The Hopf bifurcation with bounded noise

We study Hopf-Andronov bifurcations in a class of random differential equations (RDEs) with bounded noise. We observe that when an ordinary differential equation that undergoes a Hopf bifurcation is subjected to bounded noise then the bifurcation that occurs involves a discontinuous change in the Minimal Forward Invariant set.


Introduction
We will consider Hopf-Andronov bifurcations in a class of random differential equations (RDEs) (1)ẋ = f λ (x) + εξ t as the parameter λ ∈ R is varied. Here x will belong to the plane and ξ t will be a realization of some noise. We are interested in bounded noise in which ξ t takes values in a closed disk ∆ ⊂ R 2 . More specifically we will consider Hopf-Andronov bifurcations with radially symmetric noise where ξ t takes values in the unit disk. The RDEs without noise, for ε = 0, unfold a Hopf-Andronov bifurcation as λ varies. We denote by U the collection of all possible realizations of the noise. We assume that f and U are sufficiently well-behaved that the equations (1) uniquely defines a flow Φ t (x, ξ) for all realizations ξ of the noise.
For the general framework of random dynamical systems we refer the reader to L. Arnold's book [1] (see also [5,11]). A distinctive feature of dynamical systems with bounded noise is that they may admit more than one stationary measure.
For discrete time Markov processes these measures were studied by Doob [7], who showed that their supports are precisely the minimal forward invariant (MFI) sets. In [9] the authors adapted Doob's proof to the case of the continuous time (not necessarily Markov) processes generated by RDEs on compact manifolds. It was observed ( [9,14]) that these stationary measures can experience dramatic changes, such as a change in the number of stationary measures or a discontinuous change in one of the supports of densities. We refer to such changes as hard bifurcations. Given the one-to-one correspondence between stationary measures and the MFI sets on which they are supported, in order to study hard bifurcations, it is sufficient to study the bifurcations of MFI sets themselves.
We adopt from [10] the following assumptions on (1) and its flow: The set ∆ is a closed disk with smooth boundary. For each x the map H2. There exist r 0 > 0 and t 1 > 0 such that In [10], under these noise conditions the authors provided a complete classification of bounded noise co-dimension one hard bifurcations in phase space dimensions 1 and 2. We call a set F ⊂ X forward invariant if (2) Φ t λ (F, U) ⊂ F for all t ∈ R + . Denote by F the collection of forward invariant sets. There is a partial ordering on F by inclusion, i.e. E F if E ⊂ F . We call E ⊂ F a minimal forward invariant (MFI) set if it is minimal with respect to the partial ordering . In this context, MFI sets were shown to exist in [9]. It follows easily from the definitions that an MFI set for (1) is open and connected and that the closures of distinct MFI sets are disjoint.
A natural assumption is further that we are given a θ t -invariant, ergodic probability measure P on U with the following hypothesis: H3. There exist t 2 > 0 so that Φ t λ (x, ·) * P is absolutely continuous w.r.t. a Riemannian measure m on X for all t > t 2 and all x ∈ X.
Under this assumption, the closures of MFIs are supports of stationary densities [9]. We note that the concept of MFI set is the same as the concept of positively invariant maximal control sets defined in the context of control theory [6]. We now define bifurcation of MFI sets. Definition 1.1. A bifurcation of MFI sets is said to occur in a parameterized family of random differential equations if either: B1 The number of MFI sets changes. B2 A MFI set changes discontinuously with respect to the Hausdorff metric.
In a supercritical Hopf bifurcation taking place in (1) for ε = 0, a stable limit cycle appears in the bifurcation at λ = 0. For a fixed negative value of λ, the differential equations without noise possess a stable equilibrium and the RDE with small noise has an MFI which is a disk around the equilibrium. Likewise, at a fixed positive value of λ for which (1) without noise possesses a stable limit cycle, small noise will give an annulus as MFI. A bifurcation of stationary measures takes place when varying λ. We will prove the following bifurcation scenario for small ε > 0: the RDE (1) undergoes a hard bifurcation of type B2 in which a globally attracting MFI set changes discontinuously, by suddenly developing a "hole". This hard bifurcation takes place at a delayed parameter value λ = O(ε 2/3 ). We include a brief discussion of attracting random cycles. Finally we demonstrate the results with numerical experiments on a radially symmetric Hopf bifurcation with bounded noise.
