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Ruelle-Pollicott Resonances of Stochastic Systems in Reduced State Space. Part II: Stochastic Hopf Bifurcation

Abstract

The spectrum of the generator (Kolmogorov operator) of a diffusion process, referred to as the Ruelle-Pollicott (RP) spectrum, provides a detailed characterization of correlation functions and power spectra of stochastic systems via decomposition formulas in terms of RP resonances; see Part I of this contribution (Chekroun et al. in Theory J Stat. https://doi.org/10.1007/s10955-020-02535-x, 2020). Stochastic analysis techniques relying on the theory of Markov semigroups for the study of the RP spectrum and a rigorous reduction method is presented in Part I Chekroun et al. (2020). This framework is here applied to study a stochastic Hopf bifurcation in view of characterizing the statistical properties of nonlinear oscillators perturbed by noise, depending on their stability. In light of the Hörmander theorem, it is first shown that the geometry of the unperturbed limit cycle, in particular its isochrons, i.e., the leaves of the stable manifold of the limit cycle generalizing the notion of phase, is essential to understand the effect of the noise and the phenomenon of phase diffusion. In addition, it is shown that the RP spectrum has a spectral gap, even at the bifurcation point, and that correlations decay exponentially fast. Explicit small-noise expansions of the RP eigenvalues and eigenfunctions are then obtained, away from the bifurcation point, based on the knowledge of the linearized deterministic dynamics and the characteristics of the noise. These formulas allow one to understand how the interaction of the noise with the deterministic dynamics affect the decay of correlations. Numerical results complement the study of the RP spectrum at the bifurcation point, revealing useful scaling laws. The analysis of the Markov semigroup for stochastic bifurcations is thus promising in providing a complementary approach to the more geometric random dynamical system (RDS) approach. This approach is not limited to low-dimensional systems and the reduction method presented in Chekroun et al. (2020) is applied to a stochastic model relevant to climate dynamics in the third part of this contribution (Tantet et al. in J Stat Phys. https://doi.org/10.1007/s10955-019-02444-8, 2019).

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Notes

  1. 1.

    Recall that a density \(\rho \) is a stationary solution if \(\mathcal {K}^* \rho = 0\), where \(\mathcal {K}^*\) denotes the (formal) adjoint of the Kolmogorov operator \(\mathcal {K}\); see e.g. [26, Sect. 2].

  2. 2.

    Recall that the eigenfunction associated with the first eigenvalue is constant [26, Definition 1.(i)], while the eigenfunction of the adjoint corresponds to the invariant measure.

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Acknowledgements

The programs used for this analysis are available as an open-source C++ library at https://github.com/atantet/ergoPack/ together with a link to its documentation.

The authors would like to thank the reviewers for their very useful and constructive comments. This work has been partially supported by the European Research Council under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 810370 (MDC)), by the Office of Naval Research (ONR) Multidisciplinary University Research Initiative (MURI) grant N00014-16-1-2073 (MDC), by the National Science Foundation grants OCE- 1658357, DMS-1616981(MDC), AGS-1540518 and AGS-1936810 (JDN), by the LINC project (No. 289447) funded by EC’s Marie-Curie ITN (FP7-PEOPLE-2011-ITN) program (AT and HD) and by the Utrecht University Center for Water, Climate and Ecosystems (AT).

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Appendices

A Floquet Theory Applied to the Hopf Normal Form

Floquet theory allows one to characterize the local stability properties of deterministic flows about a periodic orbit. These properties are essential to the response of the system to stochastic forcing studied in Section 3 and to the small-noise expansions of the RP spectrum obtained in Section 4. We thus review here standard results from the application of Floquet theory to the normal form (2.1) of the Hopf bifurcation.

Small deviations \(x'(t)\) from the orbit \(x_\varGamma (t)\) associated with the limit cycle \(\varGamma \), satisfy the variational equation ([56], Chap. 1.5)

$$\begin{aligned} \dot{x'}(t) = A(t) x'(t), \quad x'(t) \in {\mathbb {R}^2}, \quad t \in \mathbb {R}. \end{aligned}$$
(A.1)

Here, \(A(t): = (D F)_{x_\varGamma (t)}\) denotes the Jacobian matrix about the orbit \(x_\varGamma (t)\), of the vector field F associated with the Hopf normal form written in Cartesian coordinates, i.e. the RHS of Eq. (2.6) for \(\epsilon =0\). In other words, A(t) provides the tangent map of F along \(x_\varGamma (t)\). Thus, A is periodic, i.e., \(A(t + T) = A(t)\), for any t in \(\mathbb {R}\). Let M(t) be a fundamental solution ([46], Chap. IV.1) of (A.1), i.e.,

$$\begin{aligned} \dot{M}(t) = A(t) M(t) \quad \mathrm {and}~ \det M(t) \ne 0, \quad t \in \mathbb {R}. \end{aligned}$$
(A.2)

Then the Floquet theorem (e.g. [46], Theorem IV.6.1) ensures that M(t) has the following representation

$$\begin{aligned} M(t) = Z(t) e^{t R}, \quad \mathrm {where}~ Z(t + T) = Z(t), \quad t \in \mathbb {R}, \end{aligned}$$
(A.3)

and R is a constant matrix. Imposing, without loss of generality, that \(M(0) = I\) yields \(Z(T) = Z(0) = I\) and \(M(T) = e^{T R}\).

