Commun Nonlinear Sci Numer Simulat 44 (2017) 144–158

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Commun Nonlinear Sci Numer Simulat

journal homepage: www.elsevier.com/locate/cnsns

Research paper

Evaluation of closure strategies for a periodically-forced

Duffing oscillator with slowly modulated frequency subject to

Gaussian white noise

Jason Yalim

a , Bruno D. Welfert a , ∗, Juan M. Lopez

a

School of Mathematical and Statistical Sciences, Arizona State University, Tempe AZ, 85287, USA

a r t i c l e i n f o

Article history:

Received 18 April 2016

Revised 11 June 2016

Accepted 8 August 2016

Available online 11 August 2016

a b s t r a c t

The response of a Duffing oscillator subject to a periodic forcing with slowly and stochas-

tically modulated frequency is analyzed numerically. The results of both moment and

cumulant-based stochastic reductions are compared to Monte Carlo simulations. It is

shown how the explicit characterization of higher-order central moments of the (Gaussian)

noise source and the periodic nature of the forcing enable a reliable reduction strategy

providing a faithful description of the mean behavior of stochastic solutions. The reduced

model is then used to illustrate how a large noise level and fast frequency drift may com-

bine to sustain a strong response that is normally associated to resonance in the noiseless

static case.

© 2016 Published by Elsevier B.V.

1. Introduction

Many time-dependent phenomena in social, physical, and engineering applications, including social exchange [1] , traffic

flow [2] , epidemiology [3] , molecular dynamics [4] , weather [5] , and stock markets [6] , are subject to uncertainty, noise, or

risk, involving random fluctuations whose impact is often difficult to quantify or predict, but are critical to the evolution

process. In this work, we are more specifically interested in processes driven by external forces with a strong periodic

component, for example oscillations in ocean surface temperature deviations from seasonal averages (the El Niño effect)

[7–9] , burst patterns in neuron firing in a nerve axon triggered by small amplitude oscillations in the membrane potential

[10] , or oscillations in gas flows in nanoscale device manufacturing [4] . Quite often, the periodicity of the phenomenon is

also affected by a slow drift in operating or physical conditions, whether by design (e.g. in quasi-static experiments covering

a range of settings) or naturally (e.g. changes in the levels of CO 2 and other gases in the atmosphere).

Evolutionary systems subject to noise are typically modeled by stochastic differential equations for which the statistical

analysis of direct Monte-Carlo simulations (MCS) is expensive, if not impossible, due to the large number of realizations

necessary to obtain meaningful results. Basic mathematical or statistical paradigms can be tested on low-dimensional models

reproducing the essential characteristics of the physical problem, and for which MCS are relatively cheap. One such model

is the Duffing oscillator with temporal forcing which may have a stochastic component.

The Duffing oscillator problem has a rich history. Its dynamics are well understood in the deterministic and static case

[11] . Explicit solutions exist for the unforced case [11,12] , as well as for the harmonically forced case [13] . References

∗ Corresponding author.

E-mail addresses: yalim@asu.edu (J. Yalim), welfert@asu.edu (B.D. Welfert), jmlopez@asu.edu (J.M. Lopez).

http://dx.doi.org/10.1016/j.cnsns.2016.08.003

1007-5704/© 2016 Published by Elsevier B.V.

J. Yalim et al. / Commun Nonlinear Sci Numer Simulat 44 (2017) 144–158 145

[14,15] vary the forcing amplitude with fixed nonlinear stiffness, which is equivalent to varying the nonlinear stiffness and

holding the forcing amplitude constant with an appropriate change of variable. Without the nonlinear stiffness, the Duffing

oscillator reduces to a classical linear oscillator, exhibiting oscillations for small damping and resonance when forced near

the natural frequency of the oscillator. This resonance is characterized by an increased amplitude response which is often

referred to as the resonance response peak. For softening nonlinearity the unforced system has a stable trivial solution and

two saddles. The hardening system only has the trivial solution as a stable focus in this regime, but when subjected to nega-

tive linear stiffness the hardening system will exhibit bistable behavior with a saddle trivial solution [11, Section 3.3] . Adding

forcing causes the resonance peak to tilt toward lower or higher frequencies for softening or hardening systems, and as the

magnitude of the nonlinearity increases multiple solutions exist in a range of frequencies below or above the natural fre-

quency. The endpoints of this hysteretic interval are the frequencies at which saddle-node bifurcations occur. For sufficiently

large nonlinear stiffness, chaotic or unbounded solutions develop [11, Section 5.4, Section 5.5] , but even for more modest

stiffness, the choice of initial conditions is very important, as finite-time blow-up can occur for most initial conditions (see

the basins of attraction from [14] ).

The slow ramping of parameters in the noiseless case has been used extensively in physical experiments to investigate

a system’s response over a range of parameter values, such as frequencies, typically to save time compared to performing

multiple static experiments. The ramp rate must be small enough for the system to relax adiabatically to the long-term

state associated with a static experiment at given fixed frequency (quasi-static approximation). Lewis [16] seems to be the

first to have investigated the effect of a linear drift in forcing frequency on the behavior of harmonic linear oscillators, and

both [17] and [18] independently showed that the solution of the Duffing oscillator can be formulated explicitly in terms

of Fresnel integrals. Kevorkian [19] instead considered the modulation of the natural frequency, where the passage through

the resonance regime is typically accompanied by a smaller amplitude resonance peak relative to static experiments in the

same frequency range, with amplitude oscillations developing after the passage through the resonant frequency [20] . These

amplitude oscillations occur only when the damping is small relative to the square root of the ramping rate [21] , and are

sustained as the damping approaches zero. Slower frequency modulation rates lead to higher amplitude responses since the

system can capture more energy in the vicinity of the resonant frequency. Nonlinear stiffness introduces the possibility of

blow-up, but otherwise does not modify this heuristic. In particular, it is possible to adjust the ramping rate and maintain

large oscillations in the dynamic response beyond the resonance frequency (“auto-resonance”) [22,23] .

