))\right] \,ds\right\| _{0}\\&\le \left\| \varvec{\mathcal {E}}({\mathbf {z}}_{0},t)\right\| _{0}+\left\| {\mathbf {V}}\right\| \left\| {\mathbf {V}}^{-1}\right\| L_{\delta }^{{\mathbf {z}}_{0}}\left\| {\mathbf {z}}-{\mathbf {z}}_{0}\right\| _{0}\\&\quad \int _{0}^{T}\left\| {\mathbf {G}}(t-s,T)\right\| \,ds\\&\le \left\| \varvec{\mathcal {E}}({\mathbf {z}}_{0},t)\right\| _{0}+\left\| {\mathbf {V}}\right\| \left\| {\mathbf {V}}^{-1}\right\| L_{\delta }^{{\mathbf {z}}_{0}}\delta \\&\quad \left\| \int _{0}^{T}\left\| {\mathbf {H}}(t-s,T)\right\| \,ds\right\| _{0}\\&\le \left\| \varvec{\mathcal {E}}({\mathbf {z}}_{0},t)\right\| _{0}+\delta \left\| {\mathbf {V}}\right\| \left\| {\mathbf {V}}^{-1}\right\| ^{2}L_{\delta }^{{\mathbf {z}}_{0}}\varGamma (T), \end{aligned}$$

where $L_{\delta }^{{\mathbf {z}}_{0}}$ denotes a uniform-in-time Lipschitz constant for the function ${\mathbf {S}}({\mathbf {z}})$ with respect to its argument ${\mathbf {z}}$ within the ball $\left| {\mathbf {z}}-{\mathbf {z}}_{0}\right| \le \delta ,$ and $\varGamma (T)$ is the constant defined in (10). The initial error term $\varvec{\mathcal {E}}(t)$ is defined in Eq. (42). Taking the sup norm of both sides, we obtain that $\left\| {{\varvec{\mathcal {H}}}}({\mathbf {z}})\right\| _{0}\le \delta ,$ and hence

$$\begin{aligned} \varvec{\mathcal {H}}:C_{\delta }^{{\mathbf {z}}_{0}}[0,T]\rightarrow C_{\delta }^{{\mathbf {z}}_{0}}[0,T] \end{aligned}$$

holds, whenever condition (44) holds.

Similarly, for two functions ${\mathbf {z}},\tilde{{\mathbf {z}}}\in C_{\delta }^{{\mathbf {z}}_{0}}[0,T],$ Eq. (37) gives the estimate

$$\begin{aligned}&\left| {\varvec{\mathcal {H}}}({\mathbf {z}}(t)) -\varvec{\mathcal {H}}(\tilde{{\mathbf {z}}}(t))\right| \\&\quad \le \left| \int _{0}^{T}{\mathbf {V}}{\mathbf {G}}(t-s,T) {\mathbf {V}}^{-1}\left[ {\mathbf {R}}({\mathbf {z}}(s)) -{\mathbf {R}}(\tilde{{\mathbf {z}}}(s))\right] \,ds\right| \\&\quad \le 2\left\| {\mathbf {V}}\right\| \left\| {\mathbf {V}}^{-1}\right\| L_{\delta }^{{\mathbf {z}}_{0}}\int _{0}^{T}\left\| {\mathbf {G}}(t-s,T)\right\| \,ds\left| {\mathbf {z}}(t)-\tilde{{\mathbf {z}}}(t)\right| \\&\quad \le 2\left\| {\mathbf {V}}\right\| \left\| {\mathbf {V}}^{-1}\right\| L_{\delta }^{{\mathbf {z}}_{0}}\left\| \int _{0}^{T}\left\| {\mathbf {G}}(t-s,T)\right\| \,ds\right\| _{0}\left\| {\mathbf {z}}-\tilde{{\mathbf {z}}}\right\| _{0}\\&\quad \le 2\left\| {\mathbf {V}}\right\| \left\| {\mathbf {V}}^{-1}\right\| L_{\delta }^{{\mathbf {z}}_{0}}\varGamma (T)\left\| {\mathbf {z}}-\tilde{{\mathbf {z}}}\right\| _{0}. \end{aligned}$$

Taking the sum norm of both sides then gives that $\varvec{\mathcal {H}}$ is a contraction mapping on $C_{\delta }^{{\mathbf {z}}_{0}}[0,T]$ if

$$\begin{aligned} 2\left\| {\mathbf {V}}\right\| \left\| {\mathbf {V}}^{-1}\right\| L_{\delta }^{{\mathbf {z}}_{0}}\varGamma (T)<1/a, \end{aligned}$$

(79)

holds for some real number $a\ge 1$. Solving equation (79) for $L_{\delta }^{{\mathbf {z}}_{0}}$, we obtain condition (43).