While determining the Floquet representation of a fundamental matrix is in general a difficult task, in the case of the Hopf normal form (2.1), it can easily be found from the linearization of the vector field in polar coordinates. In that respect, we assume furthermore that \(\delta \) in Eq. (2.1) is positive. The orbit \(x_\varGamma (t)\) writes then \((R, \theta _0 + \omega _f t)\), for some initial phase \(\theta _0\). The linearization about \(\varGamma \) of the vector field (2.14) in polar coordinates is given by the matrix

$$\begin{aligned} J_\varGamma (t) = \begin{pmatrix} -2\delta &{} 0 \\ -2\beta R &{} 0 \end{pmatrix} \end{aligned}$$
(A.4)

and depends implicitely on time only through the evolution of the tangent space on which it acts with the reference solution \(x_\varGamma (t)\), so that the time argument will be dropped in the sequel.

To proceed, let us introduce the Jacobian matrix of the transformation \((x, y) \rightarrow (r, \theta )\) and its inverse, respectively given by

$$\begin{aligned} J_{\mathrm{polar}}(r, \theta )&= \begin{pmatrix} \cos \theta &{} \sin \theta \\ - r^{-1} \sin \theta &{} r^{-1} \cos \theta \end{pmatrix} =~ S^{-1}(r) L(-\theta ), \quad r > 0 \\ J_{\mathrm{polar}}^{-1}(r, \theta )&= \begin{pmatrix} \cos \theta &{} - r \sin \theta \\ \sin \theta &{} r \cos \theta \end{pmatrix} =~ L(\theta ) S(r), \end{aligned}$$

where we have used the rotation and diagonal matrices

$$\begin{aligned} L(\theta ) = \begin{pmatrix} \cos \theta &{} -\sin \theta \\ \sin \theta &{} \cos \theta \end{pmatrix} \quad \mathrm {and} \quad S(r) = \begin{pmatrix} 1 &{} 0 \\ 0 &{} r \\ \end{pmatrix}. \end{aligned}$$

The matrix \(J_\varGamma \) is then related to the matrix A(t) of the tangent map \((DF)_{x_\varGamma (t)}\) in Cartesian coordinates by

$$\begin{aligned} A(t) = J_{\mathrm{polar}}^{-1}(R, \theta _0 + \omega _f t) ~ J_\varGamma ~ J_{\mathrm{polar}}(R, \theta _0) + \omega _f L\left( \frac{\pi }{2}\right) , \quad t \in \mathbb {R}. \end{aligned}$$
(A.5)

That the conversion of \(J_\varGamma \) to Cartesian coordinates coincides with the matrix A(t) of the tangent map but for the term \(\omega _f L(\pi /2)\) is due to the rotation of the polar frame along the limit cycle \(\varGamma \), which was not taken into account when calculating \(J_\varGamma \).

One can then verify that the matrix

$$\begin{aligned} M(t) = J_{\mathrm{polar}}^{-1}(R, \theta _0 + \omega _f t) ~ e^{t J_\varGamma } ~ J_{\mathrm{polar}}(R, \theta _0), \quad t \in \mathbb {R}, \end{aligned}$$
(A.6)

is a solution to (A.2), for the reference solution \(x_\varGamma (t)\) on \(\varGamma \). Since

$$\begin{aligned} {J_{\mathrm{polar}}^{-1}(R, \theta _0 + \omega _f t) = L(\omega _f t) ~ J_{\mathrm{polar}}^{-1}(R, \theta _0),} \end{aligned}$$

it follows that the fundamental matrix M(t) has a Floquet representation

$$\begin{aligned}&M(t) = Z(t) ~ e^{t R}, \quad t \in \mathbb {R}, \nonumber \\ \mathrm {with} Z(t) = L(\omega _f t) \quad&\mathrm {and} R = J_{\mathrm{polar}}^{-1}(R, \theta _0) ~ J_\varGamma ~ J_{\mathrm{polar}}(R, \theta _0). \end{aligned}$$
(A.7)

Applying M(t) to a vector \(x'\) at time 0 thus corresponds to converting this vector to polar coordinates, integrating to a time t according to the generator \(J_\varGamma \) and converting back from polar coordinates at time t. In other words, the polar frame at \(x_\varGamma (t) = (R, \theta _0 + \omega _f t)\) constitutes a co-moving frame adapted to the Floquet representation of M(t).

