# Existence of Periodic Solutions for the Duffing Equation with Impulses

Authors: Xuxin Yang; Weibing Wang; Jianhua Shen
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

We study the existence of solutions to the Duffing equation with
impulses. By means of the Poincaré-Birkhoff fixed point theorem under given conditions, we
obtain the sufficient condition of existence of infinitely many solutions. Our results generalize
those of T. R. Ding. An example is presented to demonstrate applications of our main result.


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## Body

## 1. Introduction

Many evolution processes are characterized by the fact that at certain moments of time they experience abrupt changes in the value of the state variable. Theses processes are subject to short-time perturbations whose duration is negligible in comparison with the duration of the process. Consequently, it is natural to assume that these perturbations act instantaneously, that is, in the form of impulses. It is known, for example, that many biological phenomena involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, pharmacokinetics, and frequency-modulated systems do exhibit impulsive effects. Thus impulsive differential equations, that is, differential equations involving impulse effects, appear as a natural description of observed evolution phenomena of several real world problems [1].In its original form, the Duffing equation is a nonlinear second-order ODE which is used in physics to model oscillators. It is a well known example of a dynamical system exhibiting chaotic behavior. The equation is given by(1.1)x′′+δx′+αx+βx3=γcos⁡(ωt),
where the function x=x(t) is the displacement at time t, and numbers δ,α,β,γ and ω are constants.In this paper, we study the following Duffing type model given by Ding in [2](1.2)x′′+g(x)=f(x,t).
Clearly this equation contains the Duffing equation (1.1) without friction (i.e., δ=0) as a special case. In [3], Ding considers (1.2) in the case where g∈C(ℝ,ℝ) is superlinear at infinity:(1.3)lim⁡x→∞g(x)x=+∞,
and the function f∈C(ℝ×ℝ,ℝ) is T-periodic in t. By using the Poincaré-Birkhoff fixed point theorem, Ding showed that there exist infinitely many periodic solutions to this equation.By using a similar technique, Ming et al. [4] gave results concerning existence of infinitely many periodic solutions to the p-Laplace equation(1.4)(|x′|p-2x′)′+g(x)=f(t,x),p>1,
where g∈C(ℝ,ℝ) is p-sublinear in the sense(1.5)lim⁡|x|→0g(x)|x|p-2x=+∞,
and f∈C(ℝ×ℝ,ℝ) is 1-periodic in t. For this problem, only partial results are known. For example, the conjecture is true if g is even and superlinear at infinity and f≡0 [5]. For f≠0, more restrictions on g are required [6, 7].In this paper, by developing ideas of [6, 8, 9], we extend this technique to the situation where x is allowed to have impulses at given points. We consider the periodic solutions to the Duffing equation with impulses:(1.6)x′′+g(x)=0,t≠tk,x(tk+)=akx(tk),k∈Z,x′(tk+)=bkx′(tk),k∈Z,
where ak>0, akbk=1, ak+q=ak,tk+q=tk+T,0<t1<t2<⋯<tq<T,g∈C(ℝ,ℝ), and ℤ denotes the set of integers, x(tk+) and x′(tk+) are right limits of x(t) and x′(t) at t=tk, respectively. Let PC(ℝ)={x:ℝ→ℝ;x(t) is continuous everywhere except for tk at which x(tk+) and x(tk-) exist and x(tk-)=x(tk),k∈ℤ};PC1(ℝ)={x∈PC(ℝ);x′(t) is continuous differentiable everywhere except for tk at which x′(tk+) and x′(tk-) exist and x′(tk-)=x′(tk),k∈ℤ}.The paper is organized as follows. In Section2, some preliminary results are given. In Section 3, we construct the twist map which is necessary in order to apply the fixed-point results. Then, we obtain the main result for the existence of periodic solutions for the system by using the Poincaré-Birkhoff fixed-point theorem. In Section 4, an example is given to demonstrate the main result.


