Stochastic modeling of nonlinear oscillators under combined Gaussian and Poisson White noise

Transcription

1 Nonlinear Dynaics anuscrip No. will be insered by he edior Sochasic odeling of nonlinear oscillaors under cobined Gaussian Poisson Whie noise A viewpoin based on he energy conservaion law Xu Sun Jinqiao Duan Xiaofan Li Received: dae / Acceped: dae Absrac A sochasic differenial equaion odel is considered for nonlinear oscillaors under exciaions of cobined Gaussian Poisson whie noise. Since he soluions of sochasic differenial equaions can be inerpreed in ers of several ypes of sochasic inegrals, i is soeies confusing abou which inegral is acually appropriae. In order for he energy conservaion law o hold under cobined Gaussian Poisson whie noise exciaions, an appropriae sochasic inegral is inroduced in his paper. This sochasic inegral reduces o he Di Paola-Falsone inegral when he uliplicaive noise inensiy is infiniely differeniable wih respec o he sae. The sochasic inegral inroduced in his paper is applicable in ore general siuaions. Nuerical exaples are presened o illusrae he heoreical conclusion. Keywor nonlinear oscillaor non-gaussian whie noise Poisson whie noise sochasic differenial equaions sochasic inegrals X. Sun Deparen of Applied Maheaics Huazhong Universiy of Science Technology Wuhan 4374, Hubei, China Tel.: E-ail: xsun5@gail.co; xsun@hus.edu.cn J. Duan Deparen of Applied Maheaics Illinois Insiue of Technology, Chicago, IL 666, USA X. Li Deparen of Applied Maheaics Illinois Insiue of Technology, Chicago, IL 666, USA Sybols Descripion ie ass k siffness coefficien x displaceen depending on ie ẋ firs derivaive of x wih respec o ie. ẋ second derivaive of x wih respec o ie gx, ẋ a funcion of x ẋ which appears as a generalized force er fx, ẋ a funcion of x ẋ which appears as he coefficien of noise er B Brownian oion Ḃ Gaussian whie noise defined as he generalized ie derivaive of he Brownian oion B C Copound Poisson process Ċ Copound Poisson whie noise defined as he generalized ie derivaive of he copound Poisson process C N Poisson process λ inensiy paraeer of he Poisson process N b a consan represening he weigh of B in L c a consan represening he weigh of C in L U i a uni sep funcion a ie i δ i Dela funcion a ie i R i Ro variable represening he i-h ipulse in C L a sochasic process defined as bbcc L cobined Gaussian Poisson whie noise defined as he generalized ie derivaive of he sochasic process L x y sae variable defined as ẋ A A arix defined as k k ω a variable defined as Io calculus Sraonovich calculus Cs he lef lii of Cs a s Cs he jup size of Cs a s, defined as Cs Cs ẋ i, r he value of ẋs a s i as Cs juped fro C i o r