Proof of Theorem 6

We show here that the mapping $\varvec{\mathcal {H}}$ defined in the quasi-periodic case (cf. equation (48)) is a contraction on the space (49) if the conditions (50) and (51) hold. The convergence estimate for the iteration (52) is then similar in spirit to the periodic case (cf. “Appendix G”).

We rewrite (38) as

$$\begin{aligned}&{\mathbf {z}}(t)=\varUpsilon _{Q}({\mathbf {F}}(t)-{\mathbf {R}} ({\mathbf {z}}(t)))\\&\quad :={\mathbf {V}}\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}{\mathbf {H}}(T_{\kappa }){\mathbf {V}}^{-1}\left( {\mathbf {F}}_{\varvec{\kappa }}-{\mathbf {R}}_{\varvec{\kappa }}\{{\mathbf {z}}\}\right) e^{i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle t}, \end{aligned}$$

where $\varUpsilon _{Q}$ is a linear map representing the convolution operation with the Green’s function. Similarly to the periodic case, we define the space of n-dimensional quasi-periodic functions with frequency base vector $\varvec{\Omega }$ as

$$\begin{aligned} {\mathcal {Q}}_{2n}:=\{{\mathbf {p}}:{\mathbb {T}}^{k}\rightarrow {\mathbb {R}}^{n},{\mathbf {p}}\in C^{0}\}. \end{aligned}$$

(80)

Furthermore, we note that under the non-resonance condition (7), the linear map

$$\begin{aligned}&\varUpsilon _{Q}:{\mathcal {Q}}_{2n}\rightarrow {\mathcal {Q}}_{2n}, \quad \varUpsilon _{Q}{\mathbf {q}}\\&\quad ={\mathbf {V}}\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}\int _{0}^{T_{\varvec{\kappa }}}{\mathbf {G}}(t-s,T_{\varvec{\kappa }}){\mathbf {V}}^{-1}{\mathbf {q}}(s)\,\mathrm{d}s \end{aligned}$$

is well defined, i.e., $\varUpsilon _{Q}$ maps any quasi-periodic function ${\mathbf {q}}$ with frequency base vector $\varvec{\Omega }$ to quasi-periodic functions with the same frequency base vector $\varvec{\Omega }$. This is a direct consequence of the linearity of the $\varUpsilon _{Q}$ and definition of $\varUpsilon _{P}$ in “Appendix G”.

Since the mapping (48) is well defined in the space $C_{\delta }^{{\mathbf {z}}_{0}}(\varvec{\Omega })$ defined in (49), we have by the Banach fixed point theorem that the integral equation (48) has a unique solution if the mapping $\varvec{\mathcal {H}}$ is a contraction of the complete metric space $C_{\delta }^{{\mathbf {z}}_{0}}(\varvec{\Omega })$ into itself for an appropriate choice of the radius $\delta >0.$ In a similar spirit as in the periodic case we search for conditions under which the space $C_{\delta }^{{\mathbf {z}}_{0}}(\varvec{\Omega })$ is mapped to itself. Therefore, we take the sup norm of the mapping (48) applied to an element from $C_{\delta }^{{\mathbf {z}}_{0}}(\varvec{\Omega })$ and obtain

