2 The trajectory equation as a nonlinear differential system in $\mathbb {R}^3$

The following theorem expresses (1.1) as an infinite dimensional system of nonlinear ODEs. Referring to [Reference Valani15, Reference Valani, Slim, Paganin, Samula and Vo16], this form of (1.1) is more tractable and, in special cases, such as when considering a waveform in the absence of spatial decay, the infinite system reduces to a finite dimensional system.

Theorem 2.1. Let

$$ \begin{align*}M_n=-\int_{-\infty}^t W^{(n+1)}(x(t)-x(s))e^{s-t}\,ds\end{align*} $$

for all $n\in \mathbb {N}$ . Equation (1.1) holds if and only if

(2.1) $$ \begin{align} \dot{x}&=v,\nonumber\\ \dot{v}&=\frac{1}{\kappa}(\,\beta M_0-v+\mathcal{F}),\nonumber\\ \dot{M}_n&=-M_n-W^{(n+1)}(0)+vM_{n+1}. \end{align} $$

Proof. The first two linear ODEs are easily verified, since (1.1) may be expressed as

$$ \begin{align*} \ddot{x}=\frac{1}{\kappa}(\,\beta M_0-\dot{x}+\mathcal{F}). \end{align*} $$

Given that the integrand of $M_n$ is analytic, we may apply the Leibniz integral rule. Observe that for all $n\geq 0$ ,

$$ \begin{align*} \begin{split} \dot{M_n}&=-\frac{\partial}{\partial t}\int_{-\infty}^t W^{(n+1)}(x(t)-x(s))e^{s-t}\mathop{ds}\\&=-W^{(n+1)}(0)-\int_{-\infty}^t e^{s-t}(\dot{x}W^{(n+2)}(x(t)-x(s))-W^{(n+1)}(x(t)-x(s)))\mathop{ds}\\&=-W^{(n+1)}(0)+\int_{-\infty}^t W^{(n+1)}(x(t)-x(s))e^{s-t}\mathop{ds}-\dot{x}\int_{-\infty}^t W^{(n+2)}(x(t)-x(s))e^{s-t}\mathop{ds}\\&=-M_{n}-W^{(n+1)}(0)+\dot{x}M_{n+1}, \end{split} \end{align*} $$

as claimed.

2.1 The linear stability of a sinusoidal waveform system

According to Durey [Reference Durey4], we may define $W(s)=\cos (s)$ for the high memory regime, such that letting $\sigma =1/\kappa $ , $r=\beta $ and $b=1$ , (2.1) is reduced to the vector field $V:\mathbb {R}^3\rightarrow \mathbb {R}^3$ , defined by

(2.2) $$ \begin{align} \dot{\mathbf{P}}=V(\mathbf{P})=(\sigma(Y-X+\mathcal{F}),rX-Y-XZ,XY-bZ), \end{align} $$

where $X=v$ , $Y=\beta M_0$ , $Z=\beta (1-M_1)$ and $\mathbf {P}=(X,Y,Z)$ . For all $\mathcal {F}$ such that $\mathcal {F}(0)=0$ , an equilibrium point occurs at $\mathbf {P}^*=(0,0,0)$ and the Jacobian matrix of V at $\mathbf {P}^*$ , letting $\mathcal {F}'(X)=\partial \mathcal {F}/\partial X$ , is

$$ \begin{align*} JV_{\mathbf{P}^*}= \begin{pmatrix} \sigma(\mathcal{F}'(0)-1) & \sigma & 0 \\ r & -1 & 0 \\ 0 & 0 & -b \end{pmatrix}. \end{align*} $$

So, we now consider the linearised system to first order in $\epsilon \in \mathbb {R}$ , where $\mathbf {P}=\mathbf {P}^*+\epsilon \mathbf {u}$ :

(2.3) $$ \begin{align} \dot{\mathbf{u}}=JV_{\mathbf{P}^*}\mathbf{u}. \end{align} $$

Setting $\det (\lambda I-JV_{\mathbf {P}^*})=0$ for $\lambda \in \mathbb {R}$ yields the characteristic equation

(2.4) $$ \begin{align} 0=(\lambda+b)(\lambda^2+\lambda(1+\sigma-\sigma\mathcal{F}'(0))+\sigma(1-\mathcal{F}'(0)-r)). \end{align} $$