2 Xu Sun e al. Inroducion Differenial equaions are exensively used in odeling dynaical syses in science engineering. When dynaical syses are under ro influences, sochasic differenial equaions SDEs are ofen appropriae odels. The soluions of SDEs are inerpreed in ers of sochasic inegrals 5. Dynaical syses subjec o Gaussian whie noise are ofen odeled by SDEs wih Brownian oion, he soluions are popularly inerpreed by he Io inegral,. Alhough he Io inegral is self-consisen aheaically, i is no he only ype of sochasic inegrals for inerpreing SDEs. Oher sochasic inegrals, such as he Sraonovich inegral,, have also been used o inerpre an SDE as a sochasic inegral equaion. There is no righ or wrong choice when choosing eiher Io or Sraonovich inegrals aheaically, since he wo inegrals can be convered ino each oher, provided ha he inegr saisfies cerain soohness condiions,. However, hese sochasic inegrals have differen definiions, one ay be ore direcly relaed o a pracical siuaion han he oher. While he Io inegral is a reasonable choice in any applicaions including finance biology, he Sraonovich inegral is observed o be ore appropriae in physical engineering applicaions. The S- raonovich inegral has an exra er coparing wih he corresponding Io inegral: he so-called correcion er 6. Soe auhors 7, 8 aribue his correcion er o he conversion fro physical whie noise o ideal whie noise. Beyond he Io inegral Sraonovich inegral, i is recenly found ha he A-inegral he general α-inegral 9, are ore relevan o realiy in soe applicaions. Dynaical syses driven by non-gaussian whie noise, especially Poisson whie noise, have araced a lo of aenion recenly. Correcion ers for convering Io SDEs o Sraonovich SDEs wih Poisson whie noise are presened in, 3. The DiPaola-Falsone inegral has been deonsraed discussed by any auhors 4 also soe confusions 3, 4. In our recen work5, i is shown ha DiPaola-Falsone inegral is aheaically equivalen o he soluion of an ODE syse. Theoreically, any differen kin of sochasic inegrals ay be defined for Poisson whie noise, as in he siuaion for Gaussian whie noise. I is desirable o know which sochasic inegral is ore relevan o realiy. In his paper, we consider ro vibraion under exciaions of a class of non-gaussian whie noise in ers of sochasic processes wih jups, including Poisson whie noise, we will define a sochasic inegral such ha soe fundaenal physical law e.g., energy conservaion law is saisfied. To his end, we consider a vibraion syse as a ass-spring oscillaor wih ro exciaion.ẋ kx gx, ẋ fx, ẋ L, where represens he ass, k is he siffness coefficien of he spring, x is he displaceen depending on ie. Moreover, gx, ẋ fx, ẋ L represen he generalized force ers, which ay originae fro exernal or paraeric exciaions. Finally, L is a whie noise defined as he generalized iederivaive of a sochasic process L bb cc, where b c are consans, B is a Gaussian process, C is a copound Poisson process as follows C N i R i U i. 3 In Eq. 3, N represens he nuber of jups up o ie is a Poisson process wih inensiy paraeer λ, U i is a uni sep funcion a Heaviside funcion a i, R i is a ro variable represening he i-h ipulse. I follows fro ha L bḃ cċ, 4 where Ḃ is he Gaussian whie noise, Ċ is he Poisson whie noise expressed as Ċ N i R i δ i. 5 Noe ha 4 is a general noise odel including he Gaussian whie noise b, c, he Poisson whie noise b, c, he cobined Gaussian Poisson whie noise b c. The second-order equaion can be rewrien as a syse of firs order SDEs x x d ẋ k d ẋ d dl. gx, ẋ fx, ẋ 6 Since L is non-differeniable alos everywhere, 6 canno be inerpreed in he fraework of classical calculus. In fac he soluion of 6 is inerpreed wih a

3 Sochasic odeling of nonlinear oscillaors under cobined Gaussian Poisson Whie noise 3 sochasic inegral, x x xs ẋ ẋ k ẋs gxs, ẋs dls. 7 fxs, ẋs Defining y x, A ẋ k using he variaion of paraeers forula, he soluion o Eq., or Eq. 6, can also be rewrien as, y e A y e A s gxs, ẋs e A s dls, 8 fxs, ẋs x x where y is he iniial condiion. ẋ ẋ I can be verified ha e A I A! A cosω sinω ω,! ω sinω cosω 9 k where ω. Subsiuing 9 ino 8, we ge x cosωx sinω ẋ ω sin ω s gxs, ẋs ω sin ω s fxs, ẋs dls, ω ẋ ω sinωx cosωẋ cos ω s gxs, ẋs cos ω s fxs, ẋs dls. Noe ha he sochasic inegrals in Eqs. 7 or are ye o be defined. As saed earlier, he sochasic inegrals which can be used o inerpreed SDEs ay no be unique. In order o consruc a SDE odel ha is physically relevan o he real syse, we will propose a sochasic inegral such ha he fundaenal physical law e.g., energy conservaion law is saisfied. The inegral defined in his way urns ou o be ore general han he DiPaola-Falsone inegral, in fac, he laer is a special case when he noise inensiy f is infiniely differeniable wih respec o he sae. This paper is organized as follows. In Sec., we derive a sochasic inegral, saring fro he energy conservaion law, ha is suiable for he SDE odel of he nonlinear ro oscillaor 6. The relaion beween he proposed sochasic inegral he DiPaola- Falsone sochasic inegral is discussed in Sec. 3. Nuerical eho wih an illusraive exaple are presened in Sec. 4. Sochasic inegrals for nonlinear ro vibraion Now we define or derive a sochasic inegral so ha fundaenal physical laws, such as he energy conservaion or energy-work conservaion law, hold. As for he nonlinear oscillaor described by he SDE, we expec he energy-work conservaion be saisfied ẋ kx ẋ kx gxs, ẋs fxs, ẋs Ls dxs, where ẋ kx represens he oal echanical energy of he syse a ie, he inegr in he righ h side is he forcing er of. Equaion expresses ha he change in he oal echanical energy is equal o he work done by he exernal forces. Wriing in he for of sochasic inegral, we have ẋ kx gxs, ẋsẋs gxs, ẋsẋs ẋ kx fxs, ẋs Lsẋs fxs, ẋsẋs dls. As saed before, he sochasic inegral wih respec o L should be defined such ha he soluion of saisfies he energy conservaion law. I follows fro ha a sochasic inegral wih respec o L can be decoposed ino wo ers: sochasic inegral wih respec o B sochasic inegral wih respec o C. We define he wo ers separaely in he following subsecions.. Poisson whie noise When b c, he sochasic process as expressed in reduces o a copound Poisson process.