$$\begin{aligned}&\left| {\varvec{\mathcal {H}}}({\mathbf {z}}(t))\right| \\&\quad =\left| {\mathbf {V}}\sum _{\kappa \in {\mathbb {Z}}^{k}}{\mathbf {H}}(T_{\kappa }){\mathbf {V}}^{-1}\left[ \mathbf {F_{\varvec{\kappa }}}-{\mathbf {R}}_{\varvec{\kappa }}\{{\mathbf {z}}_{0}\}\right. \right. \\&\quad \quad \left. \left. +\,{\mathbf {R}}_{\varvec{\kappa }}\{{\mathbf {z}}_{0}\}-{\mathbf {R}}_{\varvec{\kappa }}\{{\mathbf {z}}\}\right] e^{i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle t}\right| \\&\quad \le \left\| {\mathbf {V}}\sum _{\kappa \in {\mathbb {Z}}^{k}} {\mathbf {H}}(T_{\kappa }){\mathbf {V}}^{-1} \left[ {\mathbf {F}}_{\varvec{\kappa }}-{\mathbf {R}}_{\varvec{\kappa }} \{{\mathbf {z}}_{0}\}\right] e^{i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle t}\right\| _{0}\\&\quad \quad +\left\| {\mathbf {V}}\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}{\mathbf {H}}(T_{\kappa })\mathbf {V^{-1}}\left[ {\mathbf {R}}_{\varvec{\kappa }}\{{\mathbf {z}}_{0}\}-{\mathbf {R}}_{\varvec{\kappa }}\{{\mathbf {z}}\}\right] e^{i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle t}\right\| _{0}\\&\quad \le \left\| \varvec{\mathcal {E}}({\mathbf {z}}_{0},t)\right\| _{0}+\left\| {\mathbf {V}}\right\| \left\| {\mathbf {V}}^{-1}\right\| h_{max}\\&\quad \quad \times \left\| \sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}\left[ {\mathbf {R}}_{\varvec{\kappa }}\{{\mathbf {z}}_{0}\}-{\mathbf {R}}_{\varvec{\kappa }}\{{\mathbf {z}}\}\right] e^{i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle t}\right\| _{0}\\&\quad \le \left\| \varvec{\mathcal {E}}({\mathbf {z}}_{0},t)\right\| _{0}+\left\| {\mathbf {V}}\right\| \left\| {\mathbf {V}}^{-1}\right\| h_\mathrm{max}\\&\quad \quad \times \left\| {\mathbf {R}}({\mathbf {z}}_{0}(s),s)-{\mathbf {R}}({\mathbf {z}}(s),s)\right\| _{0}\\&\quad \le \left\| \varvec{\mathcal {E}}({\mathbf {z}}_{0},t)\right\| _{0}+\delta \left\| {\mathbf {V}}\right\| \left\| {\mathbf {V}}^{-1}\right\| h_\mathrm{max}L_{\delta }^{{\mathbf {z}}_{0}}, \end{aligned}$$

where we have used that the Fourier series of the nonlinearity $\sum _{k}\mathbf {R_{\varvec{\kappa }}}\{{\mathbf {z}}\}e^{i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle t}$converges to the function ${\mathbf {R}}({\mathbf {z}},t)$. Due to the Lipschitz continuity of the nonlinearity and the forcing, this holds. We finally conclude that $C_{\delta }^{{\mathbf {z}}_{0}}(\varvec{\Omega })$ is mapped to itself, if condition (51) holds.

Similarly, for two function ${\mathbf {z}},\tilde{{\mathbf {z}}}$ in $C_{\delta }^{{\mathbf {z}}_{0}}(\varvec{\Omega })$, we obtain

$$\begin{aligned}&\left| \varvec{\mathcal {H}}({\mathbf {z}}(t)) -\varvec{\mathcal {H}} ({\tilde{\mathbf {z}}}(t))\right| \\&\quad \le \left| \sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}} {\mathbf {V}}{\mathbf {H}}(T_{\kappa }) {\mathbf {V}}^{-1}\left[ \mathbf {{\mathbf {R}}_{\varvec{\kappa }} \{{\mathbf {z}}\}}-{\mathbf {R}}_{\varvec{\kappa }} \{\tilde{\mathbf {z}}\}\right] e^{i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle t}\right| \\&\quad \le 2\left\| {\mathbf {V}}\right\| \left\| {\mathbf {V}}^{-1}\right\| h_\mathrm{max}L_{\delta }^{{\mathbf {z}}_{0}}\left\| {\mathbf {z}}-\tilde{{\mathbf {z}}}\right\| _{0}\,. \end{aligned}$$

Therefore, the iteration (48) is a contraction on the space $C_{\delta }^{{\mathbf {z}}_{0}}(\varvec{\Omega })$, if the condition

$$\begin{aligned} 2\left\| {\mathbf {V}}\right\| \left\| {\mathbf {V}}^{-1}\right\| h_\mathrm{max}L_{\delta }^{{\mathbf {z}}_{0}}<\frac{1}{a}\qquad a\in {\mathbb {R}},\,a>1, \end{aligned}$$

(81)

holds, which we reformulate in (50).