Hence, we have $\lambda _0=-b$ and may derive the remaining eigenvalues explicitly:

$$ \begin{align*} \lambda_{1,2}=\tfrac{1}{2}\bigg(\sigma\mathcal{F}'(0)-\sigma-1\pm\sqrt{4\sigma(\mathcal{F}'(0)+r-1)+(\sigma \mathcal{F}'(0)-\sigma-1)^2}\bigg). \end{align*} $$

This explicit formula for eigenvalues is useful when calculating the linearised approximation $\mathbf {P}_t\approx \sum _{i=0}^2 C_i e^{\lambda _i t} \mathbf {v}_i$ for the stable region given in Theorem 2.3, where $\mathbf {v}_i$ is the corresponding eigenvector to $\lambda _i$ and $C_i\in \mathbb {R}$ . Note that the accuracy of the approximation is determined by how close the initial conditions are to $\mathbf {P}^*$ , which follows from Lemma 2.2.

Lemma 2.2 (Hartman–Grobman theorem, [Reference Hartman6])

If all the eigenvalues $\lambda _j$ of $JV_{\mathbf {P}^*}$ have nonzero real part, then the nonlinear flow is topologically conjugate to the flow of the linearised system in a neighbourhood of $\mathbf {P}^*$ . If $\Re (\lambda _j)>0$ for at least one j, the nonlinear flow is asymptotically unstable. If $\Re (\lambda _j)<0$ for all j, the nonlinear flow is asymptotically stable.

Theorem 2.3. The asymptotically stable region S of (2.2) is

$$ \begin{align*} r<1-\mathcal{F}'(0), \end{align*} $$

such that $\sigma (\mathcal {F}'(0)-1)<1$ and $r,\sigma \geq 0$ . Letting $\zeta $ denote the stability boundary, the remaining asymptotically unstable region $U:=\mathbb {R}^3\setminus (S\cup \zeta )$ satisfies $r>1-\mathcal {F}'(0)$ .

Proof. If $\lambda =i\omega $ is a root with $\omega \in \mathbb {R}$ , then according to (2.4),

(2.5) $$ \begin{align} 0 = (\sigma-\sigma\mathcal{F}'(0)-r\sigma-\omega^2)+i\omega(1+\sigma-\sigma\mathcal{F}'(0)). \end{align} $$

Separately considering real and imaginary parts, the stability boundary satisfies

$$ \begin{align*} \begin{split} 0&=\sigma-\sigma\mathcal{F}'(0)-r\sigma-\omega^2\\ &=\omega(1+\sigma-\sigma\mathcal{F}'(0)). \end{split} \end{align*} $$

Suppose that $\sigma (\mathcal {F}'(0)-1)<1$ . For $\omega \lessgtr \pm \sqrt {\sigma (1-\mathcal {F}'(0)-r)},$ we must have each real part of $\lambda _{1,2}$ less than zero, which is the criterion for asymptotic stability according to Lemma 2.2 (the converse applies for asymptotic instability). Hence, simplifying the imaginary part of (2.5), we have

$$ \begin{align*} \begin{split} 0&\lessgtr\pm(1+\sigma-\sigma\mathcal{F}'(0))\sqrt{\sigma(1-\mathcal{F}'(0)-r)},\\ r&<1-\mathcal{F}'(0).\\[-2.3pc] \end{split} \end{align*} $$

3 Nonlinear differential Galois theory and an integrability criterion

We will present an argument for algebraic B-integrability introduced by Morales-Ruiz [Reference Morales-Ruiz13] and subsequently developed by Huang et al. [Reference Huang, Shi and Li7] for its application to the Lorenz system. From (2.2), we consider the system

(3.1) $$ \begin{align} \dot{\mathbf{P}}=V(\mathbf{P}), \end{align} $$

such that $\mathbf {P}\in \mathcal {M}$ is an $(n=3)$ -dimensional complex analytic manifold with $t\in \mathbb {C}$ .

Definition 3.2. System (3.1) is B-integrable if it possesses k functionally independent first integrals $\Phi _1,\ldots ,\Phi _k$ and $(n-k)$ vector fields $w_1=V,\ldots ,w_{n-k}$ such that

$$ \begin{align*} \begin{split} 0&=[w_i,w_j]\\ &=w_j[\Phi_i], \end{split} \end{align*} $$

with $1\leq i\leq k\leq n$ and $1\leq j \leq n-k$ . Note that $[\cdot ,\cdot ]:\mathfrak {g}\times \mathfrak {g}\rightarrow \mathfrak {g}$ denotes the Lie bracket for a vector space $\mathfrak {g}$ .