## 2. Preliminaries

The Poincaré-Birkhoff fixed point theorem is a powerful tool in studying periodic solutions for the planar ODE’s of the second order. There are several versions of this theorem, see [8, 10–13].Let(r,θ) be a polar coordinates on ℝ2 and A={x∈ℝ2:r1≤|x|≤r2} an annulus on ℝ2.Definition 2.1.
A mappingT is called as twist map if
(2.1)T:(r,θ)⟶(h(r,θ),θ+l(r,θ)),
where h(r,θ),l(r,θ) are continuous on A, 2π-periodic in θ, and l(r1,θ)l(r2,θ)<0.The proof of our main results is based on the following version of the Poincaré-Birkhoff theorem, due to Ding [2] (see also [10]).Lemma 2.2 (see [2]).
LetR>r>0. Suppose that T:ℝ2→ℝ2 is an area preserving homeomorphism, such that T is a twist map in an annulus
(2.2)A={x∈R2:r≤|x|≤R},
and 0∈T(D), and D={x:|x|<r}. Then T has at least two fixed points in A.For our convenience, we introduce the following condition:(H1)
g(x)x>0 for x≠0 and(2.3)lim⁡x→0g(x)x=+∞.Lemma 2.3.
Assume thatg∈C(ℝ,ℝ) and (H1) holds. Then for given M>0, there exists a constant δ=δ(M) such that for any ɛ>0 there is a continuous function gɛ such that(1)
(2.4)|gɛ(x)-g(x)|≤ɛ,(2)
gɛ has a form gɛ(x)=Mx near x=0, that is, there exists δ>0 such that
(2.5)gɛ(x)x≥Mfor0<|x|≤δ,
and gɛ∈C1([-T,T]∖{0},R), where 0<T<+∞.(3)
For a fixedλ>0,
(2.6)lim⁡x→0G(λx)G(x)=λ2,
where G(x)=∫0xgɛ(s)ds.Proof.
Chooseρ>0, and suppose G(x)≥ν>0 for |x|≥ρ. Then by (2.3), there is a constant δ=δ(M) such that
(2.7)g(x)x≥2M,for0<|x|≤2δ.
Choose μ>0 such that μ≤ɛ, and let γ>1, such that 2γμ2+μ≤ν2. Since g(0)=0, we can choose an interval I0=(-δ0,δ0) with δ0<δ such that
(2.8)|g(x)-Mx|≤μ2,∀x∈I0.
For x≠0, we choose an interval Ix⊂[-T,T] such that
(2.9)|g(y)-g(x)|≤μ2,∀y∈Ix,
and Ix⊂((1/2)x,2x) if x>0, Ix⊂(2x,(1/2)x) if x<0. We claim that there exists a covering of the interval [-T,T], say I0, Ii≡Ixi such that Ii satisfies (2.9) for all i=0,1,2,…,n. Here we only consider the case of [-T,-δ0](δ0<T). Because g is uniformly continuous on the closed interval [-T,-δ0], there exists a σ>0 independent of x,y∈[-T,-δ0] such that |g(y)-g(x)|≤μ/2 for |x-y|≤σ. Define a sequence σn(n≥1) by
(2.10)σn=sup⁡{σ:|g(y)-g(-T+∑i=1n-1σi)|≤12μ,∀y∈[-T+∑i=1n-1σi,-T+σ+∑i=1n-1σi]},
and σ0=0. It is easy to check that σn>0,n=1,2,…. Now we assume that ∑i=1∞σi≤T-δ0. Since g is continuous on the closed interval [-T,T], we obtain that