For recent studies of Hopf bifurcations in stochastic differential equations (SDEs) we refer to [4,3,2,13]. In such systems there is a unique stationary measure, with support equal to the entire state space. Bifurcations of supports of stationary measures, as arising in RDEs with bounded noise, do not occur in the context of SDEs.

Random perturbations of a planar Hopf-Andronov bifurcation
Consider a smooth family of planar random differential equations where λ ∈ R is a parameter and u, v are noise terms from ∆ = {u 2 + v 2 ≤ 1}, representing radially symmetric noise. Hypothesis H1 is therefore fulfilled, we consider noise such that also H2 is satisfied. We assume that without the noise terms, i.e. for ε = 0, the family of differential equations unfolds a supercritical Hopf bifurcation at λ = 0 [12]. The Hopf bifurcation in planar RDEs with small bounded noise is described in the following result.
Theorem 2.1. Consider a family of RDEs (3) depending on one parameter λ, that unfolds, when ε = 0, a supercritical Hopf bifurcation at λ = 0. For small ε > 0 and λ near 0, there is a unique MFI set E λ . There is a single hard bifurcation at λ bif = O(ε 2/3 ) as ε ↓ 0. At λ = λ bif the MFI set E λ changes from a set diffeomorphic to a disk for λ < λ bif to a set diffeomorphic to an annulus for λ ≥ λ bif . At λ bif the inner radius of this annulus is r * = O(ε 1/3 ).
Proof. We first bring the system without the noise terms, into normal form. Note that a smooth coordinate transformation h λ and a multiplication by a smooth positive function α λ (which is equivalent to a time reparameterization), change (4) into A smooth normal form transformation consists of a smooth coordinate transformation h λ , a time reparameterization, and an additional reparametrization of the parameter λ. There exists a smooth normal form transformation that changes (4) into ẋ see [12] (here r = x 2 + y 2 ). Applying this normal form transformation to (3) yields where A(x, y, λ) = A 0 (λ) + O(r) is a two by two matrix that depends smoothly on its arguments. Note that expression (5) implies that the noise terms (u, v) are the same as in (3); they only get multiplied by a matrix A(x, y, λ). Changing to polar coordinates we finḋ We perform a rescaling in ε by putting r = ε 1/3r and λ = ε 2/3λ . This bringṡ Multiplying by a factor ε −2/3 and taking the limit for the radial component.
Noting that (u, v) lies in a unit disk, A(u, v) lies in an ellipse. If we let a and b be the major and minor axes of this ellipse, then by a rotation of coordinates we may transform the last equation intȱ r =λr −r 3 + au cos θ + bv sin θ.
In [10] the authors showed that the boundary of an MFI set consists of solutions of the extremal vector fields defined by the bounded noise differential equations. Observe that for ε = 0, (8) readsθ = 1 andṙ = 0 and its right hand varies continuously with ε. Hence for ε small, ther components of the extremal vector fields are approximately the extremal values of ther component. A simple calculation shows that the extremal values of the last two terms in (9) are given by: Thus the boundaries of all solutions of (9) are given by the solution r ± of the equations: ± ± a 2 cos 2 θ + b 2 sin 2 θ. Now note thatθ is of order ε −2/3 . This allows us to average equations (10) to obtain the averaged equations: Note that the integral may be transformed into: where E(k) ≡ π/2 0 1 − k 2 sin 2 θ dθ is the complete elliptic integral of the second kind. Thus the averaged equations for r ± are simply: The analysis of the MFI bifurcation is now straightforward. First consider the equation forr + . For all parameters, this has a hyperbolic attracting fixed point at the largest real solution of r 3 − λr − c = 0. According to Theorem 4.1.1 in [8] the original equation has a hyperbolic attracting periodic orbit that passes close to the hyperbolic fixed point on a Poincaré section. This periodic orbit γ + encloses an MFI set. Further it is can be shown that any orbit beginning outside of γ + will converge to the MFI set.
