LYAPUNOV EXPONENTS AND STABILITY FOR THE STOCHASTIC DUFFING-VAN DER POL OSCILLATOR

Transcription

1 LYAPUNOV EXPONENTS AND STABILITY FOR THE STOCHASTIC DUFFING-VAN DER POL OSCILLATOR Peter H. Baxendale Department of Mathematics University of Southern California Los Angeles, CA USA Abstract Let λ denote the almost sure Lyapunov exponent obtained by linearizing the stochastic Duffing-van der Pol oscillator ẍ = ω 2 x + βẋ Ax 3 Bx 2 ẋ + σxẇt at the origin x = ẋ = 0 in phase space. If λ > 0 then the process {(x t, ẋ t ) : t 0} is positive recurrent on R 2 \ {(0, 0)} with stationary probability measure µ, say. For λ > 0 let λ denote the almost sure Lyapunov exponent obtained by linearizing the same equation along a typical stationary trajectory in R 2 \ {(0, 0)}. The sign of λ is important for stability properties of the two point motion associated with the original equation. We use stochastic averaging techniques to estimate the value of λ in the presence of small noise and small viscous damping. Keywords: stochastic Duffing-van der Pol oscillator, stability, Lyapunov exponent, stochastic averaging. Introduction The stochastic Duffing-van der Pol oscillator is currently the object of much study in the theory of stability and bifurcations for stochastic dynamical systems. See for example Arnold [2], Arnold, Sri Namachchivaya and Schenk-Hoppé [4], Keller and Ochs [3], and Schenk-Hoppé [8]. In this paper we study the system with multiplicative white noise. To be specific we consider the stochastic Duffing-van der Pol oscillator ẍ = ω 2 x + βẋ Ax 3 Bx 2 ẋ + σx W t

2 2 with ω > 0 and A 0 and B 0. Here β is th coefficient of viscous damping and σ is the intensity of the white noise forcing. To ensure that the system is truly nonlinear we assume that at least one of A and B is strictly positive; if B = 0 we assume also that β < 0 so as to obtain some dissipation in the system. Putting y = ẋ/ω we get the 2-dimensional stochastic differential equation dx t = ωy t dt, ) dy t = (ωx t + βy t + (A/ω)x 3 t Bx 2 t y t dt (σ/ω)x t dw t. Notice that the system has multiplicative noise and hence that (0, 0) is a fixed point for (). Linearizing at (0, 0) gives [ ] 0 ω du t = u ω β t dt σ [ ] 0 0 u ω 0 t dw t. (2) The (top) almost sure Lyapunov exponent λ = λ(β, σ, ω) for the system () linearized at (0, 0) is defined as the almost sure limit () λ(β, σ, ω) = lim t t log u t (3) where {u t : t 0} is a solution of (2). It is easy to show that the limit exists almost surely and does not depend on u 0 so long as u 0 0. Clearly the sign of λ determines the almost sure stability or instability of the linearized process {u t : t 0}. It also determines the behavior near (0, 0) of the nonlinear process (x t, y t ). This in turn controls the behavior of the process (x t, y t ) considered on the state space R 2 \{(0, 0)}. See Section 2 for details. If λ > 0 then the solution {(x t, y t ) : t 0} of () is positive recurrent on R 2 \ {(0, 0)} with a unique invariant probability measure µ = µ(β, σ, ω, A, B), say. In this case we are interested in the question of stability along trajectories. Let (x 0, y 0 ) and ( x 0, ỹ 0 ) be distinct initial points in R 2 \ {(0, 0)}, and let {(x t, y t ) : t 0} and {( x t, ỹ t ) : t 0} be the corresponding solutions of () generated by the same noise {W t : t 0}. This produces a coupling of the trajectories {(x t, y t ) : t 0} and {( x t, ỹ t ) : t 0}. We wish to consider the behavior of ( x t, ỹ t ) (x t, y t ) as t. In this paper we shall consider the distinct but closely related question of linearized stability along trajectories. Linearizing the system along a trajectory {(x t, y t ) : t 0} in R 2 gives dv t = [ 0 ω ω + 3A ω x2 t 2Bx t y t β Bx 2 t ] v t dt [ 0 0 σ ω 0 ] v t dw t. (4)