Note next that \(J_\varGamma \) can be diagonalized as

$$\begin{aligned}&J_\varGamma = E ~ \varLambda ~ F^* \nonumber \\ \mathrm {with}~ E = \begin{pmatrix} 1 &{} 0 \\ \frac{\tilde{\beta }}{R} &{} 1 \end{pmatrix}, \quad&\varLambda = \begin{pmatrix} -2 \delta &{} 0 \\ 0 &{} 0 \end{pmatrix} \quad \mathrm {and} \quad F^* = E^{-1} = \begin{pmatrix} 1 &{} 0 \\ -\frac{\tilde{\beta }}{R} &{} 1 \end{pmatrix}, \end{aligned}$$
(A.8)

where \(F^*\) denotes the complex conjugate of the matrix F. Then, from the definition (A.7) of R,

$$\begin{aligned}&R = E_R ~ \varLambda ~ F_R^* \nonumber \\ \mathrm {with}~ E_R = J_{\mathrm{polar}}^{-1}(R, \theta _0) ~ E \quad&\mathrm {and} \quad F^*_R = E_R^{-1} = F^* ~ J_{\mathrm{polar}}(R, \theta _0). \end{aligned}$$
(A.9)

Thus, the eigenvalues of R coincide with those of \(J_\varGamma \) and its eigenvectors are given by converting those of \(J_\varGamma \) from polar coordinates.

The eigenvalues \(\alpha _1\) and \(\alpha _2\) of R are called the characteristic exponents of \(\varGamma \) and the eigenvalues of \(e^{TR}\) its characteristic multipliers ([40], Chap. 1.5). The eigenvector associated with \(\alpha _2\) is in the direction of the flow, so that \(e^{T \alpha _2}\) is always unity. On the other hand, the other eigenvalue \(\alpha _1 = -2 \delta \) determines the stability of the periodic orbit. It is in fact the eigenvalue of the tangent map \(D S_T\) of the Poincaré map.

A.1 Calculation of the Phase Diffusion Coefficient from the Correlation Matrix

In the case of the stochastic Hopf bifurcation considered here, the diffusion matrix \(D_\varGamma \) in (6.2) for any point on \(\varGamma \) is given in polar coordinates by

$$\begin{aligned} D_\varGamma&= \begin{pmatrix} 1 &{} 0 \\ 0 &{} \frac{1}{R^2} \end{pmatrix}, \quad t \in \mathbb {R} \end{aligned}$$
(A.10)

and is hence constant in time. Since \(\mathbf {f}^R_2\) is a left eigenvector of the matrix \(M(T) = e^{TR}\) with R given by (A.7), it follows that

$$\begin{aligned} \varPhi = -\epsilon ^2 \omega _f^2 \left\langle D_\varGamma \mathbf {f}_2, \mathbf {f}_2 \right\rangle = -\epsilon ^2 \frac{1 + \tilde{\beta }^2}{R^2} = -\epsilon ^2 \frac{1 + \beta ^2}{\delta \kappa }, \end{aligned}$$
(A.11)

where \(\mathbf {f}_2 = \omega _f^{-1} (-\tilde{\beta } / R), 1)\) is the conversion to polar coordinates of the left eigenvector \(\mathbf {f}^R_2\) of R and, according to (A.9), coincides with the left eigenvector of the polar Jacobian matrix \(J_\varGamma \) in (2.15) at initial time. The factor \(\omega _f^{-1}\) in \(\mathbf {f}_2 = (-\tilde{\beta } / (\omega _f R), 1)\) is due to the normalization of \(\mathbf {e}^R_2\) to the magnitude of the vector field F on \(\varGamma \), which is essential for (6.1) to hold.

B Proofs of the Stochastic Analysis Results of Section 3

B.1 Proof of Theorem 1: Isochrons and Hörmander Condition

For two arbitrary smooth vector fields V and W, recall that the Lie bracket [VW] coincides with the Lie derivative \(\mathcal {L}_V W\) of W along V. The Lie derivative can be defined in terms of pullback of a vector field by a diffeomorphism. The pullback, or Lie transport, \((S_t^{V*} W)(q)\) at a point q of a vector field W by the flow \(S_t^{V}\) generated by V can be defined as the vector at q tangent to the image by \(S_{-t}^{V}\) of any curve to which \(W(S_t^{V} q)\) is tangent. The Lie derivative at a point q is then defined in terms of pullback of a vector field, by

$$\begin{aligned} \mathcal {L}_V W = \left. \frac{\, \mathrm {d}}{\, \mathrm {d}t}\right| _{0} S_t^{V*} W. \end{aligned}$$
(B.1)

This expression says that \(\mathcal {L}_V W\) measures the rate of change of W due to the Lie transport [36, Chap. 3-4]. Note that the Lie derivative is well defined because both the vector field at some point and its pullback at the same point live in the tangent space to the manifold at this point. The following is derived from the fact that the isochrons are permuted by the flow \(S_t\) generated by \(V_0\) (Proposition 2-(ii)): if a vector field \(V_i\) is tangent to an isochron \(W_{ss}(S_t p)\) at some point \(S_t q\), i.e. if \(V_i(S_t q) \in T W_{ss}(S_t p)\), where \(T W_{ss}(p)\) denotes the tangent space to \(W_{ss}(p)\), then its pullback to a point q in \( U_\varGamma \) by \(S_t\) is necessarily tangent to the isochron \(W_{ss}(p)\), i.e \(V_i(q) \in T W_{ss}(p)\). Thus, as a linear combination of vectors in the tangent space \(T W_{ss}(p)\), the Lie derivative \((\mathcal {L}_{V_0} V_i)(q) = [V_0, V_i](q)\) is also in \(T W_{ss}(p)\). The same argument holds for the Lie derivative \(\mathcal {L}_{V_i} V_j\) between two vector fields tangent to the isochrons everywhere in \(U_\varGamma \), with the difference that the vector fields are Lie transported along the same isochron, in this case. Lastly, any iteration of Lie brackets between the family \(\{V_i, 0 \le i \le m\}\), where \(V_0\) is the vector field of the deterministic system with a hyperbolic limit cycle and the \(\{V_i, 0 < i \le m\}\) are vector fields tangent to the isochrons of the limit cycle, yields the same outcome. It follows that

$$\begin{aligned} \cup _{k \ge 1} \mathrm {span}~\{V(q): V \in \mathcal {V}_k\} = TW_{ss}(p), \quad \text {for any} q \in U_\varGamma , \end{aligned}$$

where \(W_{ss}(p)\) is the isochron passing through q.