4 4 Xu Sun e al. Noe ha he jup size of Cs a ie s can be expressed as Cs Cs Cs, where Cs is he lef lii of Cs a s. Suppose Cs jups a ies i i,,, hen he soluion 7 can be wrien as x x ẋ ẋ k N i i ẋs, i xs fxs, ẋs dcs, gxs, ẋs 3 where N, as shown in 3, represens he nuber of jups upo ie. In he following, we shall derive he sochasic inegral wih respec o jups such ha he energy conservaion law is saisfied. Firs, le us exaine he changes in he syse a i-h jup occurred a ie i i N. Fro 3, he displaceen x is coninuous while he velociy ẋ undergoes an jup given by x i x i, ẋ i ẋ i i 4 fxs, ẋs dcs. i The change in he oal energy due o he i-h jup is ha in he kineic energy given by ẋ i ẋ i i i fxs, ẋsẋs dcs, 5 due o he coninuiy of he displaceen x across an jup. If he inegrals wih respec o jups are defined in he sense of Io, hen 4 5 becoe ẋ i ẋ i fx i, ẋ i C i, 6 ẋ i ẋ i fx i, ẋ i ẋ i C i, 7 respecively. Since C i, i is clear ha 6 conradic wih 7. This indicaes ha he energy conservaion law canno be saisfied when he inegrals wih respec o jups are inerpreed in he sense of Io. In he following, we shall show ha he inegrals should be inerpreed as a kind of Rieann inegral on he iaginary pah along he jup o saisfy he energy conservaion law. Le ẋ i, r be he value of ẋs a ie i as Cs juped fro C i o r. Then ẋ i, C i ẋ i ẋ i, C i ẋ i. Wih he inegrals being inerpreed as he Rieann inegral on he iaginary pah along he jup, he energy-work law 5 can be wrien as ẋ i, C i ẋ i, C i Ci C i fx i, ẋ i, rẋ i, r dr. 8 he soluion 4 becoes ẋ i, C i ẋ i, C i Ci C i fx i, ẋ i, r dr. 9 Since he jup size can be any value, i follows fro 8 9ha for any λ R, i is rue ha ẋ i, λ ẋ i, C i λ C i fx i, ẋ i, rẋ i, r dr, ẋ i, λ ẋ i, C i λ C i fx i, ẋ i, r dr. Taking derivaives of boh sides of wih respec o λ, respecively, we ge he idenical ordinary differenial equaion ODE d i, λ fx i, ẋ i, λ. Using he fac ha ẋ i, C i ẋ i ẋ i, C i ẋ i, i follows fro ha ẋ i ẋ i Y i C i, 3 where Y i C i is deerined by he iniial or erinal value proble of he ODE d dλ Y iλ fx i, Y iλ ẋ i, 4 Y i. Noe ha in 4, λ C i for C i > or C i λ for C i <. Coparing he original soluion expression 4 wih he new forula 3, i can be seen ha he las er in 4 should be defined as Y i C i, hence, he sochasic inegral fxs, ẋs dcs should be defined as fxs, ẋs dcs N i Y i C i, 5 where Y i C i is he soluion o he ODE 4, N represens he nuber of jups up o ie.