Multiple studies have investigated the effects of noise in the static case, usually with Gaussian white noise as an addi-

tional forcing term rather than in the phase [24–26] , superposed on the harmonic forcing [26–32] . A common observation is

that large white noise tends to weaken nonlinear effects, such as the tilting of the resonance peak. A few studies considered

random perturbations directly in the evolution of the phase [15,33,34] . Again, noise has a stabilizing effect on the damped

oscillator, but introduces chaos (destroys basins of attraction) in the undamped case [15,35,36] . The additional uniformly dis-

tributed phase component in [15] appears to inhibit the Gaussian noise. This is not surprising, since the (wrapped) normal

distribution on the circle tends to the uniform distribution for sufficiently large noise.

Here, we address the effect and modelization of noise in the context of slow frequency modulations, with a small fixed

damping parameter. This does not seem to have been studied previously. The amplitude response of the Duffing oscillator

can be statistically characterized by determining a mean behavior, and possibly variance and covariance with the velocity

response or phase, using one of three approaches: ( a ) via Monte Carlo Simulations (MCS) from a large number of (time)

realizations of the noise, and computing sample (arithmetic) means and covariances, ( b ) via estimation of a probability

density function for each feasible state of the system, as the solution of a (deterministic) Fokker–Planck partial differential

equation (PDE), from which ensemble means and covariances of the states can be recovered, or ( c ) via a system of ordinary

differential equations (ODE) satisfied by ensemble means and covariances. This system requires closure assumptions, which

we discuss below.

Our goals are twofold: ( i ) evaluate moment-based stochastic reduction strategies in approach ( c ) and their viability in

providing a faithful account of the amplitude of the solution of the Duffing oscillator under various ramping conditions,

and ( ii ) investigate the interaction between the noise level and the frequency ramping rate for a large range of parameter

regimes. The approach used here is similar to that adopted in [33] , namely strategy ( c ) verified by MCS ( a ), with two

major differences: the phase is now drifting and no assumption is made on the noise level, which is possible because of a

statistically more suitable characterization of the amplitude response.

2. Background in stochastic modelization and reduction

We consider a non-dimensional Duffing oscillator described by the second-order differential equation

d 2 x t

dt 2 + γ

dx t

dt + (1 − ηx 2 t ) x t = f (θt ) , (1)

modeling the deviation x t from the equilibrium position of a mass in a forced nonlinear spring–mass system with (small)

positive damping parameter γ and positive stiffness parameter η (softening spring). The forcing term

f (θt ) := A sin θt , (2)

146 J. Yalim et al. / Commun Nonlinear Sci Numer Simulat 44 (2017) 144–158

depends on an angle θ t which evolves at an angular rate that is slowly drifting in time, at (chirp) rate ε such that

dθt

dt := ω t = ω 0 + εt + ξt , | ε| � 1 , (3)

and is subject to Gaussian white noise ξ t with variance σ 2 , i.e. ξt = σd W t /d t, where W t is a standard Wiener process,

with delta-correlated increments dW t normally distributed with mean 0 and variance dt [37] . The sign of ε determines

the direction of the frequency sweep. The periodicity of the forcing f captures the cyclic variations of the physical system,

such as climatic conditions in climate models, seasonal conditions in epidemiological systems, or operational conditions in

power-driven systems.

Defining v t := dx t / dt and k (x ) := x (1 − ηx 2 ) , Eqs. (1) and (3) can be reformulated as the first-order system

d

[

x t v t θt

]

=

[

v t f (θt ) − k (x t ) − γ v t

ω 0 + εt

]

d t +

[

0

0

σ

]

d W t . (4)

This is in the form of a standard Langevin stochastic differentiation equation (SDE) with additive noise

du t = F (u t , t) dt + G dW t , (5)

with appropriate initial conditions u 0 , and

u =

[

x v θ

]

, F (u , t) =

[

v f (θ ) − k (x ) − γ v

ω 0 + εt

]

, G =

[

0

0

σ

]

. (6)

The notations used in (5) and (6) emphasize the distinction between u t and u : u t represents the value of the stochastic pro-

cess at time t , while u represents a possible state of u t among all possible configurations. Whereas the noise in (4) appears

in the form of additive unbounded noise, in (1) the noise appears in the bounded forcing term and as a result (1) is often

referred to as a system subjected to bounded noise [38] .

Numerical Monte Carlo simulations (MCS) of the resulting set of equations yields a different solution u t for each realiza-

tion of ξ t . Arithmetic averaging provides the mean solution behavior and deviations of sample solution paths from this mean

provide input for further statistical descriptions via higher-order moments (e.g. variance). While being relatively straightfor-

ward to implement, MCS requires a large number of independent realizations, N , to achieve statistical meaningfulness. In

practice, N depends on the desired accuracy tolerance, τ , such that N ≈ Cτ−2 , with C dependent on || G || = σ . The value of N

may also be limited by computational considerations, and MCS may be impractical for more complex SDEs, such as systems

arising from the discretization of PDEs.