(2.11)|g(-T+∑i=1N-1σi+1N+1)-g(-T+∑i=1N-1σi)|≤12μ,
where N is a sufficiently large positive integer, which implies σN≥(N+1)-1 for sufficiently large positive integer N. On the other hands, ∑k=N∞k-1=∞, which implies ∑i=1∞σi=∞, a contradiction. We also assume that Ii∩Ij=∅ for |i-j|>1.
Letϕ0,ϕ1,ϕ2,…,ϕn be resolution of unit that corresponds to Ii(0≤i≤n) such that ∑i=0nϕi(x)=1, for x∈(xi,xi+1), and suppϕi⊂Ii for each i=0,1,…,n. Then, for x∈[-T,T], we define
(2.12)gɛ(x)=ϕ0(x)Mx+∑i=1nϕi(x)g(xi).
By (2.9) and (2.12), we have
(2.13)|gɛ(x)-g(x)|≤ϕ0(x)|Mx-g(x)|+∑i=1nϕi(x)|g(x)-g(xi)|≤μ≤ɛ,
proving the claim (2.4). Furthermore
(2.14)|gɛ(x)-g(x)|≤μ.
To see (2.5), we suppose that 0<|x|<δ and that, for i≥1, we have ϕi(x)≠0 and 1/2≤xi/x≤2. It follows that |xi|≤2|x|≤2δ and, by (2.7),
(2.15)g(xi)xi≥2M.
Then, by the above inequality and (2.12), we have
(2.16)gɛ(x)x=ϕ0(x)M+∑i=1nϕi(x)g(xi)xixix≥ϕ0(x)M+∑i=1nϕi(x)2M2=M.
Since gɛ has the form gɛ(x)=Mx near x=0, G(x) has the form G(x)=Mx2/2 near x=0, which implies (2.6).Letg∈C(ℝ,ℝ) and satisfies the condition (H1), and let gɛ be as Lemma 2.3. Consider the initial value problem for the following system:(2.17)x′=-y,y′=gɛ(x),t≠tk,x(tk+)=akx(tk),y(tk+)=bky(tk),k=1,2,…,x(0)=x0,y(0)=y0.Lemma 2.4.
For any sufficiently small(x0,y0) the system (2.17) has a unique solution pair x(t)=x(t,x0,y0), y(t)=y(t,x0,y0). Moreover, the functions x,y continuously depend on x0,y0.Proof.
Obviously, we only need to prove that the conclusion holds for the system without impulses(2.18)x′=-y,y′=gɛ(x),x(0)=x0,y(0)=y0.
Since gɛ is continuous, the initial value problem (2.18) has a solution by the Peano’s method.
Next, we show that the solution of (2.17) is unique. For γ>0 is sufficiently small, the initial value problem:
(2.19)x′=-y,y′=gɛ(x),t≥γ,x(γ)=α,y(γ)=β,
where α∈ℝ,β∈ℝ, has a unique solution since gɛ∈C1([γ,d],ℝ). Hence we need only to show that the solution of (2.18) is unique for 0≤t≤γ. If x0=0, (2.18) reduces to
(2.20)x′′+Mx=0,x(0)=0,x′(0)=-y0,
because gɛ has the form gɛ(x)=Mx near x=0, and thus our claim holds. If x0≠0, gɛ∈C1(Ω(x0),ℝ), where Ω(x0)={x:|x-x0|≤γ}. So the solution of (2.18) is unique for 0≤t≤γ. Continuous dependence on the initial data follows from the uniqueness.