Explicit expressions for Fourier coefficients in Remark 6

To obtain the amplifications factors given in (35), we carry out the integration explicitly, we diagonalize the system with the matrix of the undamped mode shapes ${\mathbf {U}}$, (i.e., let ${\mathbf {x}}={\mathbf {U}}{\mathbf {y}}$) and introduce the notation $\varvec{\psi _{\varvec{\kappa }}}={\mathbf {U}}^{\top }{\mathbf {f}}_{\varvec{\kappa }}$. Assuming an underdamped configuration ($\zeta _{j}<1$), we obtain for the jth degree of freedom

$$\begin{aligned} w_{j}(t)&=\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}\int _{0}^{T_{\varvec{\kappa }}}L_{j}(t-s,T_{\varvec{\kappa }})\psi _{j,\varvec{\kappa }}e^{i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle s}\,\mathrm{d}s\\&=\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}\int _{0}^{T_{\varvec{\kappa }}}\frac{e^{\alpha _{j}(t-s)}}{\omega _{j}}\left[ \frac{e^{\alpha _{j}T_{\varvec{\kappa }}}\left[ \sin \omega _{j}(T_{\varvec{\kappa }}+t-s)-e^{\alpha _{j}T_{\varvec{\kappa }}}\sin \omega _{j}(t-s)\right] }{1+e^{2\alpha _{j}T_{\varvec{\kappa }}}-2e^{\alpha _{j}T_{\varvec{\kappa }}}\cos \omega _{j}T_{\varvec{\kappa }}}+h(t-s)\sin \omega _{j}(t-s)\right] \psi _{j,\varvec{\kappa }}e^{i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle s}\,\mathrm{d}s\\&=\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}\int _{0}^{t}\frac{e^{\alpha _{j}(t-s)}}{\omega _{j}}\sin \omega _{j}(t-s)\psi _{j,\varvec{\kappa }}e^{i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle s}\,\mathrm{d}s\\&\quad +\,\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}\int _{0}^{T_{\varvec{\kappa }}}\frac{e^{\alpha _{j}(t-s)}}{\omega _{j}}\left[ \frac{e^{\alpha _{j}T_{\varvec{\kappa }}}\left[ \sin \omega _{j}(T_{\varvec{\kappa }}+t-s)-e^{\alpha _{j}T_{\varvec{\kappa }}}\sin \omega _{j}(t-s)\right] }{1+e^{2\alpha _{j}T_{\varvec{\kappa }}}-2e^{\alpha _{j}T_{\varvec{\kappa }}}\cos \omega _{j}T_{\varvec{\kappa }}}\right] \psi _{j,\varvec{\kappa }}e^{i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle s}\,\mathrm{d}s\\&=\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}\frac{e^{\alpha _{j}t}}{\omega _{j}}\psi _{j,\varvec{\kappa }}\int _{0}^{t}\sin \omega _{j}(t-s)e^{(i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\alpha _{j})s}\,\mathrm{d}s+\,\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}\frac{e^{\alpha _{j}(T_{\varvec{\kappa }}+t)}}{\omega _{j}}\left( \frac{1}{1+e^{2\alpha _{j}T_{\varvec{\kappa }}}-2e^{\alpha _{j}T_{\varvec{\kappa }}}\cos \omega _{j}T_{\varvec{\kappa }}}\right) \psi _{j,\varvec{\kappa }}\\&\quad \times \left[ \int _{0}^{T_{\varvec{\kappa }}}e^{(i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\alpha _{j})s}\sin \omega _{j}(T_{\varvec{\kappa }}+t-s)\,\mathrm{d}s \right. \left. -\,e^{\alpha _{j}T_{\varvec{\kappa }}}\int _{0}^{T_{\varvec{\kappa }}}e^{(i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\alpha _{j})s}\sin \omega _{j}(t-s)\,\mathrm{d}s\right] \\&=\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}\frac{e^{\alpha _{j}t}}{\omega _{j}}\psi _{j,\varvec{\kappa }}\left. \frac{e^{(i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\alpha _{j})s}\left( \omega _{j}\cos \omega _{j}(t-s)+(i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\alpha _{j})\sin \omega _{j}(t-s)\right) }{((i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\alpha _{j})^{2}+\omega _{j}^{2})}\right| _{s=0}^{s=t}\\&\quad +\,\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}\frac{e^{\alpha _{j}(T_{\varvec{\kappa }}+t)}}{\omega _{j}}\left[ \frac{\left. e^{(i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\alpha _{j})s}\left( \omega _{j}\cos \omega _{j}(T_{\varvec{\kappa }}+t-s)+(i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\alpha _{j})\sin \omega _{j}(T_{\varvec{\kappa }}+t-s)\right) \right| _{s=0}^{s=T_{\varvec{\kappa }}}}{\left( 1+e^{2\alpha _{j}T_{\varvec{\kappa }}}-2e^{\alpha _{j}T_{\varvec{\kappa }}}\cos \omega _{j}T_{\varvec{\kappa }}\right) \left( (i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\alpha _{j})^{2}+\omega _{j}^{2}\right) }\right] \psi _{j,\varvec{\kappa }}\\&\quad -\,\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}\frac{e^{\alpha _{j}(2T_{\varvec{\kappa }}+t)}}{\omega _{j}}\left[ \frac{\left. e^{(i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\alpha _{j})s}\left( \omega _{j}\cos \omega _{j}(t-s)+(i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\alpha _{j})\sin \omega _{j}(t-s)\right) \right| _{s=0}^{s=T_{\varvec{\kappa }}}}{\left( 1+e^{2\alpha _{j}T_{\varvec{\kappa }}}-2e^{\alpha _{j}T_{\varvec{\kappa }}}\cos \omega _{j}T_{\varvec{\kappa }}\right) \left( (i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\alpha _{j})^{2}+\omega _{j}^{2}\right) }\right] \psi _{j,\varvec{\kappa }}\\&=\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}\frac{1}{(i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\alpha _{j})^{2}+\omega _{j}^{2}}\psi _{j,\varvec{\kappa }}e^{i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle t}. \end{aligned}$$