B.2 Proof of Proposition 3: Spectral Gap

Proposition 3 can be obtained as application of [26, Theorem 6] which provides conditions ensuring existence of a spectral gap and exponential decay of correlations. Since, as shown in Section 3.1, the Markov semigroup \((P_t)_{t\ge 0}\) associated with the SHE (2.6) is irreducible and strong Feller in it is thus sufficient to check the ultimate bound condition of [26, Theorem 6] to conlude, which we do hereafter.

More specifically, denoting by \(X_t^x\) the stochastic process solving the SHE (2.8) and emanating from \(x = (r, \theta )\), we show that there exists \(k, c, d > 0\) such that

$$\begin{aligned} \mathbb {E} |X_t^x|^2 = \mathbb {E}[r_t^2] < k r^2 e^{-c t} + d, t \ge 0, r \ge 0, \end{aligned}$$
(B.2)

for any value of the control parameters \(\delta \) in \(\mathbb {R}\), \(\beta \) in \(\mathbb {R}\), \(\kappa > 0\), and \(\epsilon > 0\).

As evolution of the observable \(\varphi (r,\theta )=r^2\) by the Markov semigroup \(P_t\), note that the function \(t \rightarrow \mathbb {E}[r_t^2]\) solves the Kolmogorov equation (2.9), which leads here to the differential equation

$$\begin{aligned} \frac{d}{dt} \mathbb {E}[r_t^2] = 2 \epsilon ^2 + 2 \left( \delta \mathbb {E}[r_t^2] - \kappa \mathbb {E}[r_t^4]\right) . \end{aligned}$$
(B.3)

To derive a bound (B.2), we bound the right-hand side of the ODE (B.3) in \(\mathbb {E}[r_t^2]\) and to apply comparison results of Gronwall-Bellman-Bihari type; see e.g. [13].

For \(\delta < 0\), below the bifurcation, the estimate

$$\begin{aligned} \frac{d}{dt} \mathbb {E}[r_t^2] \le 2 \epsilon ^2 + 2 \delta \mathbb {E}[r_t^2], \end{aligned}$$

holds, since \(\mathbb {E}[r_t^{4}] > 0\). It follows from the standard Gronwall inequality for linear differential inequalities (e.g. [13, Chap. 1, Lemma 1.1]) that

$$\begin{aligned} \mathbb {E}[r_t^2] \le r^2 e^{2\delta t} + \frac{\epsilon ^2}{\delta } (1 - e^{2\delta t}) \le r^2 e^{2\delta t} + \frac{\epsilon ^2}{\delta }, \quad t \ge 0. \end{aligned}$$
(B.4)

Thus, one can choose \(k = 1\), \(c = -2\delta \) and \(d > -\epsilon ^2 / \delta \), for the ultimate bound (B.2) to be satisfied.

Next, for \(\delta \ge 0\), above the bifurcation, (B.3) is equivalent to

$$\begin{aligned} \frac{d}{dt} \mathbb {E}[r_t^2] = 2 \epsilon ^2 - 2 \mathbb {E}\left[ r_t^2 (\kappa r_t^{2} - \delta )\right] , \end{aligned}$$

and it follows, by applying Jensen’s inequality (e.g. [51], Lemma 2.5), that

$$\begin{aligned} \frac{d}{dt} \mathbb {E}[r_t^2] \le 2 \epsilon ^2 - 2 \mathbb {E}[r_t^2] (\kappa \mathbb {E}[r_t^{2}] - \delta ), \quad t \ge 0. \end{aligned}$$
(B.5)

A classical comparison theorem on differential inequalities [13, Chap. 2, Theorem 6.3] ensures that the inequality (B.5) implies boundedness from above of the 2nd moment \(\mathbb {E}[r_t^2]\) by a maximal solution y of the scalar ODE

$$\begin{aligned} y'(t) = 2 \epsilon ^2 - 2 y(t) (\kappa y(t) - \delta ), \quad y(0) = r^2, \quad t \ge 0. \end{aligned}$$

By solving this equation, one finds the maximal solution

$$\begin{aligned} y(t) = R_\epsilon (\delta , \kappa )^2 - \frac{w \sqrt{\varDelta }}{w - \exp \left( 2 \sqrt{\varDelta } t \right) }, \quad t \ge 0, \end{aligned}$$

where w is a constant of integration, \(\varDelta = R^2 + 4 \epsilon ^2 / \kappa \) and \(R_\epsilon (\delta , \kappa )^2 = (R + \sqrt{\varDelta }) / 2\) is the equilibrium to which y(t) converges as t goes to infinity. For the initial condition \(y(0) = r^2\), one finds

$$\begin{aligned} w = (r^2 - R_\epsilon (\delta , \kappa )^2) (r^2 - R_\epsilon (\delta , \kappa )^2 + \sqrt{\varDelta })^{-1}, \end{aligned}$$