A seemingly more efficient strategy consists in evaluating the conditional joint probability density

p = p(u , t) := Prob (u t = u | u 0 ) ,

of a specific configuration u ∈ R

n at time t via the Fokker–Planck (or forward Kolmogorov) FPK equation

∂ p

∂t =

1

2

n ∑

i, j=1

∂ 2 p

∂ u i ∂ u j

G i G j −n ∑

i =1

∂

∂u i ( pF (u , t) i ) , (7)

with initial condition p(u , 0) = Dirac (u − u 0 ) and vanishing boundary conditions at ‖ u ‖ → ∞ . In the case of (4) , the FPK

equation becomes ( n = 3 )

∂ p

∂t =

1

2

σ 2 ∂ 2 p

∂θ2 − (ω 0 + εt)

∂ p

∂θ− v

∂ p

∂x +

∂

∂v (k (x ) − f (θ ) + γ v ) p, (8)

which does not possess an obvious analytic solution. Integrating (8) for x, v ∈ (−∞ , ∞ ) , yields a drift diffusion equation

whose solution for t > 0 is the marginal probability distribution ∫ R 2

p(x, v , θ, t) dx dv =

1 √

2 πσ 2 t exp

(− [ θ − (θ0 + ω 0 t +

1 2 εt 2 )] 2

2 σ 2 t

), (9)

of the Gaussian process θ t centered at θ0 + ω 0 t +

1 2 εt 2 with variance σ 2 t , as could be expected from the expression

θt = θ0 + ω 0 t +

1 2 εt 2 + σW t obtained directly from (4) . However, the full distribution p itself is not Gaussian because of

the presence of the nonlinear terms k ( x ) and f ( θ ).

In general, analytic treatments of (7) are limited to linear F (in u ), or nonlinear problems with strong conservation

properties, such as Hamiltonian systems [39] , and are often restricted to steady-state distributions p ( u , ∞ ) (for autonomous

SDEs). The numerical treatment of the FPK equation requires either a finite discretization approach in state space [40–

42] , assigning possible outcomes to each state variable. Conventionally, this is only practical for small dimensional systems

and a small number of outcomes u , but improves when p ( u , t ) has a spectral model [43,44] involving a limited number

of unknown coefficients. These coefficients are typically fitted via Galerkin-type projection techniques using experimental

observations or MCS. Exponential models [31,45,46] are popular because of the role of the exponential function as a building

J. Yalim et al. / Commun Nonlinear Sci Numer Simulat 44 (2017) 144–158 147

block in the solution of linear differential equations with constant coefficients, and the role played by the characteristic

function in standard probability theory. We also note the use, with some degree of success, of approximations of p based on

the Gram–Charlier or Edgeworth expansions [47] , and equivalent linearization techniques, which aim at replacing (5) with

a statistically equivalent local linear model [4 8,4 9] .

An intermediate strategy between MCS and the FPK approach is to determine selected moments of u t ,

q (u ) (t) :=

∫ R n

q (u ) p(u , t) du 1 · · · du n , (10)

such as the ensemble mean u or the ensemble covariance matrix � with q (u ) = u or q (u ) = (u − u )(u − u ) T , respectively.

Note that if q depends only on a subset of variables, say u m +1 , . . . , u n , then (10) reduces to the mean ∫ R n −m

q (u m +1 , . . . , u n ) (∫

R m

p(u , t) d u 1 · · · d u m

)d u m +1 · · · d u n ,

obtained from the joint marginal probability distribution of u m +1 , . . . , u n . This may be substantially easier to evaluate in

some cases, such as when u m +1 , . . . , u n are (possibly correlated) Gaussian. A similar observation was also exploited in [50] for

non-Gaussian noise, directly at the level of the (generalized) FPK equation.

A deterministic equation describing the time evolution of q (u ) can be obtained from (7) via integration(s) by parts (with

vanishing boundary terms):

d q (u )

dt =

1

2

n ∑

i, j=1

∂ 2 q (u )

∂ u i ∂ u j

G i G j +

n ∑

i =1

∂q (u )

∂u i

F (u , t) i . (11)

In particular, the ensemble mean and covariance matrix evolution satisfy the differential equations

d u

dt = F (u , t) , u (0) = u (0) , (12)

and

d�

dt = Q + Q

T + G G

T , �(0) = 0 n ×n , with Q := F (u , t)(u − u ) T . (13)

For affine F (u , t) = A (t) u + b(t) , one gets F (u , t) = F ( u , t) and

Q = A (t)(u − u )(u − u ) T = A (t)�,

so that (12) and (13) are (individually) self-contained equations for u and �, with (13) being of Lyapunov type [Eq. (3.49)

of 49] . For nonlinear F , it is convenient to define δu := u − u and expand in Taylor series

F (u , t) = F ( u + δu , t) =

∑

| k |≥0

1

k ! F (k ) ( u , t)(δu ) k ,

where k = (k 1 , . . . , k n ) is a multi-index of length | k | = k 1 + . . . + k n , k ! = k 1 ! · · · k n ! , and F (k ) ( u , t) is a multi-linear map from

R

| k | n × R to R

n defined by (F (k ) ( u , t)(δu ) k

)i = (δu ) k 1

1 · · · (δu ) k n n

∂ | k | F (u , t) i

∂ u

k 1 1

· · · ∂ u

k n n

∣∣∣∣u = u

.