## 3. Main Result

Our main result is the following theorem. This result generalizes Theorem  2.1 of Ding [3].Theorem 3.1.
Under the assumption (2.3), the Duffing equation (1.6) has an infinite sequence of solutions {xn} and ∥xn∥PC1[0,T]→0,asn→∞, where
(3.1)‖xn‖PC1[0,T]=max⁡{sup⁡t∈[0,T]{|xn(t)|},sup⁡t∈[0,T]{|xn′(t)|}}.In order to prove the main result, we need the following lemmas. First we construct the twist map. Next we consider the system (2.17), our goal is to control the behavior of the norms r(t)=x(t)2+y(t)2 at the points p(t)=(x(t),y(t))∈ℝ2.Lemma 3.2.
Suppose thatp(t)=(x(t),y(t)), where t∈[0,T] is a solution to the system (2.17). There exists a constant R0>0 and functions d1,d2,ɛ:(0,R0]→ℝ+ such that if ɛ≤ɛ(R), and r(0)=R<R0, then(1)
d1(R)≤r(t)≤d2(R),(2)
limR→0d1(R)=limR→0d2(R)=0.Proof.
Define the Liapunov function(3.2)F(x,y)=12y2+∫0xgɛ(s)ds.
We compute the differential of F along the curve p(t)=(x(t),y(t)), where t∈[0,T]:
(3.3)dF(p(t))dt=yy′+gɛ(x)x′=ygɛ(x)+gɛ(x)(-y)=0,
for t≠tk.
From Lemma2.3, we see that for a fixed λ>0, there exist c1(λ)≥0,c2(λ)≥0 such that
(3.4)c2(λ)G(x)≤G(λx)≤c1(λ)G(x),
where G(x)=∫0xgɛ(s)ds, when |x| is sufficiently small. We have
(3.5)F(p(tk+))=12y(tk+)2+∫0x(tk+)gɛ(s)ds=12bk2y(tk)2+∫0akx(tk)gɛ(s)ds.
From (3.4), we have
(3.6)min⁡{bk2,c2(ak)}F(p(tk))≤F(p(tk+))≤max⁡{bk2,c1(ak)}(12y(tk)2+∫0x(tk)gɛ(s)ds)≤max⁡{bk2,c1(ak)}F(p(tk)).
By using an impulsive integer inequality, we obtain
(3.7)F(p(0))∏0<tk<tmin⁡{bk2,c2(ak)}≤F(p(tk))≤F(p(0))∏0<tk<tmax⁡{bk2,c1(ak)}.
It follows that(3.8)B1F(p(0))≤F(p(t))≤A1F(p(0)),∀t∈[0,T],
where A1=∏0<tk<1max{bk2,c1(ak)}, and B1=∏0<tk<1min{bk2,c2(ak)}. We also have
(3.9)B1min⁡|p|=RF(p)≤F(p(t))≤A1max⁡|p|=RF(p),∀t∈[0,T],
and thus there exist positive constants d1(R),d2(R) such that
(3.10)d1(R)≤r(t)≤d2(R).
Furthermore, we have
(3.11)lim⁡R→0d1(R)=lim⁡R→0d2(R)=0.Next, we consider the following system:(3.12)x′=-y,y′=g(x),t≠tk,x(tk+)=akx(tk),y(tk+)=bky(tk),k=1,2,and so forth.It follows from Lemma3.2 that we can obtain a solution of (3.12) which can be extended to the interval [0,T] if |p0|=|(x0,y0)|≤R0. Now we can define the Poincaré map ϕ:(3.13)(x0,y0)⟼(x(T,x0,y0),y(T,x0,y0))=(xT,yT),
where (x(t,x0,y0),y(t,x0,y0) is the solution of (3.12) corresponding to the initial data (x0,y0). Denote by r(t), θ(t) the norm and the polar angle of p(t)=(x(t),y(t))∈ℝ2, r(t)=x(t)2+y(t)2, respectively. Then(3.14)cos⁡θ(t)=x(t)x2(t)+y2(t),sin⁡θ(t)=y(t)x2(t)+y2(t),t≠tk,cos⁡θ(tk+)=x(tk+)x(tk+)2+y(tk+)2=akx(tk)ak2x(tk)2+bk2y(tk)2,sin⁡θ(tk+)=y(tk+)x(tk+)2+y(tk+)2=bky(tk)ak2x(tk)2+bk2y(tk)2,k=1,2,….