5 Sochasic odeling of nonlinear oscillaors under cobined Gaussian Poisson Whie noise 5. Gaussian whie noise I has been observed ha Sraonovich inegral, insead of Io inegral, is ore appropriae in soe physical syses under Gaussian whie noise. In his subsecion, we verify his known observaion in he conex of he above nonlinear ro oscillaor odel wih Gaussian whie noise, by deonsraing ha in order for he energy conservaion law o hold, he sochasic inegral us be Sraonovich. Noe ha his conclusion for Gaussian whie noise is no new, bu he derivaion is presened here o derive he observaion fro a differen perspecive i.e., energy conservaion law. Wih b c, i follows fro ha he sochasic process L reduces o a Brownian oion, becoe x cosωx sinω ẋ ω sin ω s gxs, ẋs ω sin ω s fxs, ẋs dbs, ω ẋ ω sinωx cosωẋ cos ω s gxs, ẋs cos ω s fxs, ẋs dbs. ẋ kx respecively. gxs, ẋsẋs ẋ kx 6 fxs, ẋsẋs dbs, 7 There are wo ypes of sochasic inegral exensively used for SDEs driven by Brownian oions: Io inegral Sraonovich inegral. Noe ha differen lieraure ay use differen noaion o denoe Io Sraonovich calculus. Throughou his paper, we use o denoe Io calculus, for Sraonovich calculus. In he sense of Io, 6 7 can be wrien as x cosωx sinω ẋ ω sin ω s gxs, ẋs ω sin ω s fxs, ẋs dbs, ω ẋ ω sinωx cosωẋ cos ω s gxs, ẋs cos ω s fxs, ẋs dbs. ẋ kx gxs, ẋsẋs ẋ kx 8 fxs, ẋsẋs dbs, 9 respecively. In he sense of Sraonovich, 6 7 can be wrien as x cosωx sinω ẋ ω sin ω s gxs, ẋs ω sin ω s fxs, ẋs dbs, ω ẋ ω sinωx cosωẋ cos ω s gxs, ẋs ẋ kx cos ω s fxs, ẋs dbs. gxs, ẋsẋs ẋ kx 3 fxs, ẋsẋs dbs, 3 respecively. Provided ha he funcion f is sufficien s- ooh, he soluions in Sraonovich inegrals, 3 3, can be convered ino he following fors wih Io

6 6 Xu Sun e al. inegrals x cosωx sinω ẋ ω sin ω s gxs, ẋs ω fxs, ẋsf ẋxs, ẋs sin ω s fxs, ẋs dbs, ω ẋ ω sinωx cosωẋ cos ω s gxs, ẋs ẋ kx fxs, ẋsf ẋxs, ẋs cos ω s fxs, ẋs dbs, ẋ kx gxs, ẋs f xs, ẋs fxs, ẋsf ẋxs, ẋsẋs 3 fxs, ẋsẋs dbs. 33 As shown in he Appendix, he soluion 3 saisfies he energy-work relaion 33, suggesing ha when he roness is odeled in sense of Sraonovich, he energy-work conservaion law is saisfied. On he oher h, in a siilar procedure as in he Appendix, i can be shown ha he energy-work law 9 conradics wih he soluion 8. Therefore, Sraonovich inegral insead of Io inegral should be used so ha his nonlinear ro oscillaor odel saisfies he energy conservaion law..3 Cobined Gaussian Poisson whie noise When boh b c, he exciaion is a cobined Gaussian Poisson whie noise. Cobining he resuls in he subsecions.., we find ha, in order o saisfy he energy-work conservion law, one has o inerpre he sochasic inegrals wih respec o Brownian oions as Sraonovich inegrals, he inegrals wih respec o jups as he su of soluions of soe firs order ODEs. Therefore, he soluion o is given by he expression 7, where he sochasic inegral is defined as fxs, ẋs dls b N fxs, ẋs dbs Y i L i, 34 i where L i cc i C i c C i. Recall ha, in 34, denoe inegrals in he Sraonovich sense, N represens he nuber of jups up o ie, Y i L i is he soluion o he ODE 4, where λ akes value of λ L i for L i > or L i λ for L i <. To suarize, he ain resul of his secion is o deonsrae ha in order o saisfy he energy conservaion law, he sochasic inegral wih respec o Poisson whie noise should be inerpreed as in equaion 5, he sochasic inegral wih respec o he cobined Gaussian Poisson whie noise should be inerpreed as in equaion 34. Relaionship beween he above sochasic inegrals soe oher sochasic inegrals will be revealed in he nex secion. 3 Relaion wih he Di Paola-Falsone sochasic inegral In his secion, we shall show ha he correcion er Y i L i, as given by he soluion o he ODE 4, is consisen wih he one proposed in he work by Di Paola Falsone,3, when he noise inensiy f is infiniely differeniable wih respec o he sae variable x. In oher wor, when he uliplicaive noise inensiy is infiniely sooh, he sochasic inegral inroduced in his paper reduces o he Di Paola-Falsone sochasic inegral. This eans ha he Di Paola-Falsone inegral is a special case of he s- ochasic inegral ha guaranees he energy conservaion law. The sochasic inegral inroduced in his paper is hus applicable in ore general siuaions. If f is infiniely differeniable wih respec o x so ha he he unique soluion of 4 is analyic e.g., expressible as a convergen Taylor series, hen we have he following convergen Taylor expansion for Y i L i Y i L i Y i d dλ Y iλ λ L i d! dλ Y iλ λ L i. 35

