We study the dynamics of an atomic force microscope (AFM) model, under the Lennard-Jones force with nonlinear damping and harmonic forcing. We establish the bifurcation diagrams for equilibria in a conservative system. Particularly, we present conditions that guarantee the local existence of saddle-node bifurcations. By using the Melnikov method, the region in the space parameters where the homoclinic orbits persist is determined in a nonconservative system.
Universiti Teknologi PetronasCIE 3-17-41. Introduction
Atomic force microscopes (AFMs) were developed in 1986 by Bining and coworkers [1]. They are based on the tunneling microscope and the needle profilometer principles. Generally, AFMs measure the interactions between particles, thus allowing the nanoscale study of the surfaces for different materials [2–4]. In fact, a wide variety of applications in the analysis of pharmaceutical products, the study of the properties of fluids and fluids in cellular detection, and studies on medicines, among others, can be found in [5–8].
In the model presented in [9, 10], the authors study the interaction between the sample and the device’s tip (Figure 1). The associated differential equation is(1)y¨+Cy+a3y˙+y=b1y+a8−b2y+a2+ft,where b1,b2, and a are positive constants and f is a continuous T-periodic function with zero average; that is, f¯=1/T∫0Tftdt=0.(2)FLJ≔b1y+a8−b2y+a2,where FLJ is known as the Lennard-Jones force, which can be considered as a simple mathematical model to explain the interaction between a pair of neutral atoms or molecules (see [11, 12] for the standard formulation). The first term describes the short-range repulsive force due to overlapping electron orbits, known as Pauli’s repulsion, whereas the second term simulates the long-range attraction due to van der Waals’ forces. This is a special case of a wide family of Mie forces:(3)Fn,mx=Axn−Bxm,where n and m are positive integers with n>m, also known as the n−m Lennard-Jones force [13]. On the contrary, the dissipative term of (1)(4)Fr=Cy+a3y˙,is associated with a damping force of compression squeeze-film type. In specialized literature, compression film type damping can be considered as the most common and dominant dissipation in different mechanisms (see [14, 15] and their bibliography).
Mechanical model associated with the AFM’s devices.
For the conservative system, two main results were obtained, Theorems 1 and 2, where we establish analytically the bifurcation diagram of the equilibria for specific regions with the involved parameters in contrast to the one obtained in [16]. In particular, Theorem 2 proves the local existence of two saddle-node bifurcations that can be related to the hysteresis phenomenon [17, 18].
In the nonconservative system, we present as a main result Theorem 4, which gives a thorough and rigorous condition for the persistence of homoclinic orbit when the external forcing is of the form ft=BcosΩt. The condition found relates the amplitude of the external forcing B with the damping constant C, which in practice can be used to prevent the AFM device from becoming decalibrated.
This article is structured in the following way: Section 1 is an introduction, Section 2 is dedicated to prove the main results in the conservative system, and Section 3 contains the proof for the main result of the nonconservative system along with some illustrative examples.
2. Bifurcation Diagrams
With the change of variable x=y+a, (1) is rewritten as(5)x¨=mx+a+εft−Cx3x˙,where mx=b1/x8−x−b2/x2 is the total force acting over the system, which is a combination of the Lennard-Jones force and the restoring force of the oscillator. The change of the singularity from −a to 0 will facilitate the study of the bifurcation diagram for equilibria in the conservative system (ε=0). Note that the classification of the equilibrium solutions of (5) plays an important role when the full equation is studied. We now describe some properties of the function mx:(6)limx⟶0+mx=∞,limx⟶∞mxx=−1.
Moreover, m has only one positive root, and a direct analysis provides a critical value(7)b1∗=427b23,such that
If b1>b1∗, then mx is decreasing
If b1=b1∗, then mx is nonincreasing and has an inflection point in xc=4/3b21/3
Finally, if b1<b1∗, then mx has a local maximum (resp., minimum) in xr (resp., xl) and mxr,mxl<0
Therefore, the equilibrium set G=x∈ℝ+:mx+a=0 is finite, not empty, and the number of equilibria depends on the parameter a. Figure 2 shows the possible variants of the m function in terms of b1, b2, and a.
The m function in terms of parameters b1 and b2. (a) m is decreasing monotonously if b_{1} > b1∗. (b) m has a maximum and a local minimum if b_{1} < b1∗.
The proof of Theorem 1 will be made by establishing the equilibria for system (5). Let us define the energy function be(8)Ex,v≔v22+x22+17b1x7−b2x−ax.