Let us look for exponential bounds on y(t). First,

$$\begin{aligned} r \le R_\epsilon (\delta , \kappa ) \Rightarrow w \le 0 \Rightarrow y(t) \le R_\epsilon (\delta , \kappa )^2, \quad \text {for} t \ge 0, \end{aligned}$$

while

$$\begin{aligned} r&\ge R_\epsilon (\delta , \kappa ) \Rightarrow 0 \le w \le 1 \Rightarrow y(t) \\&\le R_\epsilon (\delta , \kappa )^2 + (r^2 - R_\epsilon (\delta , \kappa )^2) \exp \left( -2 \sqrt{\varDelta } t \right) , \quad \text {for} t \ge 0. \end{aligned}$$

We have thus shown that the second moment \(\mathbb {E}[r_t^2]\) satisfies the inequality

$$\begin{aligned} \mathbb {E}[r_t^2] \le y(t) \le R_\epsilon (\delta , \kappa )^2 + r^2 \exp \left( -2 \sqrt{\varDelta } t \right) , \quad r \ge 0, \quad t \ge 0. \end{aligned}$$

Thus, for \(\delta \ge 0\) and \(\epsilon > 0\), the second moment satisfies the ultimate bound (B.2) with \(k = 1\), \(c = 2 \sqrt{\varDelta }\) and \(d = R_\epsilon (\delta , \kappa )^2\). This estimate is valid even at the critical value 0 of \(\delta \), as long as the noise level \(\epsilon \) is nonzero. In this case, the exponential decay rate \(a = 4 \epsilon \) is proportional to the noise level.

C Proofs of the Small-Noise Expansions of Section 4

C.1 Proof of Proposition 4: Expansions for \(\delta < 0\) About the Stationary Point

We proceed to the small-noise expansion of the Kolmogrov equation corresponding to the SHE (4.1) in adimensional Cartesian coordinates, \(x' = x / L_\epsilon (\delta )\), \(y' = y / L_\epsilon (\delta )\) and \(t' = \delta t\),

$$\begin{aligned} \partial _{t'} u&= \left[ \left( -1 - \sigma _\epsilon ^2 \left( x'^2 + y'^2\right) \right) x' - \left( \tilde{\gamma } - \tilde{\beta } \sigma _\epsilon ^2 \left( x'^2 + y'^2\right) \right) y'\right] \partial _{x'} u + \frac{1}{2} \partial ^2_{x'x'} u\\&\quad + \left[ \left( \tilde{\gamma } - \tilde{\beta } \sigma _\epsilon ^2 \left( x'^2 + y'^2\right) \right) x' + \left( -1 - \sigma _\epsilon ^2 \left( x'^2 + y'^2\right) \right) y\right] \partial _{y'} u + \frac{1}{2} \partial ^2_{y'y'} u. \end{aligned}$$

Since the small parameter \(\sigma _\epsilon = 1 / r_\epsilon \) appears squared only, we can expand the eigenvalues and eigenfunctions in \(\sigma _\epsilon ^2\). To zeroth order, we have

$$\begin{aligned} \lambda ^{(0)} \psi ^{(0)}&= \mathcal {K}_{x_*}^{(0)} \psi ^{(0)}, \nonumber \\ \mathrm {with}\mathcal {K}_{x_*}^{(0)}&= \left( -x' - \tilde{\gamma } y'\right) \partial _{x'} + \left( \tilde{\gamma } x' - y'\right) \partial _{y'} + \frac{1}{2} \partial _{x'x'} + \frac{1}{2} \partial _{y'y'} \end{aligned}$$
(C.1)

This equation yields to the eigenvalue problem of a two-dimensional nonsymmetric Ornstein-Uhlenbeck process with Kolmogorov operator \(\mathcal {K}_{x_*}^{(0)}\). Its linear drift and diffusion have the following matrix representation in adimensional Cartesian coordinates \((x', y')\):

$$\begin{aligned} J_{x_*} = \begin{pmatrix} -1 &{} -\tilde{\gamma } \\ \tilde{\gamma } &{} -1 \end{pmatrix}, \quad \text {and} \quad D = \frac{1}{2} I. \end{aligned}$$

Here, \(J_{x_*}\) corresponds also to the tangent map at the origin of the vector field F associated with the Hopf normal form (2.6) for \(\epsilon = 0\), while I denotes the \(2\times 2\) identity matrix. The stationary density of this Ornstein-Uhlenbeck process is given in adimensional polar coordinates \((r', \theta ')\) by

$$\begin{aligned} \rho _{x_*}(r') = \frac{1}{\pi } r' e^{-r'^2}. \end{aligned}$$
(C.2)

For the weighted inner-product \(\langle \cdot ,\cdot \rangle _{\rho _{x_*}}\) with respect to this density, the Kolmogorov operator associated with this Ornstein-Uhlenbeck process is asymmetric. This asymmetry comes from the anti-symmetry of the rotation operator

$$\begin{aligned} \varOmega = -\tilde{\gamma } y' \partial _{x'} + \tilde{\gamma } x' \partial _{y'}, \end{aligned}$$

i.e. \(\langle \varOmega f, g \rangle _{\rho _{x_*}} = -\langle f, \varOmega g\rangle _{\rho _{x_*}}\), while the operator

$$\begin{aligned} -x' \partial _{x'} - y' \partial _{y'} + \frac{1}{2} \partial _{x'x'} + \frac{1}{2} \partial _{y'y'}, \end{aligned}$$

encapsulating the diffusion and contraction effects, is symmetric.