7 Sochasic odeling of nonlinear oscillaors under cobined Gaussian Poisson Whie noise 7 I follows fro 4 ha for any n, d n dλ n Y iλ d dy i λ { } d n dλ n Y iλ f x i, Y iλ ẋ i. 36 Subsiuing 36 ino 35, using he fac ha Y i, we ge Y i L i j f j x i, ẋ i j! L i j, 37 where f x, ẋ i fx i, ẋ i f j x, ẋ i f i x i, ẋ i λ fx i, ẋ i λ λ for j. 38 Thus he correcion er given by 37 is exacly he sae as he one proposed in,3. We have shown ha he correcion er Y i L i can be obained in wo ways: Eiher solving he iniial value proble o he ODE 4 or copuing he expansion 37. Noe ha he forer approach of solving 4 is applicable under uch ore general condiion han he laer one of evaluaing he infinie series 37, because he exisence of he soluion o he ODE only require fx, y o be Lipschiz coninuous, bu 37 des fx, y o be infiniely differeniable. Readers are referred o lieraure 5 for a deailed coparison of hese wo fors of expressions. The sochasic inegral, as defined in 34 4, is derived direcly fro he energy conservaion law. Alhough his inegral for is he essenially he sae as ha in our previous work 5, he derivaion here in he curren paper does no rely on DiPao-Flasones forula while 5 focuses on he rigorous proof of he relaionship beween he proposed sochasic inegral he DiPaola-Falsone forula. This inegral is applicable beyond he field of ro vibraion, as shown in 5. 4 Nuerical experiens Soluions of he SDE for a nonlinear oscillaor, defined by 7 34, can hardly be obained wih analyical eho. In his secion, he SDE is nuerically solved o verify he conclusion obained in secion. Consider he case wih boh Gaussian Poisson whie noises, dl b db c dc, wih he copound Poisson process given by 5. The nuerical procedure of he SDE for a nonlinear oscillaor defined by 7 34 is as follows. On each ie subinerval i < < i, i.e. when no jups occur, becoes dx ẋ d dẋ k x d gx, ẋ d b fx, ẋ dbs. 39 The above equaion can be convered ino Io SDE hen copued by convenional algorihs for Io S- DEs, such as Euler ehod, Milsein ehod, or oher high-order algorihs 6. A he ie i when an jup occurs, i follows fro 7 34 ha { x i x i, ẋ i ẋ i Y 4 i L i, where Y i L i is obained by solving he deerinisic ODE 4 using Runge-Kua or ulisep eho. Consider he following sochasic Duffing-van der Pol equaion.. x x ẋ x x 3 ẋ L, 4 wih he iniial condiion x ẋ. The SDE 4 can be wrien in for of wih, k, gx, ẋ x ẋ x 3 fx, ẋ ẋ. In he siulaion, we ake L as in wih b c, i.e. L B C, where C is a pure jup process given by 3 wih N being a Poisson process wih inensiy paraeer as λ 3.4 R i i,, N being ro nubers of he sard noral disribuion. Figure shows a saple pah of he driven process L. Case In his case, 4 or is inerpreed by In he siulaion, o inegrae 7 34 nuerically, we use Euler ehod o advance 39 when no jups occur, while evaluae 4 by solving 4 wih Euler ehod when jups arrive. Noe ha o apply Euler eho, he sochasic inegral in 39 need o conver ino Io inegral. The sep size of Euler ehod for solving boh 39 4 is.. Figs. 3 show he nuerical soluion of he displaceen x he velociy ẋ respecively, corresponding o he pah shown in Fig.. By coparing Fig. wih 3, i is ineresing o see ha while here are jups for he velociy, here is no jups for he displaceen. This is a consequence of equaion 4. Based on he nuerical soluions of x ẋ, as shown in Figs. 3, we