Using δu = 0 and � j� = (δu ) j (δu ) � , one obtains the expansions

F (u , t) i = F ( u , t) i +

1

2

n ∑

j,� =1

� j�

∂ 2 F (u , t) i ∂ u j ∂ u �

∣∣∣∣u = u

+

∑

| k |≥3

1

k ! (δu ) k 1

1 · · · (δu ) k n n

∂ | k | F (u , t) i

∂u

k 1 1

· · · ∂u

k n n

∣∣∣∣u = u

, (14)

and

Q i j =

n ∑

� =1

� j�

∂F (u , t) i ∂u �

∣∣∣∣u = u

+

∑

| k |≥2

1

k ! (δu ) k 1

1 · · · (δu ) k n n (δu ) j

∂ | k | F (u , t) i

∂ u

k 1 1

· · · ∂ u

k n n

∣∣∣∣u = u

. (15)

The series (14) and (15) may or may not converge. Moreover, their approximations, e.g. via truncation, typically result in the

system (12) –(13) needing closure assumptions.

3. Closure strategies

The central moments in (14) for | k | ≥ 3,

(δu ) k 1 · · · (δu ) k n n ,

1

148 J. Yalim et al. / Commun Nonlinear Sci Numer Simulat 44 (2017) 144–158

and in (15) for | k | ≥ 2,

(δu ) k 1 1

· · · (δu ) k n n (δu ) j ,

must in general be approximated for the system (12) and (13) to be self-contained or closed. For autonomous systems with

polynomial nonlinearities, these moments can be obtained via finite recursion relations [49,51] . However, our system is non-

autonomous and non-polynomial due to the harmonic forcing. Here we present the construction of four closure strategies,

each improving on the previous one in some fashion.

3.1. Strategy 1: the moment-neglect approach

The simplest strategy is to set (δu ) k 1 1

· · · (δu ) k n n = 0 for any | k | ≥ 3, so that (δu ) k 1 1

· · · (δu ) k n n (δu ) j = 0 for | k | ≥ 2. In the

Duffing case (4) with (6) , the truncations

f (θ ) ≈ f ( θ ) +

1 2 �θθ f ′′ ( θ ) = (1 − 1

2 �θθ ) sin θ, (16)

and

k (x ) ≈ k ( x ) +

1 2 �xx k

′′ ( x ) = k ( x ) − 3 ηx �xx , (17)

in (14) yield the approximation

d

dt

[

x v θ

]

≈[ v

(1 − 1 2 �θθ ) sin θ − (1 − ηx

2 − 3 η�xx ) x − γ v ω 0 + εt

]

, (18)

of (12) , while the truncations for q = x, v , θ lead to

a (q ) := f (θ ) δq − k (x ) δq ≈ f ′ ( θ )�qθ − k ′ ( x )�xq = �qθ cos θ − (1 − 3 ηx 2 )�xq , (19)

in (15) , yielding the approximation

d

dt

⎡

⎢ ⎢ ⎢ ⎢ ⎣

�xx

�x v �xθ

�vv �v θ�θθ

⎤

⎥ ⎥ ⎥ ⎥ ⎦

≈

⎡

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

2�x v

�xθ cos θ − (1 − 3 ηx 2 )�xx − γ�x v + �vv

�v θ2�v θ cos θ − 2(1 − 3 ηx

2 )�x v − 2 γ�vv

�θθ cos θ

−(1 − 3 ηx 2 )�xθ − γ�v θσ 2

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

, (20)

of (13) . Eqs. (18) –(20) form a coupled system of nine equations for the three means and six covariances of the state variables

x, v , and θ . In the noiseless case σ = 0 the covariance matrix � = 0 and (18) reverts to the deterministic system (4) , a trait

that all four strategies share.

3.2. Strategies 2 and 3: non-Gaussian-moment-neglect approaches

Since the marginal distribution (9) of θ is Gaussian, i.e. completely characterized by its mean θ and variance �θθ , all

moments of θ can be evaluated exactly in terms of �θθ . In particular,

f (θ ) =

∑

k ≥0

f (k ) ( θ )

k ! (δθ ) k =

∑

� ≥0

f (2 � ) ( θ )

(2 � )! (δθ ) 2 � =

∑

� ≥0

(−1) � sin θ

(2 � )!

(2 � )!

2

� � ! ��

θθ

= e −�θθ / 2 sin θ,

(21)

and

f (θ ) δθ =

∑

k ≥0

f (k ) ( θ )

k ! (δθ ) k +1 =

∑

� ≥0

f (2 � +1) ( θ )

(2 � + 1)! (δθ ) 2 � +2

=

∑

� ≥0

(−1) � cos θ

(2 � + 1)!

(2 � + 2)!

2

� +1 (� + 1)! �� +1

θθ

= �θθ e −�θθ / 2 cos θ.

(22)

Expansions of the form (21) and (22) were already considered in [33] , but without an explicit evaluation of the series, and

are equivalent to a formula used in [29, Eq.(8)] . The expression of the central moments, (δθ ) 2 � , used in (21) and (22) can

be found, for example, in [49, p.22] , or by induction from the Furutsu–Novikov formula [52]

g(δθ ) δθ = � g ′ (δθ ) , (23)

θθ

J. Yalim et al. / Commun Nonlinear Sci Numer Simulat 44 (2017) 144–158 149

valid for any scalar differentiable function g . Note that (21) is simply the imaginary part of the characteristic function e iθ =e i θ−�θθ / 2 [49, p.20] , while (22) also follows directly from (23) with g(δθ ) = e iδθ . For general problems where u m +1 , . . . , u n are

Gaussian, the Furutsu–Novikov formula can be expressed as

g( v ) v = �v g ′ ( v ) T , (24)

where g is now a (nonlinear) functional of v = [ δu m +1 , . . . , δu n ] T , and �v is the restriction of � to v . This formula can be

understood as an integration by parts for Gaussian processes in stochastic calculus [53] .