B-integrability was introduced by Bogoyavlenskij [Reference Bogoyavlenskij2] and it was shown that if a system is B-integrable, then it is integrable by quadrature. Intuitively, B-integrability is a generalisation of Liouvillian integrability from Hamiltonian systems to dynamical systems (see [Reference Bogoyavlenskij2]). Referring to Llibre et al. [Reference Llibre, Valls and Zhang11], a completely integrable system is orbitally equivalent to a linear differential system.

Let $\hat {\mathbf {P}}=\hat {\mathbf {P}}(t)$ denote a nonequilibrium solution of (3.1). A form of the variational equation was briefly introduced as (2.3). For the known particular solution, the variational equation along the phase curve $\Gamma $ of $\hat {\mathbf {P}}$ is

$$ \begin{align*} \dot{\nu}=JV_{\hat{\mathbf{P}}}\cdot\nu, \end{align*} $$

such that $\nu \in T_{\Gamma }\mathcal {M}$ and $T_{\Gamma }\mathcal {M}$ is a vector bundle of $T \mathcal {M}$ restricted on $\Gamma $ . With the normal bundle $N=T_{\Gamma } \mathcal {M}/\Gamma $ , a natural projection $\pi : T_{\Gamma }\mathcal {M}\rightarrow N$ and $\eta \in N$ , the variational equations may be reduced to

(3.2) $$ \begin{align} \dot{\eta}=\pi_{*}(T(V)(\pi^{-1} \eta)). \end{align} $$

The differential Galois group of (3.2) can be defined as a matrix group G, with $G\subset \mathrm{GL} (n-1,\mathbb {C})$ acting on the fundamental solutions of (3.2) such that it does not change polynomial and differential relations between them.

The following theorem is a powerful result which extends Galoisian obstructions to mermorphic integrability from Hamiltonian systems, provided in [Reference Morales-Ruiz13], to the more general non-Hamiltonian case.

The following theorem provides necessary conditions for (3.1) to possess a certain number of first integrals. In the proof of Theorem 3.7, we will show that (3.1) violates condition (1) and provide a criterion for when condition (3) is violated.

The point $\hat {\mathbf {P}}=(0,0,e^{-bt})$ is a nonequilibrium solution to (3.1). Hence, the variational equations along $\Gamma $ are

(3.3) $$ \begin{align} \begin{pmatrix} \dot{\nu} \\ \dot{\eta} \\ \dot{\vartheta} \end{pmatrix}=\begin{pmatrix} \sigma(\mathcal{F}'(0)-1) & \sigma & 0 \\ r-e^{-bt} & -1 & 0 \\ 0 & 0 & -b \end{pmatrix} \begin{pmatrix} \nu \\ \eta \\ \vartheta \end{pmatrix}, \end{align} $$

such that $\mathbf {P}=(\nu ,\eta ,\vartheta +e^{-bt})$ is in (3.1). Trivially, this may be reduced to the closed subsystem

$$ \begin{align*} \begin{split} \begin{pmatrix} \dot{\nu} \\ \dot{\eta} \\ \end{pmatrix}&=\begin{pmatrix} \sigma(\mathcal{F}'(0)-1) & \sigma\\ r-e^{-bt} & -1 \\ \end{pmatrix} \begin{pmatrix} \nu \\ \eta \\ \end{pmatrix}, \\ 0&=\ddot{\nu}+(\sigma-\sigma\mathcal{F}'(0)+1)\dot{\nu}-\sigma(r+\mathcal{F}'(0)-1-e^{-bt})\nu. \end{split} \end{align*} $$

Applying the change of variable $\tau =e^{-bt}$ , the above equation is expressed with rational coefficients:

$$ \begin{align*} \frac{d^2 \nu}{d\tau^2}+\frac{b+\sigma(\mathcal{F}'(0)-1)-1}{b\tau}\frac{d \nu}{d\tau}-\frac{\sigma(r+\mathcal{F}'(0)-\tau-1)}{b^2\tau^2}\nu=0. \end{align*} $$

Hence, letting $\nu (\tau )=\chi (\tau )\tau ^{(1-\sigma (\mathcal {F}'(0)-1)-b)/2b}$ ,

(3.4) $$ \begin{align} \frac{d^2\chi}{\mathop{d\tau^2}}=\frac{(\sigma-\sigma\mathcal{F}'(0)+1)^2+4\sigma(r+\mathcal{F}'(0)-1)-b^2-4\sigma\tau}{4b^2\tau^2}\chi. \end{align} $$

Using the convention of Kovačič [Reference Kovačič9], since the coefficient of $\chi $ is a rational function, when we refer to the poles of the coefficient, we mean the poles in $\mathbb {C}$ . If $r=z_1/z_2$ with $z_1,z_2\in \mathbb {C}[\tau ]$ relatively prime, then the poles of r are the zeros of $z_2$ and the order of the pole is the multiplicity of the zero of $z_2$ . By the order of the coefficient at $\infty $ , we mean the order of $\infty $ as a zero of the coefficient so that the order of the coefficient of $\chi $ at $\infty $ is $\deg z_2-\deg z_1$ .