8 8 Xu Sun e al. can copue he energy increen E he work W K, which are defined as E W K x ẋ x ẋ, 4 gx, ẋẋs gx, ẋẋs N i fx, ẋẋs dls fx, ẋẋs dbs Ȳ i L i, 43 respecively, where L i cc i C i c C i, denoe inegrals in he Sraonovich sense, N is he nuber of jups upo ie, Ȳi L i is he soluion o he following ODE, d dλȳiλ fx i, Ȳiλ ẋ i Ȳiλ ẋ i, Ȳ i. 44 wih λ aking value of λ L i for L i > or L i λ for L i <. The resuls of E W K are copared in Fig. 4. For convenience of coparison, par of he daa in Fig. 4 for ie.,.4,.6,.8 are lised in Table. As shown in Fig. 4 or Table, he energy increen E appears o agree well wih he work W K. The difference beween E W K is ainly due o he runcaion error caused by Euler ehod, which has accuracy of order in copuing sochasic inegral wih respec o brownian oion 6, where is he sep size, as saed earlier. Noe ha any eho such as Milsein ehod, are available o copue he sochasic inegral wih high order accuracy 6. However, hese eho are ou of he scope of his paper will no be used here. Case In his case, 4 or is inerpreed by using Io sochasic inegrals. Now 39 4 becoe dx ẋ d dẋ k x d gx, ẋ d 45 b fx, ẋ db, respecively. In he siulaion, he driving process L all he siulaion paraeers are aken he sae as in case. Figures 5 6 presen he corresponding nuerical soluions of he displaceen velociy, respecively. Coparison of he energy increen, defined by 4, he work done, now defined by W K gx, ẋẋs b N i fx, ẋẋs dbs fxs, ẋs ẋs L i, 47 is presened in Fig. 7. Par of Daa in Fig. 7 for ie.,.4,.6,.8, are lised in Table. Noe ha all he curves in Fig. 7 end o have a very sall variance in he ie span.84 <. This is he consequence of he fac ha he velociy is very sall for.84 <, as shown in Fig. 6. Since fx, ẋ gx, ẋ ẋ, i follows fro 4, ha boh he energy increen he work change slowly for very sall velociy ẋ. Coparing Fig. 7 wih Fig. 4, Table wih Table, we can see ha wih he sae siulaion paraeers, he proposed sochasic inegral lea o uch s- aller difference beween he energy increen he corresponding work han is Io inegral counerpar, as can also be seen ore clearly fro Fig. 8, where he above wo differences are jus shown ogeher. This suggess ha he proposed inegral is a beer choice han Io inegral in order for he energy conservaion law o be saisfied in sochasic odeling under cobined Gaussian Poisson whie noise. { x i x i, ẋ i ẋ i fxs, ẋs L i, 46

9 Sochasic odeling of nonlinear oscillaors under cobined Gaussian Poisson Whie noise 9 L Tie Fig. A saple pah of he driving process L as a cobinaion of a Gaussian process a copound Poisson process given in 3. Velociy Tie Fig. 3 The evoluion of he velociy ẋ as he soluion of Duffing-van der Pol equaion 4 defined by he proposed inegral 7 34, where he driving process L is given in Fig.. 8 Increen of Energy Work Displaceen.5 Magniude Tie Fig. The evoluion of he displaceen x as he soluion of Duffing-van der Pol equaion 4 defined by he proposed inegral 7 34, where he driving process L is given in Fig Tie Fig. 4 Coparison of he change in oal energy, defined by 4, he work done by he force, defined by 43, where he proposed inegral 7 34 is used. Table Energy increen E vs. he work W K for he case wih he proposed inegral, where he Err is defined as he difference beween E W K ie E W K Err

10 Xu Sun e al Increen of Energy Work Displaceen.4 Magniude Tie Fig. 5 The evoluion of he displaceen x as he soluion of Duffing-van der Pol equaion 4 defined by Io inegral, where he driving process L is given in Fig Tie Fig. 7 Coparison of he change in oal energy, defined by 4, he work done by he force, defined by 47, where Io inegral is used..5.5 for he proposed inegral for Io inegral.5 Velociy.5.5 Err Tie Fig. 6 The evoluion of he velociy ẋ as he soluion of Duffing-van der Pol equaion 4 defined by Io inegral, where he driving process L is given in Fig Tie Fig. 8 Coparison of he difference beween he energy increen he work for cases wih he proposed inegral Io inegral, respecively. Table Energy increen E vs. he work W K for he case wih Io inegral, where he Err is defined as he difference beween E W K ie E W K Err