The use of exact expressions (21) and (22) , rather than approximations (16) and (19) (with q = θ ), introduces corrections

into (18) and (20) , which now read

d

dt

[

x v θ

]

≈[ v

e −�θθ / 2 sin θ − (1 − ηx 2 − 3 η�xx ) x − γ v

ω 0 + εt

]

, (25)

and

d

dt

⎡

⎢ ⎢ ⎢ ⎢ ⎣

�xx

�x v �xθ

�vv �v θ�θθ

⎤

⎥ ⎥ ⎥ ⎥ ⎦

≈

⎡

⎢ ⎢ ⎢ ⎢ ⎢ ⎣

2�x v

�xθ cos θ − (1 − 3 ηx 2 )�xx − γ�x v + �vv

�v θ2�v θ cos θ − 2(1 − 3 ηx

2 )�x v − 2 γ�vv

�θθ e −�θθ / 2 cos θ − (1 − 3 ηx 2 )�xθ − γ�v θ

σ 2

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎦

. (26)

The factor e −�θθ / 2 = e −σ 2 t/ 2 effectively damps the forcing sin θ on a time scale of order σ−2 , i.e. on long time windows

corresponding to slow sweeps of the frequency for small σ , or increasing noise level σ on fixed time windows.

Strategy 2 makes use of (25) but still uses (20) , while Strategy 3 replaces (20) with (26) .

3.3. Strategy 4: the Gaussian (cumulant-neglect) approach

Under the assumption that δu is (nearly) Gaussian, all moments of δu are evaluated using the Novikov formula (24) .

In particular, (δu ) k 1 1

· · · (δu ) k n n vanishes for | k | odd and is a function of coefficients of � for | k | even. In the context of the

Duffing problem, this does not affect (25) , but it modifies (26) via the new approximation of (19) :

a (q ) =

∑

k ≥0

f (k ) ( θ )

k ! (δθ ) k (δq ) − k ′ ( x )�xq − k ′′ ( x )

2

(δx ) 2 (δq ) − k ′′′ ( x ) 6

(δx ) 3 (δq )

≈∑

� ≥0

f (2 � +1) ( θ )

(2 � + 1)! (2 � + 1)�qθ (δθ ) 2 � − k ′ ( x )�xq − k ′′′ ( x )

6

�xq �xx

= �qθ e −�θθ / 2 cos θ − (1 − 3 ηx 2 − η�xx )�xq .

Therefore, (26) becomes

d

dt

⎡

⎢ ⎢ ⎢ ⎢ ⎣

�xx

�x v �xθ

�vv �v θ�θθ

⎤

⎥ ⎥ ⎥ ⎥ ⎦

≈

⎡

⎢ ⎢ ⎢ ⎢ ⎢ ⎣

2�x v

�xθ e −�θθ / 2 cos θ − (1 − 3 ηx 2 −η�xx )�xx − γ�x v + �vv

�v θ2�v θ e −�θθ / 2 cos θ − 2(1 − 3 ηx

2 −η�xx )�x v − 2 γ�vv

�θθ e −�θθ / 2 cos θ − (1 − 3 ηx 2 −η�xx )�xθ − γ�v θ

σ 2

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎦

. (27)

This cumulant-neglect technique is widely used to close non-Gaussian systems, and in particular for Duffing equations

[26,37,49,54–62] . In particular, an expression similar to (22) has been considered in [62] for a simpler randomized harmonic

process. Note however, that none of the cited works have considered ramped frequency.

4. Numerics

The Monte Carlo simulations (MCS) of (5) with (6) were obtained by integrating in time using the stochastic integrator

described in [63] . This method is of strong first-order when noise is present and reverts to the second-order Heun’s method

in the deterministic case. The ensemble states were approximated by the arithmetic mean and covariance of the simulations.

The second-order Heun’s method was also used to time evolve the reduced model, as it is a deterministic system.

For both the MCS and reduced model, a fixed frequency step, �ω, was used to maintain consistency across solutions,

such that,

�ω =

��,

N

150 J. Yalim et al. / Commun Nonlinear Sci Numer Simulat 44 (2017) 144–158

Fig. 1. Response and envelopes of the response for x in the deterministic case ( σ = 0 ): (a) ε = ±10 −3 (fast), (b) ε = ±10 −4 , and (c) ε = ±10 −5 (slow) ramp,

the left panels show the forward sweep and the right panels the backward sweep, as indicated by the arrows.

where �� is the mesh size of the frequency interval and N is the number of partitions. The timestep, h , is the ratio of the

frequency step and the ramping rate, h = �ω/ε. N was chosen such that the smallest sampled ramping speed, | ε| = 10 −7 ,

had a converged solution. However, for ramping speeds larger than this, a much less refined mesh is sufficient, allowing for

the practical use of high-level software simulation packages.

5. Numerical results

All the numerical experiments presented in this study have a unit forcing amplitude A = 1 , a small damping γ = 0 . 05 ,

and a large softening cubic stiffness η = 7 × 10 −4 . This choice of η ensures a hysteretic response, but is small enough so that

no finite-time blow-up occurs for the initial conditions employed. All cases used the initial condition (x 0 , v 0 , θ0 ) = (0 , 0 , 0) ,

with ω 0 = 0 . 4 for the forward sweep and ω 0 = 1 . 3 for the backward sweep.

5.1. Typical deterministic results

Before exploring how noise in the phase affects the ramped Duffing oscillator, we first demonstrate the effect of the

choice of ramping speed on the amplitude response of the deterministic system.