3 Stochastic Duffing-van der Pol oscillator 3 In the same way that the process u t gives information about the behavior of (x t, y t ) near (0, 0), we expect that the process v t will give information about the behavior of ( x t, ỹ t ) (x t, y t ) when ( x t, ỹ t ) (x t, y t ) is small. Thus we consider the stability of the solution v t of (4) when (x 0, y 0 ) (0, 0). In particular we consider the (top) almost sure Lyapunov exponent λ = λ(β, σ, ω, A, B), say, for the system () linearized along a stationary trajectory {(x t, y t ) : t 0} in R 2 \ {0}, defined as the almost sure limit λ(β, σ, ω, A, B) = lim t t log v t. (5) Notice that the almost sure limit is with respect to the product measure P µ where P represents the probability measure for {W t : t 0} and µ is the distribution of (x 0, y 0 ) R 2 \ {(0, 0)}. Recall that µ and thus λ exist only when λ > 0. An exact formula for λ(β, σ, ω) is given by Imkeller and Lederer [2]; earlier numerical calculations by Kozin and Prodromou [5] gave criteria in terms of β, σ and ω for λ(β, σ, ω) to be positive or negative. However the evaluation of λ(β, σ, ω, A, B) is much more complicated, see Section 3, and to date there is no closed form formula. In this paper we use stochastic averaging techniques to obtain estimates for λ in the case of small noise and small viscous damping. We replace σ by εσ and β by ε 2 β. A result of Auslender and Milstein [5] gives ( ) β λ(ε 2 β, εσ, ω) ε σ2 8ω 2 as ε 0, so we look for estimates on λ(ε 2 β, εσ, ω, A, B) when β > σ 2 /4ω 2. In Section 4 we consider the case when B = 0. In this case the system can be regarded as a small perturbation of a Hamiltonian system, and we have λ(ε 2 β, εσ, ω, A, 0) ε 2/3 λ 0 where λ 0 > 0. This implies (linearized) instability along trajectories. In Section 5 we consider the case when A > 0 and B > 0. Now we have λ(ε 2 β, εσ, ω, A, 0) ε 2 λ ave where λ is the top Lyapunov exponent along trajectories for a stochastically averaged version of the original system. The value of λ ave is given by a two dimensional integral formula. The sign of λ ave depends only on the dimensionless quantities βω 2 /σ 2 > /4 and ωb/a. Numerical

4 4 simulations presented in Section 5. show that for σ 2 /4ω 2 < β < 0 the Lyapunov exponent λ ave is positive if ωb/a and is negative if ωb/a Behavior near 0 The following result makes precise the idea that the behavior of the linearization u t at (0, 0) determines the behavior of (x t, y t ) near (0, 0). It is essentially an application of Theorems 2.2 and 2.3 of Baxendale [7], see also Theorem 2.8 of Baxendale [8]. The verification of condition (2.3) for [7] (or condition H(z) of [8]) is carried out in [8] for the case B = 0, β < 0, and in [9] for the case B > 0. Theorem For the system () (i) If λ < 0 then lim sup t t log (x t, y t ) λ almost surely, for all (x 0, y 0 ) 0. (ii) If λ > 0 there exists a unique probability measure µ on R 2 \ {(0, 0)} such that t lim φ(x s, y s ) ds = φ dµ t t 0 almost surely, for all bounded measurable φ : R 2 \ {(0, 0)} R and all (x 0, y 0 ) (0, 0). In particular µ is the unique invariant measure for {(x t, y t ) : t 0} on R 2 \ {(0, 0)}. Notice that if λ > 0 then the linearized process u t almost surely, whereas the original process (x t, y t ) is affected by the cubic nonlinearity and does not tend to infinity. 3. Khasminskii-Carverhill formula In this section we assume that λ > 0, so that the invariant probability measure µ on R 2 \ {(0, 0)} exists. We apply the method developed by Khasminskii [4] for linear stochastic differential equations and extended to the nonlinear setting by Carverhill []. For v t given by (4) write [ ] cos θt v t = v t. Then sin θ t ( dθ t = ω + ( 3A ω x2 t 2Bx t y t ) cos 2 θ t + (β Bx 2 t ) sin θ t cos θ t ) σ2 ω 2 sin θ t cos 3 θ t dt σ ω cos2 θ t dw t. (6)

5 Stochastic Duffing-van der Pol oscillator 5 Moreover, Itô s formula for log v t and the ergodic theorem give λ = lim t t log v t = Q(x, y, θ) dν(x, y, θ) (7) (R 2 \{0}) S where Q(x, y, θ) = ( 3A ω x2 2Bxy) sin θ cos θ + (β Bx 2 ) sin 2 θ + σ2 2ω 2 cos 2θ cos2 θ and ν is the invariant probability measure for the diffusion process {(x t, y t, θ t ) : t 0} on (R 2 \ {(0, 0)}) S given by () and (6). The existence of ν is implied by the existence of its marginal µ, which in turn is guaranteed by λ > 0. The uniqueness of ν can be verified using the methods of Arnold and San Martin [3]. The formula for λ is obtained similarly, using just the original setting of Khasminskii. Putting x = y = 0 reduces the three dimensional problem in (7) to the one dimensional problem handled by [2] and [5]. 4. The Duffing case B = 0, β < 0 In this section we consider the system dx t = ωy ( t dt dy t = ωx t + ε 2 βy t + A ) ω x3 t dt εσ ω x t dw t which may be regarded as a small perturbation of the Hamiltonian system with H(x, y) = ω ( x 2 + y 2) + A 2 4ω x4. Theorem 2 (Baxendale and Goukasian [0]) If σ 2 /4ω 2 < β < 0 then λ(ε 2 β, εσ, ω, A, 0) = ε 2/3 λ 0 + O(ε 4/3 ) as ε 0 where λ 0 = λ 0 (β, σ, ω) > 0. There are three main ideas in the proof. The first one is to write the equation for v t in the moving frame given by the vectors H(x, y) and H(x, y)/ H(x, y) 2 (where H(x, y) denotes the symplectic gradient of H). The resulting equation has the form dv t = [ 0 J(xt, y t ) 0 0 ] v t dt + ε 2 M 0 (x t, y t )v t dt + εm (x t, y t )v t dw t (8)