Next consider the equation forr − . A saddle-node bifurcation occurs in the averaged equation when (13) in which a stable and an unstable fixed point are formed. It follows from Theorem 4.3.1 in [8] that the Poincaré map for original equation also undergoes a saddle-node bifurcation near the one for the averaged equation. Thus a stable periodic orbit γ − for r − is created. The stable orbits γ + for r + and γ − for r − enclose an annular MFI set. We summarize the above calculations. For λ < λ bif the disk region inside γ + is a minimal forward invariant set. For λ ≥ λ bif the MFI set is the annular region bounded by γ − and γ + . Thus the MFI set changes discontinuously at λ bif and so a hard bifurcation of type B2 occurs.

Random cycles
The deterministic Hopf bifurcation involves the creation of a limit cycle. In this section we comment on the occurrence of attracting random cycles.
Random cycles are defined in analogy with random fixed points [1]; for its definition we need to consider the noise realizations ξ and the flow Φ t λ (x, ξ) for two sided time t ∈ R. We henceforth consider the skew product flow We formulate a result with a relaxed notion of attracting random cycle, as we allow for time reparameterizations. The notion implies though that a Poincaré return map on a section {θ = 0} has an attracting random fixed point.
Recall that a random fixed point is a map R : U → R 2 that is flow invariant, Φ t λ (R(ξ), ξ) = R(θ t ξ) for P almost all ξ. A random cycle is defined as a continuous map S : U × S 1 → R 2 that gives an embedding of a circle for P almost all ξ ∈ U and is flow invariant in the sense Different regularities of the embeddings S 1 → S(ξ, S 1 ) may be considered.
The following result establishes the occurrence of attracting random cycles following the hard bifurcation, for small noise amplitudes. We confine ourselves with a statement on continuous random cycles, thus avoiding for instance constructions of invariant cone fields to prove Lipschitz continuity.
Theorem 3.1. For values of (λ, ε) with λ > λ bif and ε small, the MFI E λ is diffeomorphic to an annulus and the flow Φ t λ admits a random cycle S : The random cycle is attracting in the sense that there is a neighborhood U λ of the MFI E λ , so that for all x ∈ U λ , the distance between (Φ t λ (x, ξ), t) and t∈R (S(θ t ξ, S 1 ), t) goes to zero as t → ∞. Proof. Recall from the proof of Theorem 2.1 the blow-up differential equations in polar coordinates: where u, v are noise realizations. Consider a reparameterization of time, τ = g(r, θ, u, v)t with Note that g is close to 1, in particular positive, for small values of ε. The reparameterization yields differential equations, for functionsr, θ, t of τ , Note that the time reparameterization does not preserve the flow of individual cycles S(ξ, S 1 ). However, for a fixed noise realization ξ, the graph t∈R (S(θ t ξ, S 1 ), t) is an invariant manifold for both (15) and (16). As the time parameterization has changed only by a factor 1 g(r,θ,u,v) that is bounded and bounded away from zero, to prove the existence of an attracting random cycle it suffices to do this for (16).
The random fixed point is attracting in the sense that there is a neighborhood U λ of E λ , so that for all Proof. Just note that Π λ = Ψ 2πε 2/3 λ , defined in the proof of Theorem 3.1.
A better notion of random attracting cycle would be without time reparameterizations; Φ t λ (S(ξ, S 1 ), ξ) = S(θ t ξ, S 1 ). This would allow a discussion of the dynamics on the random cycle, such as the possibility to find an attracting random fixed point on it, compare [2,3]. 4. Simulations of bounded noise, radially symmetric Hopf.

4.1.
Radially symmetric Hopf with bounded noise. In this section we reproduce invariant measures for the radially symmetric systeṁ where λ ∈ R is a parameter and u and v are noise terms satisfying: u 2 + v 2 ≤ 1, representing radially symmetric noise. In the simulations below we generate the noise by stochastic differential equations with reflective boundary conditions. For ε = 0 the differential equations (18) undergo a supercritical Hopf-Andronov bifurcation at λ = 0. For λ < 0 the origin is a stable global attractor. For λ > 0 the origin is unstable, but there is a circular globally attracting periodic orbit at r = √ λ. Changing to polar coordinates we find: where α = (u cos θ + v sin θ) = u, v · cos θ, sin θ and β = (−u sin θ + v cos θ) = v, −u · cos θ, sin θ . (20)ṙ ± = λr ± − r 3 ± ± ε. From (13) and (14), Write ρ + for the positive zero of the upper differential equation for r + . For λ ≥ λ bif , let ρ − be the largest positive zero of the lower differential equation for r − . For λ < λ bif the MFI set is a disk of radius ρ + . For λ ≥ λ bif the MFI set is an annulus bounded by circles with radii ρ − , ρ + . See Figure 1. Here ε = 0.05. The hard bifurcation takes place at λ bif ≈ 0.2565 and the inner boundary has initial radius r * ≈ 0.2924. The MFI set changes from a disk with radius ρ + for λ < λ bif to an annulus with radii ρ − < ρ + for λ ≥ λ bif . The graph labeled r s indicates the boundary of the set of fixed points which are stable for λ ≤ 0 and unstable for λ > 0.