For the critically damped configuration ($\zeta _{j}=1$), we obtain

$$\begin{aligned} w_{j}(t)&=\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}\int _{0}^{T_{\varvec{\kappa }}}L_{j}(t-s,T_{\varvec{\kappa }})\psi _{j,\varvec{\kappa }}e^{i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle s}\,\mathrm{d}s \\&=\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}\int _{0} ^{T_{\varvec{\kappa }}}\left[ \frac{e^{\alpha _{j} (T_{\varvec{\kappa }}+t-s)} \left[ \left( 1-e^{\alpha _{j}T}\right) (t-s)+T_{\varvec{\kappa }}\right] }{\left( 1-e^{\alpha _{j}T_{\varvec{\kappa }}}\right) ^{2}} \right. \\&\quad \left. +\,h(t-s)(t-s)e^{\alpha _{j}(t-s)}\right] \psi _{j,\varvec{\kappa }}e^{i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle s}\,\mathrm{d}s\\&=\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}e^{\alpha _{j}t}\psi _{j,\varvec{\kappa }}\int _{0}^{t}(t-s)e^{(i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\alpha _{j})s}\,\mathrm{d}s \\&\quad +\,\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}\frac{e^{\alpha _{j}(T_{\varvec{\kappa }}+t)}}{\left( 1-e^{\alpha _{j}T_{\varvec{\kappa }}}\right) ^{2}}\psi _{j,\varvec{\kappa }}\\&\quad \int _{0}^{T_{\varvec{\kappa }}}\left[ \left( 1-e^{\alpha _{j}T_{\varvec{\kappa }}}\right) (t-s)+T_{\varvec{\kappa }}\right] e^{(i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\alpha _{j})s}\,\mathrm{d}s \\&=\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}\frac{1}{(i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\alpha _{j})^{2}}\psi _{j,\varvec{\kappa }}e^{i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle t}. \end{aligned}$$

Finally, for the overdamped configuration ($\zeta _{j}>1$), we obtain

$$\begin{aligned} w_{j}(t)&=\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}\int _{0}^{T_{\varvec{\kappa }}}L_{j}(t-s,T_{\varvec{\kappa }})\psi _{j,\varvec{\kappa }}e^{i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle s}\,\mathrm{d}s\\&=\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}\int _{0} ^{T_{\varvec{\kappa }}}\left[ \frac{e^{\beta _{j} (T_{\varvec{\kappa }}+t-s)}\left( 1-e^{\gamma _{j} T_{\varvec{\kappa }}}\right) -e^{\gamma _{j}(T_{\varvec{\kappa }}+t-s)} \left( 1-e^{\beta _{j}T_{\varvec{\kappa }}}\right) }{(\beta _{j}-\gamma _{j})} \left( 1-e^{\gamma _{j}T_{\varvec{\kappa }}} -e^{\beta _{j}T_{\varvec{\kappa }}}+e^{\left( \gamma _{j}+\beta _{j}\right) T_{\varvec{\kappa }}}\right) \right. \\ {}&\quad \quad \left. +\frac{h(t-s)\left( e^{\beta _{j}(t-s)}-e^{\gamma _{j}(t-s)} \right) }{(\beta _{j}-\gamma _{j})}\right] \psi _{j,\varvec{\kappa }}e^{i \left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle s}\,\mathrm{d}s\\&=\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}\frac{\psi _{j,\varvec{\kappa }}}{(\beta _{j}-\gamma _{j})}\left[ e^{\beta _{j}t}\int _{0}^{t}e^{(i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\beta _{j})s}\,\mathrm{d}s-e^{\gamma _{j}t}\int _{0}^{t}e^{(i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\gamma _{j})s}\,\mathrm{d}s\right] \\&\quad +\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}\frac{\psi _{j,\varvec{\kappa }}\left[ \left( 1-e^{\gamma _{j}T}\right) e^{\beta _{j}(T+t)}\int _{0}^{T_{\varvec{\kappa }}}e^{(i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\beta _{j})s}\,\mathrm{d}s-\left( 1-e^{\beta _{j}T}\right) e^{\gamma _{j}(T+t)}\int _{0}^{T_{\varvec{\kappa }}}e^{(i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle -\gamma _{j})s}\,\mathrm{d}s\right] }{(\beta _{j}-\gamma _{j})(1-e^{\gamma _{j}T_{\varvec{\kappa }}}-e^{\beta _{j}T_{\varvec{\kappa }}}+e^{\left( \gamma _{j}+\beta _{j}\right) T_{\varvec{\kappa }}})}\\&=\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}\frac{1}{(\beta _{j}-i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle )(\gamma _{j}-i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle )}\psi _{j,\varvec{\kappa }}e^{i\left\langle \varvec{\kappa },\varvec{\Omega }\right\rangle t}. \end{aligned}$$