The RP spectrum of one-dimensional Ornstein-Uhlenbeck processes is well studied (see e.g. [72, Chap. 5]). In several dimensions, the more recent work [64] shows that the spectrum of an Ornstein-Uhlenbeck process is discrete and composed of eigenvalues —corresponding here to the set of \(\lambda _k^{(0)}\) solving (C.1) with the \(\psi _k^{(0)}\) in \(L^2_{\rho _{x_*}}(\mathbb {R}^2)\) — are given by integer linear combinations of the eigenvalues of the drift matrix \(J_{x_*}\), i.e. the complex conjugate pair \(-1 \pm i \tilde{\gamma }\), in our case. In dimensional terms, the eigenvalues of the SHE (2.6) are thus given to first order by the combiations \((l + n) \delta + i (n - l) \gamma \), with \(n, l \in \mathbb {N}\), which coincides with the eigenvalues of the deterministic normal form (2.1); c.f. [38].

In addition, it has recently been shown by [19] that the solutions to (C.1) are given by products of Laguerre polynomials with harmonic functions. In adimensional polar coordinates \((r', \theta ')\) this yields in our case,

$$\begin{aligned} \psi _{ln}^{(0)}(r', \theta ') = {\left\{ \begin{array}{ll} e^{i(n-l)\theta '} \sqrt{\frac{l!}{n!}} \left( r'\right) ^{n-l} L_l^{n-l}\left( -r'^2\right) , \quad &{}n \ge l \\ e^{i(l-n)\theta '} \sqrt{\frac{n!}{l!}} \left( r'\right) ^{l-n} L_n^{l-n}\left( -r'^2\right) , \quad &{}n < l, \end{array}\right. } \end{aligned}$$

or in dimensional polar coordinates \((r, \theta )\),

$$\begin{aligned} \psi _{ln}^{(0)}(r, \theta ) = {\left\{ \begin{array}{ll} e^{i(n-l)\theta } \sqrt{\frac{l!}{n!}} \left( \sqrt{-\frac{\delta }{\epsilon ^2}} r\right) ^{n-l} L_l^{n-l}\left( -\frac{\delta r^2}{\epsilon ^2}\right) , \quad &{}n \ge l \\ e^{i(l-n)\theta } \sqrt{\frac{n!}{l!}} \left( \sqrt{-\frac{\delta }{\epsilon ^2}}r\right) ^{l-n} L_n^{l-n}\left( -\frac{\delta r^2}{\epsilon ^2}\right) , \quad &{}n < l. \end{array}\right. } \end{aligned}$$

From the orthogonality of the Laguerre polynomials [59, p. 84] and of the harmonic functions, it follows that the appropriately normalised eigenfunctions form a complete orthonormal family of \(L^2_{\rho _{x_*}}(\mathbb {R}^2)\). The product of these eigenfunctions with the density \(\rho _{x_*}\) thus yield the eigenfunctions of the Fokker-Planck equation dual to the Kolmogorov equation (C.1).

To first order in \(\sigma _\epsilon ^2\),

$$\begin{aligned} \lambda ^{(0)} \psi ^{(1)} + \lambda ^{(1)} \psi ^{(0)}&= \left[ -\left( x'^2 + y'^2\right) x' + \tilde{\beta } \left( x'^2 + y'^2\right) y'\right] \partial _{x'} \psi ^{(0)}\\&\quad + \left[ -\tilde{\beta } \left( x'^2 + y'^2\right) x' -\left( x'^2 + y'^2\right) y'\right] \partial _{y'} \psi ^{(0)} \\&\quad + \left( -x' - \tilde{\gamma } y'\right) \partial _{x'} \psi ^{(1)} + \left( \tilde{\gamma } x' - y'\right) \partial _{y'} \psi ^{(1)} + \frac{1}{2} \partial _{x'x'} \psi ^{(1)} \\&\quad + \frac{1}{2} \partial _{y'y'} \psi ^{(1)} \end{aligned}$$

Thus the magnitude of this term depends on the twist factor \(\tilde{\beta } = \beta / \kappa \). For this reason, we use the asymptotic notation \(\mathcal {O}_{\tilde{\beta }}((\epsilon \sqrt{\kappa } / \delta )^2)\) to represent it.