6 6 where J(x, y) is a scalar function and M 0 (x, y) and M (x, y) are matrixvalued functions. The second main idea is to use a technique due to Pinsky and Wihstutz [7] for small perturbations of nilpotent linear systems. Their method shows that the constant coefficient linear stochastic differential equation [ ] [ ] [ ] 0 a dv t = v 0 0 t dt + ε 2 a a 2 b b v a 2 a t dt + ε 2 v 22 b 2 b t dw t (9) 22 has Lyapunov exponent λ(ε) = ε 2/3 γ 0 a 2/3 b 2 2/3 + O(ε 4/3 ) as ε 0 so long as a 0 and b 2 0, where π γ 0 = 2 /3 3 /6 [Γ( )]2 The exact value of γ 0 is due to Ariaratnam and Xie []. The third main idea is to use stochastic averaging techniques to pass from the variable coefficient equation (8) to the constant coefficient equation (9). The stochastic averaging involves three different time scales, so that the original three dimensional integral formula for λ reduces to a one dimensional integral formula for λ 0 which involves a strictly positive integrand. 5. The case A > 0, B > 0 Following Arnold, Sri Namachchivaya & Schenk-Hoppé [4] we rescale spatially x x/ε, y y/ε to obtain dx t = ωy t dt dy t = ωx t dt + ε 2 ( βy t + A ω x3 t Bx 2 t y t ) dt εσ ω x t dw t. This system is a small perturbation [ of the rigid rotation ] [ ] with constant cos ωt sin ωt xt angular velocity ω. Write z t =, then sin ωt cos ωt dz t = ε 2 F (z, t)dt + εg(z, t)dw t (0) where the vector fields F (z, ) and G(z, ) have period 2π/ω. Applying the method of stochastic averaging to the k-point motion of this SDE (rescale time by factor /ε 2 and take weak limit), we get dz t = ( β 2 B z t σ 2 2ω ) z t dt 3A 8ω z t 2 Jz t dt () ( K z t dwt + K 2 z t dwt 2 + ) 2Jz t dwt 3 y t

7 Stochastic Duffing-van der Pol oscillator 7 where J = [ 0 0 ] [ 0, K = 0 ] [ 0, K 2 = 0 ]. It is important to observe that the stochastic averaging done in the passage from (0) to () is carried out at the level of the k-point motions, for all k. Related results are contained in Section 5.6 of Kunita [6]. If the stochastic averaging is done only for the one point motion of (0) the resulting information is only the generator for the one point motion of (). There are many ways of choosing a stochastic differential equation (SDE) with a given generator for the one point motion, and these different choices of SDE can have many different stability behaviors, and many different values for the Lyapunov exponent along trajectories, see Baxendale [6]. However by doing stochastic averaging for the k-point motions, we gain information about the local characteristics of the stochastic flow associated with (). In particular the law of the stochastic flow associated with () is uniquely determined. In this setting Arnold, Sri Namachchivaya and Schenk-Hoppé [4] choose the simpler SDE ( β dz t = 2 B z t 2 ) z t dt 3A 8 8ω z t 2 Jz t dt (2) + σ 2 ( z t dwt + ) 3Jz t dwt 2 2ω which generates the same one-point motion as (). This SDE has λ = β/2 + σ 2 /8ω 2 and λ = 0 (when λ > 0). Moreover the (one-point) amplitude process r t = z t given by () or (2) satisfies an SDE of the form dr t = ( ) βrt 2 Br3 t 8 + 3σ2 r t 6ω 2 dt + σ 2 2ω r t dw t (3) which has λ = β/2+σ 2 /8ω 2 and λ = 2λ (when λ > 0). Thus equations (), (2) and (3) give three different values for the Lyapunov exponent along trajectories. The following theorem tells us that () is the correct SDE to study. Theorem 3 (Baxendale [9]) Assume A > 0, B > 0. Let λ ave and λ ave be the Lyapunov exponents for the averaged SDE (). Then as ε 0, ( ) β λ(ε 2 β, εσ, ω) ε 2 λ ave = ε σ2 8ω 2 (4)