Numerical solution procedure

The numerical approximation of the solution ${\mathbf {z}}(t)$ to integral equations such as (37) and (38) is performed via the finite sum

$$\begin{aligned} {\mathbf {z}}_{m}(t)=\sum _{j=1}^{m}{\mathbf {c}}_{mj}b_{mj}(t), \end{aligned}$$

where ${\mathbf {c}}_{mj}$ is the unknown coefficient attached to the chosen basis function $b_{mj}(t)$. The basic idea of all related numerical methods is to project the solution onto a suitable finite-dimensional subspace to facilitate numerical computations. These projection methods can be broadly divided into the categories of collocation and Galerkin methods (cf. Kress [54]).

If the basis functions $b_{mj}$ are chosen to perform a collocation-type approximation, the integral equation will be guaranteed to be satisfied at a finite number of collocation points. Specifically, if m collocation points $t_{1}^{(m)},t_{2}^{(m)}\dots ,t_{m}^{(m)}\in [0,T]$ are chosen, the integral equation is reduced to solving a finite-dimensional system of (nonlinear) algebraic equations in the coefficients ${\mathbf {c}}_{mj}$:

$$\begin{aligned} {\mathbf {z}}_{m}(t_{j})=\varvec{\mathcal {G}}_{P}\left( {\mathbf {z}}_{m}\right) \,(t_{j}),\quad j=1,\dots ,m. \end{aligned}$$

(82)

Equation (82) needs to be solved for the unknown coefficients ${\mathbf {c}}_{mj}\in {\mathbb {R}}^{2n}$ for all $j\in \{1,\dots ,m\}$. Note that under the non-resonance conditions (7), the Green’s function ${\mathbf {G}}$ is bounded in time. Then, the collocation method with linear splines will converge to the exact solution for the linear integral equation at each iteration step (cf. Kress [54], chapter 13). Furthermore, if the exact solution ${\mathbf {z}}$ is twice continuously differentiable in t, we also obtain an error estimate for the linear spline collocation approximate solution ${\mathbf {z}}_{m}$ as

$$\begin{aligned} \Vert {\mathbf {z}}_{m}-{\mathbf {z}}\Vert _{\infty }\le M\Vert \ddot{{\mathbf {z}}}\Vert _{\infty }\Delta ^{2}, \end{aligned}$$

where $\Delta$ is the spacing between the uniform collocation points and M is an order constant depending upon the Green’s function ${\mathbf {G}}$.

Alternatively, the Galerkin method can be chosen to approximate the solution of Eqs. (37) and (38) using Fourier basis with harmonics of the base frequencies, followed by a projection onto the same basis vectors, given by

$$\begin{aligned}&\int _{0}^{T_{j}}{\mathbf {z}}_{m}(s)b_{mj}(s)\,\mathrm{d}s \nonumber \\&\quad =\int _{0}^{T_{j}}\sum _{\varvec{\kappa } \in {\mathbb {Z}}^{k}}{\mathbf {V}}{\mathbf {H}}(T_{\varvec{\kappa }}) {\mathbf {V}}^{-1}\left( \mathbf {F_{\varvec{\kappa }}} -{\mathbf {R}}_{\varvec{\kappa }}\{{\mathbf {z}}_{m}\}\right) \nonumber \\&\quad \quad e^{i\left\langle \varvec{\varvec{\kappa }},\varvec{\Omega }\right\rangle s}b_{mj}(s)\,\mathrm{d}s\,,\quad j=1,\dots ,m, \end{aligned}$$