11 Sochasic odeling of nonlinear oscillaors under cobined Gaussian Poisson Whie noise 5 Conclusion We consider sochasic differenial equaion odels of nonlinear oscillaors under exciaions of cobined Gaussian Poisson whie noise. Since any differen ypes of sochasic inegral can be defined o inerpre soluions of sochasic differenial equaions, i is desirable o know which sochasic inegral is ore relaed o physical realiy. To his end, we inroduce a sochasic inegral which is defined such ha he energy conservaion law is saisfied. The relaionship of his sochasic inegral he well known DiPaola-Falsone inegral is discussed. I urns ou ha he forer is consisen wih he laer, bu copared o he laer, he forer is applicable under ore general condiions. We poin ou ha here are no absolue bes ways in choosing aong sochasic inegrals, i depen on specific applicaion. The proposed sochasic inegrals are ore appropriae in physics such as echanical syses while he Io inegral is ore suiable in finance applicaions. We will furher explore applicaions of he proposed sochasic inegral in our fuure work. Appendix: Proof of he energy-work law 33 fro he soluion 3 For convenience, we inroduce he following noaions: gs gxs, ẋs, fs fxs, ẋs, fẋs f ẋsxs, ẋs. In his Appendix, all s- ochasic inegrals wih respec o Brownian oion are in he sense of Io we have dropped noaion. Then he energy-work law 33 is equivalen o h side of 48, we ge sin ω s gs fsf ẋs LHS sinω sfs dbs cos ω s gs fsf ẋs cosω sfs dbs ω cosωx sinωẋ sin ω s gs fsf ẋs ω cosωx sinωẋ sin ω s gs fsf ẋs sinω sfs dbs sinω sfs dbs ω sinωx cosωẋ cos ω s gs fsf ẋs ω sinωx cosωẋ cosω s gs fsf ẋs cosω sfs dbs cosω sfs dbs. 49 Subsiuing 3 ino he righ-h side of 48, we ge ẋ ω x ẋ ω x gs fsf ẋs ẋs f s fsẋs dbs. 48 Nex, we show he soluion given in 3 saisfies he energy-work law 48. Denoe he righ lef h sides of 48 by RHS LHS, respecively. Subsiue 3 ino he lef RHS s s s s gs fsfẋs ω sinωsx cosωsẋ cosωs p gs fsf ẋs gp fpf ẋp cosωs p dp gs fsf ẋs fp dbp f s fs ω sinωsx cosωsẋ dbs cosωs pfs gp fpf ẋp dp dbs cosωs pfsfp dbp dbs. 5

12 Xu Sun e al. To prove LHS in 49 is equal o RHS in 5, we clai he following facs s s cosωs p gs fsf ẋs gp fpf ẋp dp sinω s gs fsf ẋs cosω s gs fsf ẋs cosωs pfsfp dbp dbs, 5 f s sinω sfs dbs cosω sfs dbs, 5 s s cosωs p gs fsf ẋs cosωs p gp fpf ẋp cosω s gs fsf ẋs fp dbp cosω sfs dbs sinω s gs fsf ẋs fs dp dbs sinω sfs dbs 53 gs fsf ẋs ω sinωsx cosωsẋ ω sinωx cosωẋ cosω s gs fsf ẋs ω cosωx sinωẋ sinω s gs fsf ẋs, 54 fs ω sinωsx cosωsẋ dbs ω sinωx cosωẋ ω cosωx sinωẋ cosω sfs dbs sinω sfs dbs. 55 One can easily see ha are rue by using he rignoeric ideniies cosωs cosω cosω s sinω sinω s, sinωs sinω cosω s cosω sinω s, In he following, we give he proofs for 5 5. The proof of 53 is siilar o hose for 5 5 is no given here. To prove 5 is rue, we rewrie he righ-h side of 5 as double inegrals sinω s gs fsf ẋs cosω s gs fsf ẋs sinω p sinω s gs fsf ẋs dp s cosω p cosω s gp fpf ẋp gs fsf ẋs dp cosωs p gp fpf ẋp gs fsf ẋs dp cosωs p gp fpf ẋp gs fsf ẋs dp. gp fpf ẋp Siilarly, o prove 5, we rewrie he righ-h side 5 as sinω sfs dbs cosω sfs dbs cosωs pfsfp dbp dbs. 56