Fig. 1 shows the solution x of (6) without noise (σ = 0 ) versus the instantaneous frequency ω( t ), for various ramping

rates in ω, using both forward and backward frequency sweeps. Resonance effects occur in the vicinity of ω = 1 . The slow

quasi-static ramp ( Fig. 1 c) closely matches static results corresponding to ε = 0 . The amplitude of x suddenly increases

around a frequency ω

∗ ≈ 0.93 ( Fig. 1 c, left panel) when ramping up ω, and suddenly decreases at ω ∗ ≈ 0.84 ( Fig. 1 c, right

panel) when ramping down ω, leading to hysteresis. The “jump” values obtained for the slow sweeps with | ε| = 10 −5 are

consistent with the values ω

∗ = 0 . 92 and ω ∗ = 0 . 84 predicted by a static analysis using the method of harmonic balance [64,

Eqs.9,13] [also see 65 ]. The maximal amplitude reached at ω ∗ also matches the static prediction A ∗ ≈ 23.9 [64, Eq.8] very

well, while the static prediction for A

∗ ≈ 9.8 [64, Eq.11] seems to underestimate the amplitude peak by close to 50%. The

fast sweep speed | ε| = ±10 −3 tends to delay the occurrence and decrease the magnitude of the peak amplitude ( Fig. 1 a),

essentially because less time is spent in the resonant regime near ω = 1 , and a similar reaction time corresponds to a larger

J. Yalim et al. / Commun Nonlinear Sci Numer Simulat 44 (2017) 144–158 151

Fig. 2. Envelopes of the MCS ensemble mean amplitude response (from 10 0 0 0 realizations) for the slow downsweep case ( ε = −10 −6 ), with different noise

levels (a) σ = 10 −4 , (b) σ = 10 −3 , and (c) σ = 10 −2 . Note that for (c), there is a magnified vertical scale and that the ensemble results have not reached a

high-order of convergence.

frequency variation. Relaxation oscillations, due to the sharp transitions around the “jump” frequencies, are also clearly

visible on the downwind side of the peak amplitude.

The response reaches larger amplitudes on the backward frequency sweep compared to the forward sweep, and so from

now on we focus on the cases with ε < 0.

5.2. Typical noisy response

As the results of the Monte Carlo simulations (MCS) will be utilized to validate the effectiveness of the reduced models,

we first overview their responses.

Fig. 2 illustrates the stochastic response for the slow down-ramp with rate ε = −10 −6 . The approximation of x obtained

by averaging 10 0 0 0 sample runs associated to different realizations of the noise d ξ t is shown for three noise levels σ .

For small σ ( Fig. 2 a), the response is similar to the deterministic case. As can be seen for the larger σ = 10 −3 in Fig. 2 (b),

noise dampens the resonance peak. Noise of a sufficient magnitude will strongly damp the resonance peak and stabilize

the system. This affect is perhaps the most clear for σ = 10 −2 ( Fig. 2 c). This strong damping can be attributed to the more

uniform distribution of θ in [0, 2 π ) (mod 2 π ), which, upon averaging successive increments, kills off the forcing term (2) .

5.3. Evaluation of closure strategies

Fig. 3 compares the mean amplitude response, x , for the four closure strategies with the downsweeping ramp rate ε =−10 −6 and high-noise level σ = 10 −2 . These results should be comparable to those obtained by MCS in Fig. 2 (c). Fig. 3 (a)

corresponds to strategy 1. The initial decay of the amplitude of x is followed by a rapid growth, due to the forcing amplitude

in the time evolution of x ,

d 2 x

dt 2 + γ

d x

dt + (1 − ηx

2 − 3 η�xx ) x = (1 − 1 2 σ 2 t) sin θ, (28)

obtained by substituting �θθ = σ 2 t in (18) , increasing linearly for t > 2/ σ 2 . Since 2 ε/σ 2 = 2 / 45 � 1 , the growth in the

amplitude of x occurs well before passage through the resonance regime. Eventually, the solution blows up when 1 − ηx 2 −3 η�xx becomes negative; the spring modeled by (28) snaps.

While the linear growth of the forcing amplitude in (28) contributes to the premature failure of strategy 1, it is not

entirely responsible for the failure. Indeed, Fig. 3 (b) shows that strategy 2 succumbs to a similar failure, as its solution x to

(20) –(25) , which satisfies

d 2 x

dt 2 + γ

d x

dt + (1 − ηx

2 − 3 η�xx ) x = e −σ 2 t/ 2 sin θ, (29)

with a decaying forcing amplitude, is also unstable. The response x increases with time due to the linear growth of �θθ in

(20) , and thus in the forcing of x , leading to the observed rapid blow-up.

152 J. Yalim et al. / Commun Nonlinear Sci Numer Simulat 44 (2017) 144–158

Fig. 3. Envelopes of the reduced model response x for the slow downsweep case ( ε = −10 −6 ) with large noise ( σ = 10 −2 ). Note that (a) and (b) exhibit

finite time blow up around ω = 1 .

Strategy 3, as observed in Fig. 3 (c), also has an exponentially damped forcing. However, realizing the additional exact

exponential damping in the dynamic evolution of �v θ removes the instability observed in the two previous strategies.

The results of the Gaussian closure method, strategy 4, are shown in Fig. 3 (d). The response is also stable and visually

no different from that provided by strategy 3. While at this scale the results of strategies 3 and 4 seem indistinguishable,

comparing outcomes over a broad parameter regime helps highlight the defining characteristics of each method. Ultimately,

the wrong choice of closure scheme can result in an indefinite covariance matrix �( u ), especially for larger noise levels or

time windows [66] .