C.2 Proof of Proposition 5: Expansions for \(\delta > 0\) About the Limit Cycle \(\varGamma \)

We are here interested in the finding the leading eigenvalues and eigenfunctions originating from the ruins of the deterministic limit cycle \(\varGamma \) when \(\sigma _\epsilon = \frac{1}{r_\epsilon }\) is small. We thus proceed to an additional change of variables from the adimensional coordinates \((r', \phi ')\) to a frame centered on \(\varGamma \) and rotating at the angular frequency \(\tilde{\omega }_f = \tilde{\gamma } - \tilde{\beta }\) of the adimensional deterministic dynamics on \(\varGamma \),

$$\begin{aligned} \hat{r}&= r' - r_\epsilon = r' - \sigma _\epsilon ^{-1}\\ \hat{\phi }&= \phi ' + \tilde{\omega }_f t'. \end{aligned}$$

The (4.1) then reads in \((\hat{r}, \hat{\phi })\) coordinates,

$$\begin{aligned} \, \mathrm {d}\hat{r}&= \left( \hat{r} + \sigma _\epsilon ^{-1}\right) \left( 1 - \left( \sigma _\epsilon \hat{r} + 1\right) ^2 + \frac{\sigma _\epsilon ^2}{2\left( \sigma _\epsilon \hat{r} + 1\right) ^2}\right) \, \mathrm {d}t' + \, \mathrm {d}W_r \nonumber \\ \mathrm {or} \, \mathrm {d}\hat{\phi }&= - \tilde{\beta } \frac{\sigma _\epsilon }{\left( \sigma _\epsilon \hat{r} + 1\right) } \, \mathrm {d}W_r + \frac{\sigma _\epsilon }{\left( \sigma _\epsilon \hat{r} + 1\right) }\, \mathrm {d}W_\theta , \end{aligned}$$
(C.3)

and the corresponding Kolmogorov equation, with \(\hat{u}(\hat{r}, \hat{\phi }) = u(r, \phi )\), is,

$$\begin{aligned} \partial _{t'} \hat{u}&= \left( \hat{r} + \sigma _\epsilon ^{-1}\right) \left( 1 - \left( \sigma _\epsilon \hat{r} + 1\right) ^2 + \frac{\sigma _\epsilon ^2}{2\left( \sigma _\epsilon \hat{r} + 1\right) ^2}\right) \partial _{\hat{r}} \hat{u} \\&\quad + \frac{1}{2} \partial _{\hat{r} \hat{r}} \hat{u} - \tilde{\beta } \frac{\sigma _\epsilon }{\left( \sigma _\epsilon \hat{r} + 1\right) } \partial _{\hat{r} \hat{\phi }} \hat{u} + \frac{\sigma _\epsilon ^2 (1 + \tilde{\beta }^2)}{2 \left( \sigma _\epsilon \hat{r} + 1\right) ^2} \partial _{\hat{\phi } \hat{\phi }} \hat{u}. \end{aligned}$$

In this case, we have no choice but to expand the eigenvalues and eigenfunctions in \(\sigma _\epsilon \). We have that

$$\begin{aligned} \frac{\sigma _\epsilon }{\sigma _\epsilon \hat{r} + 1}&= \sigma _\epsilon - \sigma _\epsilon ^2 \hat{r} + \sigma _\epsilon ^3 \hat{r}^2 + \mathcal {O}\left( \sigma _\epsilon ^4\right) \\ \frac{\sigma _\epsilon ^2}{2 \left( \sigma _\epsilon \hat{r} + 1\right) ^2}&= \frac{\sigma _\epsilon ^2}{2} - \sigma _\epsilon ^3 r + \mathcal {O}\left( \sigma _\epsilon ^4\right) , \end{aligned}$$

and the radial component of the drift expands as

$$\begin{aligned} - 2 \hat{r} + \sigma _\epsilon \left( \frac{1}{2} - 3 \hat{r}^2\right) - \sigma _\epsilon ^2 \left( \frac{\hat{r}}{2} + \hat{r}^3\right) + \mathcal {O}\left( \sigma _\epsilon ^3\right) . \end{aligned}$$

The terms of order \(1 / \sigma _\epsilon \), which are associated with the deterministic solution on the limit cycle, vanish.

The eigenvalue equation yields to zeroth order,

$$\begin{aligned} \lambda ^{(0)} \psi ^{(0)} = -2 \hat{r} \partial _{\hat{r}} \psi ^{(0)} + \frac{1}{2} \partial ^2_{\hat{r} \hat{r}} \psi ^{(0)} \end{aligned}$$
(C.4)

This Hermite equation in the \(\hat{r}\)-coordinate corresponds to the eigenvalue problem of a one-dimensional stable Ornstein-Uhlenbeck process (see e.g. [66]) with damping coefficient given by the dimensional Floquet exponent \(-2 \delta \) associated with the dimensional Floquet vector \(\mathbf {e}_1 = (1, \tilde{\beta } / R)\) transverse to \(\varGamma \); see Section 2 and Appendix A. The stationary density for this one-dimensional Ornstein-Uhlenbeck process is given in adimensional polar coordinates \((r', \theta ')\) by

$$\begin{aligned} \rho _{\varGamma }(r) = \frac{1}{2\pi }\sqrt{\frac{2}{\pi \epsilon ^2}} e^{-2 (r' - r_\epsilon )^2}. \end{aligned}$$
(C.5)

The solutions to the eigenproblem (C.4) for any \(\lambda _l^{(0)} = -2 l,\)l in \(\mathbb {N}\), are given by the rescaled Hermite polynomials [72, Chap. 5.5]

$$\begin{aligned} \psi _{l}^{(0)}(\hat{r}, \hat{\phi }) = \eta (\hat{\phi }) H_l(\sqrt{2} \hat{r}), \end{aligned}$$
(C.6)

where \(H_l\) is the \(l^{th}\) Hermite polynomial [59, p. 60] and \(\eta \) is some function of \(\hat{\phi }\) only.