(83)

where $b_{mj}(t)=e^{i\left\langle \varvec{\varvec{\kappa }}_{j},\varvec{\Omega }\right\rangle t}$ are the Fourier basis functions, with the corresponding time periods $T_{j}=\frac{2\pi }{\left\langle \varvec{\varvec{\kappa }}_{j},\varvec{\Omega }\right\rangle }$. Due to the orthogonality of these basis function, system (83) simplifies to

$$\begin{aligned}&{\mathbf {c}}_{mj}=\sum _{\varvec{\kappa }\in {\mathbb {Z}}^{k}}{\mathbf {V}}{\mathbf {H}}(T_{\varvec{\kappa }}){\mathbf {V}}^{-1}\left( \mathbf {F_{\varvec{\kappa }}}-\mathbf {R_{\varvec{\kappa }}}\{{\mathbf {z}}_{m}\}\right) ,\\&\quad \quad \quad \quad j=1,\dots ,m \end{aligned}$$

which is a system of nonlinear equations for the unknown coefficients ${\mathbf {c}}_{mj}$.

In the frequency domain, this system of coefficient equations are the same as that obtained from the multi-frequency harmonic balance method after finite truncation (see, e.g., Chua and Ushida [24] or Lau and Cheung [25]). Our explicit formulas in (35), however, should speed up the calculations relative to a general application of the harmonic balance method. The same scheme applies in the periodic case. For both cases (periodic and quasi-periodic), the error due to the omission of higher harmonics in harmonic balance procedure is not well understood (see our review of the available results on the periodic case in Introduction ). For the quasi-periodic case, no such error bounds are known to the best of our knowledge.

Equations (37) and (38) are Fredholm integral equations of the second kind (cf. Zemyan [55]). Fortunately, the theory and solution methods for integral equations of the second kind are considerably simpler than for those of the first kind. We refer to Atkinson [56] for an exhaustive treatment of numerical methods for such integral equations. Our supplementary MATLAB^® code provides the implementation details of a simple yet powerful collocation-based approach for the periodic case, and a Galerkin projection using a Fourier basis for the quasi-periodic case.

1.1 Numerical continuation

A simple approach is to use sequential continuation, in which the frequency (or time period) of oscillation is treated as a continuation parameter (multi-parameter in case of quasi-periodic oscillations). This parameter is varied in small steps and the corresponding (quasi) periodic response is iteratively computed at each step. The solution in any given step is typically used as initial guess for the next step. Such an approach will generally fail near a fold bifurcation with respect to one of the base frequencies $\varvec{\Omega }$. In such cases, more advanced continuation schemes such as the pseudo-arc-length continuation are required.

The pseudo-arc-length approach. We briefly explain the steps involved in numerical continuation for calculation of periodic orbits using the pseudo-arc-length approach, which is commonly used to track folds in a single-parameter solution branch. We seek the frequency response curve of system (22) as a family of solutions to

$$\begin{aligned} \varvec{\mathcal {N}}({\mathbf {x}},T)&:={\mathbf {x}}-\int _{0}^{T}{\mathbf {U}}{\mathbf {L}}\left( t-s;T(p)\right) {\mathbf {U}}^{\top }\nonumber \\&\quad \left( {\mathbf {f}}(s)-{\mathbf {S}}({\mathbf {x}}(s))\right) \,\mathrm{d}s={\mathbf {0}}, \end{aligned}$$

(84)

with the fictitious variable p parameterizing the solution curve $({\mathbf {x}},T)$ of equation (84). If $({\mathbf {x}},T)$ is a regular solution for equation (84), then the implicit function theorem guarantees the existence of a nearby unique solution. The tangent vector ${\mathbf {t}}$ to the solution curve at $({\mathbf {x}},T)$ is given by the null-space of the Jacobian $\left. D\varvec{\mathcal {N}}\right| _{({\mathbf {x}},T)}$, i.e., ${\mathbf {t}}$ can be obtained by solving the system of equations

$$\begin{aligned}&\left. D\varvec{\mathcal {N}}\right| _{({\mathbf {x}},T)}{\mathbf {t}} ={\mathbf {0}},\\&\left\langle {\mathbf {t}},{\mathbf {t}}\right\rangle =1, \end{aligned}$$