Lemma 3.6. The identity element $G^0$ of G is solvable if and only if

$$ \begin{align*} \frac{2\sqrt{(\sigma-\sigma\mathcal{F}'(0)-1)^2+4\sigma r}}{|b|}\in\mathbb{Z}_{\text{odd}}. \end{align*} $$

Proof. Letting $s=-2\sqrt {\sigma \tau }/|b|$ and $\hat {\chi }=\chi /s$ , (3.4) becomes the Bessel equation

(3.5) $$ \begin{align} \frac{d^2 \hat{\chi}}{ds^2}+\frac{1}{s}\frac{d\hat{\chi}}{ds}+\bigg(1-\frac{\rho^2}{s^2}\bigg)\hat{\chi}=0, \end{align} $$

such that

$$ \begin{align*} \rho=\sqrt{\frac{(\sigma-\sigma\mathcal{F}'(0)-1)^2+4\sigma r}{b^2}}. \end{align*} $$

Since $\hat {\chi }'+s^{-1}\hat {\chi }=0$ has the nontrivial solution $s^{-1}\in \mathbb {C}$ , we have $G\subset \text {SL}(2,\mathbb {C})$ . Referring to [Reference Kolchin8], when $2\rho \in \mathbb {Z}_{\text {odd}}$ , we may (replacing $\rho$ by $-\rho$ if necessary) suppose that $\mu=\rho-\frac{1}{2}\geq 0$ so that

$$ \begin{align*} \psi_{\pm}=e^{\pm is}\sum_{0\leq k\leq \mu} \frac{(\mu+k)!}{k!(\mu-k)!}(\pm i)^k 2^{-k} s^{-k-{1}/{2}} \end{align*} $$

is a fundamental system of solutions to (3.5), such that each solution is exponential over $\mathbb {C}$ and their product is in $\mathbb {C}$ .

Hence, the differential Galois group G is a diagonal group

$$ \begin{align*} \mathcal{D}=\left.\left\{ \begin{pmatrix} a & 0\\ 0 & a^{-1} \end{pmatrix}\,\right|\, a\in\mathbb{C}, a\ne 0\right\}, \end{align*} $$

which has a solvable (abelian) identity element $G^0$ if $2\rho \in \mathbb {Z}_{\text {odd}}$ . Otherwise, when $2\rho \notin \mathbb {Z}_{\text {odd}}$ , [Reference Kolchin8] demonstrates that $G=\text {SL}(2,\mathbb {C})$ . Kovačič’s algorithm may also be applied to derive the Galois group of (3.5) (see [Reference Duval and Loday-Richaud5, Reference Kovačič9]).

Theorem 3.7. If (3.1) is B-integrable in the meromorphic category, then

(3.6) $$ \begin{align} r = r_n := \frac{(b(2n+1))^2-4(\sigma(\mathcal{F}'(0)-1)+1)^2}{16\sigma}, \end{align} $$

such that $\sigma ,b\neq 0$ and $n\in \mathbb {Z}$ .

Proof. Since the differential Galois group of (3.3) is a normal subgroup of the differential Galois group of (3.4), the differential Galois group of (3.3) is also infinite by Lemma 3.5. The identity element $G^0$ is a normal subgroup of G with finite index. Hence, $G^0$ is a trivial subgroup if and only if G is finite [Reference Morales-Ruiz13], which implies that $G^0\neq \{\mathbf {1}\}$ and (3.1) is not completely integrable with meromorphic first integrals by Theorem 3.4.

According to Lemma 3.6, if

(3.7) $$ \begin{align} \frac{2\sqrt{(\sigma-\sigma\mathcal{F}'(0)-1)^2+4\sigma r}}{|b|}\neq 2n+1, \end{align} $$

the identity elements of the differential Galois groups of (3.5) and (3.3) are not solvable. Therefore, the identity element $G^0$ of (3.3) is not abelian. After rearranging (3.7) and considering the contrapositive, the criterion in (3.6) follows trivially in view of Theorems 3.3 and 3.4.