The function \(\eta \) in (C.6) is determined for \(l = 0\) by solving for the higher-order equations. In general, the first and second-order terms of the expansion yield,

$$\begin{aligned} \mathcal {O}(\sigma _\epsilon )&: \quad \lambda ^{(0)} \psi ^{(1)} + \lambda ^{(1)} \psi ^{(0)} = \left( \frac{1}{2} - 3 \hat{r}^2\right) \partial _{\hat{r}} \psi ^{(0)} - \tilde{\beta } \partial ^2_{\hat{r} \hat{\phi }} \psi ^{(0)} -2 \hat{r} \partial _{\hat{r}} \psi ^{(1)} + \frac{1}{2} \partial ^2_{\hat{r} \hat{r}} \psi ^{(1)}\\ \mathcal {O}(\sigma _\epsilon ^2)&: \quad \lambda ^{(0)} \psi ^{(2)} + \lambda ^{(1)} \psi ^{(1)} + \lambda ^{(2)} \psi ^{(0)} = -\left( \frac{\hat{r}}{2} + \hat{r}^3\right) \partial _{\hat{r}} \psi ^{(0)} - \tilde{\beta } \hat{r} \partial ^2_{\hat{r}\hat{\phi }} \psi ^{(0)}\\&\qquad + \frac{1 + \tilde{\beta }^2}{2} \partial _{\hat{\phi } \hat{\phi }} \psi ^{(0)} + \left( \frac{1}{2} - 3 \hat{r}^2\right) \partial _{\hat{r}} \psi ^{(1)} - \tilde{\beta } \partial ^2_{\hat{r} \hat{\phi }} \psi ^{(1)} -2 \hat{r} \partial _{\hat{r}} \psi ^{(2)} + \frac{1}{2} \partial ^2_{\hat{r} \hat{r}} \psi ^{(2)} \end{aligned}$$

The special case \(\psi ^{(0)} = \psi _{l}^{(0)}\) with \(l = 0\) is such that \(\partial _{\hat{r}} \psi ^{(0)} = 0\). Thus, for \(\psi ^{(1)} = \psi ^{(2)} = 0\), to first and second order,

$$\begin{aligned} \mathcal {O}\left( \sigma _\epsilon \right)&: \quad \lambda ^{(1)} \psi ^{(0)} = 0 \nonumber \\ \mathcal {O}\left( \sigma _\epsilon ^2\right)&: \quad \lambda ^{(2)} \eta = \frac{1 + \tilde{\beta }^2}{2} \partial _{\hat{\phi } \hat{\phi }} \eta . \end{aligned}$$
(C.7)

The first equation in (C.7) implies that \(\lambda ^{(1)} = 0\), while the second equation corresponds to the eigenproblem for pure diffusion on the circle with diffusion coefficient \((1 + \tilde{\beta }^2) / 2\). Its solutions for \(\lambda _n^{(2)} = -n^2 (1 + \tilde{\beta }^2) / 2, n \in \mathbb {Z}\) are given by the harmonics \(\eta _{\pm n} = \exp {(\pm i n \hat{\phi })}\), such that \(\psi _{l, \pm n}^{(0)} = H_l(\sqrt{2} \hat{r}) \exp {(\pm in\hat{\phi })}\).

Unfolding the change of variables, \(\hat{r} = \sqrt{\delta } (r - R) / \epsilon , \hat{\phi } = \theta - \tilde{\beta } \log (r / R) + \omega _f t\) and \(t' = \delta t\), yields the small-noise expansion from Proposition 5 of the eigenvalues and eigenfunctions of the SHE (2.8) for \(\delta > 0\) and \(\sigma _\epsilon \) small. A term \(\exp {(i \omega _f t)}\) appears in front of the eigenfunctions that can be canceled out since multiples of eigenfunctions are also eigenfunctions. To find the adjoint eigenfunctions \(\psi ^{(0)*}_{ln}\), orthonormal to the eigenfunctions \(\psi ^{(0)}_{ln}, l \in \mathbb {N}, n \in \mathbb {Z}\), one uses the orthogonality of the Hermite polynomials [59, p. 65]. Finally, note that higher-order terms in the expansion depend on \(\tilde{\beta }\).

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Tantet, A., Chekroun, M.D., Dijkstra, H.A. et al. Ruelle-Pollicott Resonances of Stochastic Systems in Reduced State Space. Part II: Stochastic Hopf Bifurcation. J Stat Phys 179, 1403–1448 (2020). https://doi.org/10.1007/s10955-020-02526-y

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Keywords

  • Ruelle-Pollicott resonances
  • Stochastic bifurcation
  • Markov semigroup
  • Stochastic analysis
  • Ergodic theory