Write (r0,θ0)=(r(0),θ(0)) and (rT,θT)=(r(T),θ(T)). Then, the map ϕ can be expressed in the polar coordinates as(3.15)rT=h(r0,θ0),θT=θ0+l(r0,θ0),
where h and l are continuous and 2π-periodic in θ0. In order to apply the Poincaré-Birkhoff fixed point theorem, we need to estimate the difference θT-θ0.Lemma 3.3.
ForR>0 there is a constant K(R), and r0=R, then θT-θ0=l(R,θ0)≤K(R).Proof.
We estimate the differential of the polar angleθ(t) along the solution curve (x(t),y(t)) as follows:
(3.16)dθ(t)dt=xy′-yx′x2+y2=xgɛ(x)+y2x2+y2,t≠tk.
For t=tk, we have
(3.17)tanθ(tk+)=y(tk+)x(tk+)=bkaktanθ(tk).
From (3.17), we obtain
(3.18)θ(tk+)θ(tk)=arctan((bk/ak)tanθ(tk))θ(tk).
Next we construct the function, for a>0, by
(3.19)h(x)={1xarctan(atanx),ifx≠0,a,ifx=0.
Then, it is easy to see there exist α(a) and β(a) such that
(3.20)0<α(a)≤h(x)≤β(a)<∞.
From (3.18) and (3.20), we have
(3.21)0<α(bkak)θ(tk)≤θ(tk+)≤β(bkak)θ(tk)<∞.
Furthermore, by Lemma 3.2, we have 0<d1(R)≤x2+y2≤d2(R), and (3.16), hence
(3.22)dθ(t)dt≤K(R).
From (3.21) and (3.22), we obtain
(3.23)θ(T)≤θ0∏0<tk<Tβ(bkak)+∫0T∏s<tk<Tβ(bkak)K(R)ds.
This implies that there exists an integer m=m(R) such that
(3.24)θT-θ0+2mπ=l(R,θ0)+2mπ<0,∀θ0∈[0,2π].Lemma 3.4.
There isr<R such that
(3.25)θT-θ0+2mπ=l(r,θ0)+2mπ>0,∀θ0∈[0,2π].
So the map ϕ is the twist map in the annular region A={(x,y):r≤x2+y2≤R}.Proof.
LetM be a large number to be chosen later. Let g be the approximation function defined in Lemma 2.3. In particular we have
(3.26)gɛ(x)x≥M,if0<|x|≤δ.
Choose r such that d2(r)<δ. If the initial data r0=x2+y2=r, then by Lemma 3.2 the solution (x,y)=(x(t),y(t)) satisfies x2(t)+y2(t)≤d2(r)<δ, and hence gɛ(x)≥Mx.
To estimate the differenceθT-θ0, it is convenient to introduce another angle variable Θ, defined by the formulas:
(3.27)cos⁡Θ(t)=M1/2x(t)Mx2(t)+y2(t),sin⁡Θ(t)=y(t)Mx2(t)+y2(t),t≠tk.
It follows that
(3.28)tanΘ(tk)=y(tk)M1/2x(tk),t=tk,Θ(tk+)=arctan(y(tk+)M1/2x(tk+))=arctan(bky(tk)M1/2akx(tk))=arctan(bkaktanΘ(tk)),t=tk.
Note that by (3.27) and (3.14) cosΘ and cosθ, respectively, sinΘ and sinθ, have the same sign. Thus the two angles Θ and θ always lie in the same quarter and Θ=θ+α+2kπ,k∈ℤ,-π/2≤α≤π/2, if and only if the polar angle θ is in the same interval. Hence, if |θ-Θ|<π/2 at the initial time t=0, this estimate will remain true along a continuous curve not passing through the origin.
We have(3.29)dΘ(t)dt=11+(y/M1/2x)2×y′M1/2x-yM(1/2)x′Mx2=M1/2(xgɛ(x)+y2)Mx2+y2≥M1/2.
By (3.20), (3.28), and (3.29) we have
(3.30)Θ′(tk)≥M1/2,Θ(tk+)≥α(bkak)Θ(tk),
and by (3.30), it follows that
(3.31)Θ(T)≥Θ(0)∏0<tk<Tα(bkak)+∫0T∏s<tk<Tα(bkak)M1/2ds.