5.4. Noise vs. ramp rate competition

In general, the adaptation of a system to variations in parametric conditions requires a relatively slow change in parame-

ters for the response to remain close to what one would normally observe in a static experiment with fixed parameters. This

is the so-called quasi-static approximation. For large ramping rates, one might expect the response to significantly deviate

from the static case. In the Duffing problem with a fixed noise level σ , a rapid sweep of the frequency range (large | ε|) is

expected to miss resonance effects around ω = 1 and yield a weaker response. Above a certain level, (white) noise tends to

have a similar effect. As can be seen in (27) , large values of σ lead to a highly variable phase θ (large �θθ ), thus reducing

the impact of the forcing (2) , as can be inferred from the expressions (21) and (22) .

In order to evaluate the combined effect of the noise level and the ramping rate on the response, the maximal amplitude

during the (down) sweep of the frequency range is determined, from either the arithmetic mean of 10 0 0 realizations of u

via MCS of (5) , or from the solution u of the reduced models. This is shown in a linear scale in Fig. 4 over (log 10 σ , log 10 | ε|)

space. The MCS and reduced models have results that are comparable, especially in the case of Gaussian closure.

As expected, the response amplitude decreases monotonically with ε in the absence of noise ( σ = 0 ), as well as with

noise level σ in the static case ( ε = 0 ). For small enough noise levels σ and ramping rate ε, the response is similar to

the deterministic static case, with an amplitude associated to a (white) plateau. For fixed ε, the amplitude of the response

decreases with increasing noise level, as in the static case. However, perhaps surprisingly, non-trivial ramping rates appear

to help sustain large response amplitudes within an intermediate range of noise levels, σ ∈ (10 −4 , 10 −2 ) .

To further understand why the noise appears to affect the response only when

log 10 σ � 1

2

log 10 | ε| + C, C ∈ R , (30)

recall that resonance occurs when

ω(t) = ω 0 + εt ≈ 1 , i.e. , t ≈ C/ | ε| , and observe that (21) yields

f (θ ) ≈ f ( θ ) e −Cσ 2 / | ε| .

J. Yalim et al. / Commun Nonlinear Sci Numer Simulat 44 (2017) 144–158 153

Fig. 4. Comparison of the downsweep’s maximum amplitude response for (a) the ensemble MCS with 10 0 0 realizations, (b) strategy 3, and (c) strategy

4, as a function of σ and −ε ≥ 0 . The circle (black), diamond (yellow), and square (white) markers represent samples that are investigated further. (color

online)

Thus, noise has a visible impact only for C σ 2 /| ε| � 1, which reduces to (30) upon taking logarithms. The constant C in

(30) depends mildly on ω 0 and does not contribute substantially to the right-hand side if ω 0 = 1 + O(1) .

The relation (30) is confirmed by sample MCS and reduced model simulations for selected ( σ , ε) pairs with fixed ratios

σ 2 / | ε| ∈ { 10 −2 , 1 , 10 2 } , corresponding to the black circles, yellow diamonds, and white squares in Fig. 4 ; the results are

shown in Figs. 5–7 , where the ensemble responses were computed from 10 0 0 0 realizations. In Fig. 5 the noise level is

weak and the reduced model is indistinguishable from the MCS results.

In Fig. 6 , the noise rapidly dominates the phase of the forcing and the output from the reduced model quickly vanishes.

Although individual realizations of MCS retain a finite amplitude as ω decreases, their mean converges to 0 as the number

of realizations increases (a 10-fold increase in that number typically yielding a 10 −1 / 2 -fold reduction in the mean response).

While this behavior could be construed as stochastic resonance, it is more appropriate to interpret it as more effective

damping effects of (white) noise at lower ramp rates. This cooperation between stochasticity and ramping rate in the forcing

frequency is similar to the results from [67] .

In Fig. 7 , the similarity of solutions with equivalent σ /| ε| ratios becomes apparent, thus highlighting the care that

must go into a change of the ramping rate in a noisy system. For small and intermediate noise-to-ramping-rate ratios

( 10 −2 � σ 2 / | ε| � 1 in Figs. 5 and Fig. 7 ), strategy 3 provides similar responses for different ram p rates but introduces new

dynamics in a range 0.8 � ω � 1, compared to MCS. The “double hump” in the reduced model simulation originates from

the feed-back of the variance term �xx into the evolution equation for d v /dt, i.e. d 2 x /dt 2 , in (25) . Under the quasi-static

approximation ε → 0 with τ := | ε| t (i.e. ω) fixed, the solutions x of (25) and �x θ of (26) satisfy

d 2 x

dt 2 + γ

d x

dt + (1 − ηx

2 ) x = e −τ/ 2 sin θ + 3 ηx �xx , (31)

and

d 2 �xθ2

+ γd�xθ + (1 − 3 ηx

2 )�xθ = τ e −τ/ 2 cos θ. (32)

dt dt

154 J. Yalim et al. / Commun Nonlinear Sci Numer Simulat 44 (2017) 144–158

Fig. 5. Envelopes of the (a) ensemble mean response of x from MCS (10 0 0 0 realizations), (b) strategy 3, and (c) strategy 4, all with ratio σ 2 /ε = −10 −2 ,

associated to the black circles in Fig. 4 : (σ, ε) = (10 −4 , −10 −6 ) .

Fig. 6. Envelopes of the (a) ensemble mean response of x from MCS (10 0 0 0 realizations), (b) strategy 3, and (c) strategy 4, all with ratio σ 2 /ε = −10 2 ,

associated with the white squares in Fig. 4 : (σ, ε) = (10 −2 , −10 −6 ) .