13 Sochasic odeling of nonlinear oscillaors under cobined Gaussian Poisson Whie noise 3 The inegral doain for he righ-h side of 56 is a square given by A {s, p s,, p, }. Decopose he square ino hree pars: A {s, p s < p }, A {s, p p < s }, A 3 {s, s s }, hen he righ-h side of 56 becoes cosωs pfsfp dbp dbs A A A 3 cosωs pfsfp dbp dbs. 57 Noe ha cosωs pfsfp dbp dbs A cosωs pfsfp dbp dbs A s cosωs pfsfp dbp dbs, 58 A 3 cosωs pfsfp dbp dbs f s. 59 I follows fro ha 5 is rue. Add Eqs. 5 o 55, we ge LHS RHS, hence 33 is proved. References. B. K. Oksendal. Sochasic Differenial Equaions : an Inroducion wih Applicaions. Springer, 6h Ediion, 3.. F. C. Klebaner. Inroducion o sochasic calculus wih applicaions. nd Ediion, Iperial College Press, H. T. Zhu. Muliple-peak probabiliy densiy funcion of non-linear oscillaors under gaussian whie noise. Probabilisic Engineering Mechanics, 3:46 5, Z. H. Zhao, K. Chang, J. J. Neio. Asypoic behavior of soluions o absrac sochasic fracional parial inegrodifferenial equaions. Absrac Applied Analysis, 3:3886, G. Luo, J. Liang, C. Zhu. The ransversal hooclinic soluions chaos for sochasic ordinary differenial equaions. Journal of Maheaical Analysis Applicaions, 4:3 35, E. Wong M. Zakai. On he relaion beween ordinary sochasic differenial equaions. Inernaional Journal of Engineering Science, 3:3 9, R. A. Ibrahi. Paraeric Ro Vibraion. Research Sudies Press, Y. Yong Y. K. Lin. Exac saionary response soluion for second order nonlinear syses under paraeric exernal whie noise exciaions. Journal of Applied Mechanics, Transacions of he Aerican Sociey of Mechanical Engineers, 54:44 48, J. Shi, T. Chen, R. Yuan, P. Ao. Relaion of a new inerpreaion of sochasic differenial equaions o io process. Journal of Saisical physics, 48:579 59,.. R. Yuan P. Ao. Beyond io versus sraonovich. Journal of Saisical Mechanics, page P7,.. M. Volpe, L. Helden, T. Breschneider, J. Wehr, C. Bechinger. Influence of noise on force easureen. Physcial review leers, 4:76,.. M. Di Paola G. Falsone. Io sraonovich inegrals for dela-correlaed processes. Probabilisic engineering echanics, 8:97 8, M. Di Paola G. Falsone. Sochasic dynaics of non-linear syses driven by non-noral dela-correlaed processes. ASME Journal of applied echanics, 6:4 48, S. Caddei M. Di Paola. Ideal physical whie noise in sochasic analysis. Inernaional Journal of Non-linear Mechanics, 35:58 59, C. Proppe. The wong-zakai heore for dynaical syses wih paraeric poisson whie noise exciaion. Inernaional Journal of Engineering science, 4:65 78, M. Di Paola A. Pirroa. Direc derivaion of correcive ers in sde hrough nonlinear ransforaion on fokker-planck equaion. Nonlinear Dynaics, 36:349 36, A. Pirroa. Non-linear syses under paraeric whie noise inpu: Digial siulaion response. Inernaional Journal of Non-linear Mechanics, 4:88, A. Pirroa. Muliplicaive cases fro addiive cases: Exension of kologorov-feller equaion o paraeric poisson whie noise processes. Probabilisic engineering echanics, :7 35, Y. Zeng W. Q. Zhu. Sochasic averaging of quasilinear syses driven by poisson whie noise. Probabilisic engineering echanics, 5:99 7,.. Y. Zeng W. Q. Zhu. Sochasic averaging of n- diensional non-linear dynaical syses subjec o non-gaussian wide-b ro exciaions. Inernaional Journal of Non-linear Mechanics, 45:57 586,.. J. Wan W. Zhu. Sochasic averaging of quasiinegrable non-resonan hailonian syses under cobined gaussian poisson whie noise exciaions. Nonlinear Dynaics, 76:7 89, 4.. W. Liu, W. Zhu, W. Jia. Sochasic sabiliy of quasiinegrable non-resonan hailonian syses under paraeric exciaions of cobined gaussian poisson whie noises. Inernaional journal of non-linear Mechanics, 58:9 98, M. Grigoriu. The io sraonovich inegrals for s- ochasic differenial equaions wih poisson whie noise. Probabilisic engineering echanics, 3:75 8, S. L. J. Hu. Closure on discussion by di paola,. falsone, g., on response of dynaic syses excied by non-gaussian pulse processes. ASCE Journal of engineering echanics, :47 474, X. Sun, J. Duan, X. Li. An alernaive expression for sochasic dynaical syses wih paraeric poisson

14 4 Xu Sun e al. whie noise. Probabilisic Engineering Mechanics, 3: 4, P. Kloeden E. Plaen. Nuerical Soluions of S- ochasic differenial equaions. Springer, 99.