So there exist σ1∈ℝ and σ2>0, such that
(3.32)Θ(T)≥σ1+σ2M1/2.
Recall the relations Θ=θ+α+2kπ,k∈ℤ,-π/2≤α≤π/2. Then, by (3.32), we have
(3.33)θ(T)-θ(0)+2mπ=Θ(T)-α-2kπ-θ(0)+2mπ≥σ1+σ2M1/2-α-2kπ-θ(0)+2mπ>0,∀θ0∈[0,2π].
We choose an integer m=m(r) such that
(3.34)θT-θ0+2mπ=l(r,θ0)+2mπ>0,∀θ0∈[0,2π],
provided that M is large enough.Proof of Theorem3.1.
Letϕ be the map defined (3.13), or equivalently by (3.15). Define the mappings:
(3.35)Φ0:(x0,y0)⟶(x(t1,x0,y0),y(t1,x0,y0))=(x1,y1),Φ0*:(x1,y1)⟶(a1x(t1,x0,y0),b1y(t1,x0,y0))=(x1*,y1*),Φi:(xi*,yi*)⟶(x(ti+1,x0,y0),y(ti+1,x0,y0))=(xi+1,yi+1),Φi*:(xi+1,yi+1)⟶(ai+1xi+1,bi+1yi+1)=(xi+1*,yi+1*),i=1,…,q-1,Φq:(xq*,yq*)⟶(x(T,x0,y0),y(T,x0,y0)).
When akbk=1, Φi(0≤i≤q), Φj*(0≤j≤q-1) are area-preserving mappings. Since
(3.36)ϕ=Φq∘Φq-1*∘Φq-1∘⋯∘Φ0*∘Φ0,ϕ is an area-preserving mapping. Obviously, ϕ(0,0)=(0,0)∈D={(x,y):x2+y2<r2}. Lemmas 3.3, 3.4 imply that ϕ is a twist map on the annulus A={(x,y):r2≤x2+y2≤R2} for sufficiently small ɛ. Now it follows from the result of Ding, Lemma 2.2 given in the introduction, that ϕ has at least two fixed points in A. Let (xɛi(t),yɛi(t)) be one of the corresponding periodic solutions of (3.12). By Lemma 3.2, we have
(3.37)d1(r)≤xɛi2(t)+yɛi2(t)≤d2(R).
By the Arzela-Ascoli theorem, a sequence of {xɛi(t),yɛi(t)} converges to (x(t),y(t)) as ɛi→0. Then, (x(t),y(t)) satisfies (3.12), and (x(t),y(t)) is a periodic solution for the system (1.6) with d1(r)≤x2(t)+y2(t)≤d2(R). Since R is arbitrary, we obtain an infinite sequence of periodic solutions for system (1.6) with small amplitudes.


## 4. Discussion

To the best of our knowledge, there are few papers concerned with the existence of periodic solutions for impulsive equations by means of the Poincaré-Birkhoff fixed point theorem. The goal of this paper is to fill the gap in this area. Our main result expands the applied scope of the work of Ding [3] by allowing the equation to have impulses. Application of the result obtained is illustrated by Example 4.1 given below in which the conclusion does not follow from earlier result, as it does not allow impulses.Example 4.1.
Consider the equation(4.1)x′′(t)+x(t)3=0,t≠tk,x(tk+)=akx(tk),x′(tk+)=bkx′(tk),k∈Z,
where ak>0, akbk=1,ak+q=ak,tk+q=tk+T,0<t1<t2<⋯<tq<T.
In fact, in (4.1), g(x)=x3. Then, xg(x)≥0 and limx→0x-1g(x)=+∞. The condition (H1) is satisfied and by Theorem 3.1, (4.1) has an infinite sequence of periodic solutions.In addition, this paper is the new application for the work of Ding [10] in the impulsive differential equations.



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