Setting d / dt → 0 in (26) yields the approximation

�2 xθ ≈ �xx τ e −τ/ 2 ,

which establishes a direct coupling between (31) and (32) . These equations correspond to two oscillators, and from (31) one

expects, as in the case �xx = 0 , a nonlinear resonance tongue in x peaking well below ω = 1 . On the other hand, for given

x , (32) is a linear equation, with resonance effects expected to peak at ω ≈ 1.

Finally, we highlight the similarity of the Gaussian closure method (strategy 4) to the MCS in Figs. 4–7 . While the saddle-

node bifurcation does occur at a slightly higher frequency for strategy 4 compared to the MCS in Fig. 7 , it would seem

that strategy 4 provides a good estimation of the statistics of the Duffing oscillator over a broad parameter regime. Fig. 8

compares the statistical states of the ensemble MCS to strategy 4, demonstrating that the moments are all of the same order

and are fairly well predicted by the reduced scheme. The quantities �xx (non-negative) and �xθ / √

�xx �θθ (in [ −1 , 1] )

J. Yalim et al. / Commun Nonlinear Sci Numer Simulat 44 (2017) 144–158 155

Fig. 7. Envelopes of the (a,d) ensemble mean response of x from MCS (10 0 0 0 realizations), (b,e) strategy 3, and (c,f) strategy 4, all with a ratio σ 2 /ε = −1 ,

associated to the yellow diamonds in Fig. 4 : (a–c) with (σ, ε) = (10 −3 , −10 −6 ) and (d–f) with (σ, ε) = (10 −2 , −10 −4 ) . Note that while strategy 4 does not

succumb to secondary resonances, the saddle-node bifurcation occurs at a slightly higher frequency relative to the MCS ( ω ≈ 0.84 cp. ω ≈ 0.86). (color

online)

were monitored to check for possible indefiniteness of the covariance matrix (all of Figs. 4–8 have positive semi-definite

covariance). The variances �xx and �vv have much larger magnitude than the covariances �xv , �x θ and �v θ . Perhaps the

starkest differences are in the variances of the position and velocity responses; strategy 4 underestimates the maximum

and overestimates the minimum near ω = 1 . It is possible that these results may be improved by extending strategy 4 to

higher-order moments of the system.

6. Conclusions

The determination of lower-order (central) moments is a popular way to evaluate the behavior of solutions of stochastic

equations driven by noisy conditions. In general, closure conditions must be introduced to obtain self-contained equations

satisfied by these moments. A common closure strategy is to neglect terms involving moments at and above a certain order,

typically third-order, i.e. focusing only on the mean and variance.

In this study, we demonstrated on a Duffing oscillator subject to frequency modulated forcing that ignoring higher-order

moments of certain variables, such as those directly connected to the noise source (e.g. θ ), may lead to incorrect variance

estimates and premature blow-up of other variables’ mean (e.g. x ).

With proper consideration of the higher-order moments associated with noisy variables, the reduced deterministic sys-

tem can provide a faithful representation of the mean dynamics of state variables, in particular at low and high noise-to-

ramping-rate ratios (as measured by σ 2 /| ε|), and intermediate ratios when made more robust by employing the cumulant-

closure technique (strategy 4). The low dimensionality of the model Duffing problem enabled the validation of the reduction

with Monte Carlo simulations (MCS). However, more accurate treatment of covariances is an on-going challenge.

The modified moment-neglect method (strategy 3) exhibited artifacts which were not present in the MCS or Gaussian

closure (strategy 4) results for low to intermediate noise-to-ramping-rate ratios. We connected these artifacts to the feed-

156 J. Yalim et al. / Commun Nonlinear Sci Numer Simulat 44 (2017) 144–158

Fig. 8. Envelopes of the statistical states of the Duffing oscillator for the faster yellow diamond from Fig. 4 with (σ, ε) = (10 −2 , −10 −4 ) : (a–e) show the

ensemble states from MCS with 10 0 0 0 realizations and (f–j) the reduced states with strategy 4.

back mechanism of higher-order moments with their own resonance patterns (such as �x θ ) onto lower-order moments

(such as x ).

The reduced model allowed us to quickly identify a quadratic coupling between the noise parameter and the frequency

ramping rate such that similar results would be obtained. Further, increasing the ramping rate under a fixed noise level led

to a counter-intuitive increase in the system’s response amplitude. This effect was attributed to better damping properties

of noise at lower ramp rates, rather than stochastic resonance. Besides stressing the care that must go into adjusting these

J. Yalim et al. / Commun Nonlinear Sci Numer Simulat 44 (2017) 144–158 157

parameters, these result highlight the possibility of performing equivalent quasi-static experiments at higher sweep rates

when introducing an appropriate noise source.

The class of strategies investigated here involves the solution of a coupled system of nine ODE in the case of the Duffing

problem, which is equivalent to 3 direct Monte Carlo simulations. For problems with n discrete variables the resulting ODE

system becomes n + n (n + 1) / 2 = n (n + 3) / 2 -dimensional, with a cost of about n /2 Monte Carlo simulations, which remains

manageable for discretizations of PDEs in one dimension, such as Burger’s equation. For problems in higher dimensions

however, the cost of a full simulation of the covariance system becomes prohibitive. Strong diagonal dominance of the co-

variance matrix may point towards possible remediation of the dimensionality issue, using ideas from principal component

analysis and classification analysis, such as (nonlinear) shrinkage or regularization via low-rank or sparse approximations.

Acknowledgments

This work was partially funded by a summer research block grant for J. Yalim provided by Arizona State University’s

School of Mathematical and Statistical Sciences.

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