where the second equation specifies a unity constraint on the length of the tangent vector to uniquely identify ${\mathbf {t}}$. This tangent vector gives a direction on the solution curve along which p increases. Starting with a known solution ${\mathbf {u}}_{0}:=({\mathbf {x}}_{0},T_{0})$ on the solution branch with the corresponding tangent direction vector ${\mathbf {t}}_{0}$ and a prescribed arc-length step size $\Delta p$, we obtain a nearby solution ${\mathbf {u}}$ by solving the nonlinear system of equations

$$\begin{aligned}&\varvec{\mathcal {N}}\left( {\mathbf {u}}\right) ={\mathbf {0}},\\&\left\langle {\mathbf {u}}-{\mathbf {u}}_{0},{\mathbf {t}}_{0}\right\rangle =\Delta p. \end{aligned}$$

This system can again be solved iteratively using, e.g., the Newton–Raphson procedure, with the Jacobian given by $[D\varvec{\mathcal {N}},\,{\mathbf {t}}_{0}]^{\top }$. We need the derivative of $\varvec{\mathcal {N}}({\mathbf {x}},T)$ with respect to T to evaluate this Jacobian, which can also be explicitly computed using the derivative of the Green’s function ${\mathbf {L}}$ or ${\mathbf {G}}$ with respect to T. These expressions are detailed in “Appendix E” and implemented in the supplementary MATLAB^® code.

Although this pseudo-arc-length continuation is able to capture folds in single-parameter solution branches, it is not the state-of-the-art continuation algorithm. More advanced continuation schemes, such as the atlas algorithm of $\textsc {coco}$ [37], enable continuation with multi-dimensional parameters required for quasi-periodic forcing. In this work, we have implemented our integral equations approach with the MATLAB^®-based continuation package $\textsc {coco}$ [37] .

1.1.1 Continuation of periodic orbits in the conservative autonomous setting

As remarked earlier, conservative autonomous systems have internally parameterized periodic orbits a priori unknown period. Moreover, each (quasi-) periodic orbit of such a system is part of a family of (quasi-) periodic trajectories with the solutions differing only in their phases. A unique solution is obtained using a phase condition, a scalar equation of the form

$$\begin{aligned} {\mathfrak {p}}({\mathbf {z}}(t))=0, \end{aligned}$$

(85)

which fixes a Poincare section transverse to the (quasi-) periodic orbit in the phase space. The choice of this section is arbitrary but often involves setting the initial velocity of one of the degrees of freedom to be zero, i.e., letting

$$\begin{aligned} {\mathfrak {p}}({\mathbf {z}})=\dot{x}_{k}(0)=0, \end{aligned}$$

asserting that the initial velocity at the kth degree of freedom is zero. As the solution time period T is unknown, Eqs. (84), (85) have an equal number of equations and variables $({\mathbf {x}},T)$. Thus, there is no parameter among the variables, which can be used for the continuation of a given solution.

In the literature, this issue is avoided by introducing a fictitious damping parameter, say d, and establishing the existence of a periodic orbit if and only if $d=0$, i.e., in the conservative limit (Muñoz-Almaraz et al. [59]). With this trick, we reformulate the integral equation (84) (with forcing ${\mathbf {f}}(t)\equiv {\mathbf {0}}$ ) to find periodic orbits of the system

$$\begin{aligned} {\mathbf {M}}\ddot{{\mathbf {x}}}+d{\mathbf {K}}\dot{{\mathbf {x}}}+{\mathbf {K}}{\mathbf {x}}+{\mathbf {S}}({\mathbf {x}})={\mathbf {0}}. \end{aligned}$$

(86)

Periodic solutions to this system are created through a Hopf bifurcation that occurs precisely at the conservative limit of system (86) ($d=0$). It can be shown that the damping parameter maintains a zero value along the periodic solution-branch obtained at the bifurcation point. Advanced continuation algorithms can then be used to detect such a Hopf bifurcation and to make a switch to the periodic solution branch (cf. Galán-Vioque and Vanderbauwhede [60]).

A similar continuation procedure can be carried out in our integral equation approach. The periodic solution, however, technically does not arise from a Hopf bifurcation, because the non-resonance conditions (7) and (13) are violated at the Hopf bifurcation point. Nonetheless, continuation of a given solution point on the solution branch is possible using standard continuation algorithms. In this conservative autonomous setting, a given solution of system (84), (85) may again be continued in the $({\mathbf {x}},T,d)$ space using, e.g., the pseudo-arc-length approach.