8 8 and if β > 4σ 2 /ω 2 then λ(ε 2 β, εσ, ω, A, B) ε 2 λ ave. (5) In fact the first result (4) is due to Auslender and Milstein [5]; the explicit formula for λ ave comes from the rotational symmetry of (). The result (5) is plausible because the Lyapunov exponent is not affected by spatial rescaling and rigid rotations. However since stochastic averaging guarantees only convergence on finite intervals of time there is work to be done. The proof of (5) uses the Khasminskii-Carverhill formula (7) and the weak convergence of the invariant measures ν on (R 2 \ {0}) S as ε 0. It is here that the condition B > 0 is used. If B = 0 then the sequence of invariant measures is not tight and the estimate (5) fails, see Section 4. The condition A > 0 ensures that the limit measure is unique. It will also ensure that the measure ν in (6) is unique. It remains to calculate the Lyapunov exponent λ ave. The rotational symmetry in () reduces the calculation in the Khasminskii-Carverhill formula to a two-dimensional problem. We have λ ave = β ( 2 σ2 3Ar 2 8ω 2 + 8ω sin 2ψ Br2 cos 2ψ 8 where ν is the invariant probability on (0, ) S for ( ) βrt dr t = 2 Br3 t 8 + 3σ2 r t 6ω 2 dt + σ 2 2ω r t dbt and dψ t = ) d ν(r, ψ) (6) ( ) B 8 r2 t sin 2ψ t + 3Ar2 t 4ω cos2 ψ t dt σ 2 2ω sin 2ψ t dbt + σ 2 2ω ( + cos 2ψ t) dbt 2. The processes r t and ψ t are related to the processes (x t, y t ) and θ t used in [ ] [ ] xt cos φt the derivation of the formula (7) by the formulas = r y t t sin φ t and ψ t = θ t φ t. Moreover, the dependence of λ ave on the parameters β, σ, ω, A and B satisfies λ ave (β, σ, ω, A, B) = σ2 ω 2 λ ave ( ω2 β σ 2,,,, ωb A ).

9 Stochastic Duffing-van der Pol oscillator 9 We see that the sign of λ ave depends only on the dimensionless quantities ω 2 β/σ 2 and ωb/a. The first quantity measures the strength of the viscous damping relative to the noise intensity. The second quantity measures the relative sizes of the cubic dissipation Bx 2 ẋ and cubic restoring force Ax Numerical calculations for λ ave Write s t = rt 2, then s t satisfies the affine equation ds t = ( βs t + B ) dt σ s t dbt 4 2ω The following numerical results used a first order Euler scheme to simulate the process (s t, ψ t ) on some time interval [0, T ], and this was used to approximate the integral in (6) by the corresponding discrete time average. See Talay [9] for a theoretical discussion of this method. The simulations all had ω = σ = and T The value of β was increased from to in steps of In Figures and 2 the dotted line shows the exact value λ ave = β/2 + /8. The solid lines show the results of simulation of λ ave with time step h = The dashed line shows the results of simulation of λ ave with time step h = Figure shows λ ave for the values ωb/a = 0.5,,.5, 2 and 3. Figure 2 shows λ ave for the values ωb/a =,.24 and.5. From the numerical data in Figure we see in the range σ 2 /4ω 2 < β < 0 we have λ ave > 0 λ ave < 0 } if { ωb/a ωb/a.5 Finally consider the numerical data in Figure 2 for ωb/a =.24. Here we are dealing with values of λ ave which are of the same order as the time step h, and hence of the same order as the possible error in the computed value of λ ave, see [9]. However, Figure 2 shows that halving the time step, from 0.00 to , has little effect on the data. Thus it is reasonable to believe that the graph of the true value of λ ave will be close to a smoothed out version of the solid (or dashed) line. Therefore our calculations strongly suggest that when ωb/a =.24 the averaged system () experiences two dynamic bifurcations as β is increased from to 0. One occurs when β = σ 2 /4ω 2 as the fixed point (0,0) loses stability, but the system still has stability along trajectories. The other occurs at approximately β = 0.7σ 2 /ω 2, as the property of stability along trajectories is lost.

10 Figure. Lyapunov exponent as a function of β when σ 2 /ω 2 =. Dotted line: λ = β/2 + /8. Solid lines: λ for ωb/a = 0.5,,.5, 2 and 3. 5 x Figure 2. Lyapunov exponent as a function of β when σ 2 /ω 2 =. Dotted line: λ ave = β/2 + /8. Solid lines: λave for ωb/a =,.24 and.5 with h = Dashed line: λave for ωb/a =.24 with h = Acknowledgments I would like to thank Levon Goukasian (University of Southern California) for his assistance with the numerical computations in Section 5..

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