In addition to the MFI set, another important dynamical feature of these equations is the set of equilibrium points. For λ ≤ 0 these equilibrium points are stable, but for λ > 0 they are unstable. These are the points that are fixed points of (18) when (u, v) is a fixed value. One can find that this set is a disk centered at the origin with radius r s which is a solution of: (1 − 2λ)r 6 + (1 + λ 2 )r 2 − ε 2 = 0.
The radius r s as function of λ is plotted in Figure 1.

4.2.
Simulations. In this section we simulate the bounded noise bifurcation for the sake of demonstration. We note that associated with MFI sets are invariant densities whose supports are the MFI sets [9]. In the symmetric Hopf bifurcation as in the previous subsection, we approximate these invariant densities for different values of λ as λ moves through the random bifurcation.
The generation of bounded noise is somewhat arbitrary without specific knowledge about the noise involved in a particular setting. In this work we generate the noise terms u and v via the stochastic system: where dW 1 and dW 2 are independent (of each other) normalized white noise processes and (21) are interpreted in the usual way as Itō integral equations. In order to assure boundedness and radial symmetry, we restrict (u, v) to the unit disk and impose reflective boundary conditions. Other methods of generating and bounding the noise did not produce significantly difference in the results.
The parameter σ can be interpreted as the rapidity of the noise. If σ is small, then u and v change slowly and as σ increases, they change more quickly. It turns out in our simulations that the value of σ has a strong influence on the characteristics of the invariant density of (18).
In the first set of simulations Figure 2, σ = 1.0 as an example of fast noise and in Figure 3, σ = 0.00001 to show slow noise. Note that for slow noise and λ > 0 there is a separation of time scales. Specifically, for each λ and each set of values of (u, v) the deterministic system defined by holding (u, v) fixed has an exponentially attracting periodic orbit. We expect then that the invariant density is concentrated on the set which is the union of all of these deterministic limit cycles. For fast noise, this will not be the case and as is observed in Figure 2, the approximated invariant densities tend to be smoother than for slow noise.
In both sets of simulations we use ε = .1. This leads to the following values of the bifurcation parameter and the radius of the inner boundary of the MFI set at the bifurcation: We began the simulations by selecting random initial values for (u, v) and (x, y). As we did not need an accurate solution to generate the noise we approximated solutions to the SDE in (21) using a first order Taylor method, namely Euler's method. We then added this noise into the system in (18), which we solved using a second order Adam's-Bashforth method.
After selecting the initial noise term and a starting point for the RDE, we ran the system for 1000 iterations to allow the solution to the RDE to move into the MFI sets. We then ran the system for 5 × 10 5 iterations and recorded the values of every fifth (x, y) coordinate. We repeated this process for 10 2 starting points and recorded a total of 10 7 points, which we used to generate a 2-dimensional histogram. Bright regions indicate that large numbers of samples were observed in this region indicating a higher value of the invariant density. Darker regions indicate lower density. The deterministic bifurcation takes place at λ = 0 and the random bifurcation occurs at λ bif ≈ 0.4072. In each set of figures, the densities are plotted for multiples of λ bif ; namely for λ = .01 λ bif , .05 λ bif , .1 λ bif , .5 λ bif , λ bif , and, 1.1 λ bif . The outer   circle is the outer boundary of the MFI set, the circle inside the MFI is the stable periodic orbit of the deterministic system and the inner circle that appears at λ bif is the inner boundary of the MFI set. It has initial radius r * = 0.3684. The invariant densities are positive for points inside the MFI set, but can be expected to go to zero very rapidly as the boundary is approached (compare [14]). Also note that the densities become undetectably small in the center long before the random bifurcation.