Note that the local minimums of E correspond to nonlinear centers and the local maximums correspond to saddles. However, when E has a degenerate critical point x∗,0, since the Hessian matrix A is such that Tr A=0, Det A=0, but A≠0. In this case, Andronov et al. [19] showed that the system can be written in the “normal” form:(9)x˙=y,y˙=akxk1+hx+bnxny1+gx+y2Rx,y,where hx,gx, and Rx,y are analytic in a neighborhood of the equilibrium point hx∗=gx∗=0, k≥2, ak≠0, and n≥1. Thus, the degenerate critical point x∗,0 is either a focus, a center, a node, a (topological) saddle, a saddle-node, a cup, or a critical point with an elliptic domain (see [20], Theorem 2, pp. 151; Theorem 3, pp. 151).
Theorem 1.
The equilibrium solutions of the conservative system associated with (5) are classified as follows:
A nonlinear center if eitherb1≥b1∗and a∈ℝ+orb1<b1∗anda∈ℝ+−−mxr,−mxl
Two nonlinear centers and a saddle ifb1<b1∗anda∈−mxr,−mxl
A nonlinear center and a cusp if eitherb1<b1∗anda=−mxrora=−mxl
Proof.
We present here the main steps 1−3 of the argument:
Note that G has a unique element if either b1>b1∗ and a∈ℝ+ or b1<b1∗ and a∈ℝ+−−mxr,−mxl, and the equilibrium is a nonlinear center since E reaches a local minimum at that point. For the case b1=b1∗, a=−mxc is degenerate, and using the expansion given in (9), we have k=3 and
(10)ak=24b26xc5−720b1∗6xc11<0.
Therefore, from Theorem 2 (pp. 151) of [20], this completes (1).
Under the hypothesis made, the set G has three solutions such that two are local minimums of E and the other is a local maximum of E. Consequently, two of the equilibria are nonlinear centers and the other equilibrium is a saddle.
In this case, G has two solutions such that one of them is a local minimum of E and corresponds to a nonlinear center while the other is degenerate with k=2 and b1=0 in (9). Consequently, Perko ([20], Theorem 3, pp 151) guarantees that equilibrium is a cusp.
In the next section, we focus on the persistence of homoclinic orbits present in Theorem 1 when studying equation (5).
The conservative equation associated with (5) can be written as the parametric system:(11)x′=y,y′=Fx,a,where Fx,a=mx+a. Note that Theorem 1 allows us to build the bifurcation diagram of equilibria in terms of the parameter a (Figures 2 and 3). Moreover, when b1≥b1∗, the parameter a does not modify the dynamics of the system as it does when b1<b1∗. In fact, there exists numerical evidence [10, 14], which shows that the points xi,ai, with ai=−mxi, i=r,s, are bifurcation points. In the following theorem, it will be formally shown that those points are saddle-node bifurcation points.
Bifurcation diagrams of equation (5) in the conservative system. (a) Bifurcation diagram in terms of the parameters b_{1} and b_{2}. (b) Bifurcation diagram in terms of the parameter a and the number of equilibrium solutions when setting b_{1} and b_{2} such that b_{1} < b1∗.
Theorem 2.
If b1<b1∗, then the points xi,ai, i=r,l, are local saddle-node bifurcation for the conservative system (5).
Proof.
In fact, it is enough that the following conditions are fulfilled, as shown in [21], Theorem 3.1, pp 84:(12)A1∂xxFx,axi,ai≠0,A2∂aFx,axi,ai≠0.
Indeed, we have ∂xxFx,axl,al>0 (resp., ∂xxFx,axr,ar<0) because m has relative minimum (resp., maximum) in xl (resp., xr) and ∂aFx,axi,ai=1.
To summarize, the results obtained in Theorems 1 and 2 are illustrated in the bifurcation diagram of the conservative system associated with (5). In Figure 3(a), the red curve separates the region in terms of the parameters b1 and b2, for which the conservative system has a unique equilibrium (independent of the parameter a), of the region where the number of equilibrium solutions depends on the parameter a. In fact, if we take b2,b1∈ℝ+2−b2,b1∈ℝ+2:b1≥b1∗, then the conservative system may have one, two, or three equilibria as illustrated in Figure 3(b). In this figure, the solid lines are related to the stable equilibria, while the dotted line is related to the solutions of unstable equilibria. Furthermore, it can be shown that locally around the points xi,ai, i=l,r, there is a saddle-node bifurcation.
3. Homoclinic Persistence
The discussion in this section is limited to the case b1<b1∗ and a∈−mxr,−mxl. The objective is to apply Melnikov’s method to (5); when ft=BcosΩt, it can be used to describe how the homoclinic orbits persist in the presence of the perturbation. For AFM models, the persistence of homoclinic orbits has great practical use since it can produce uncontrollable vibrations of the device, causing fail, and generate erroneous readings [9, 10, 15].
Before we address this problem, let us establish some notation. Consider the systems of the form(13)x′=fx+εgx,t,x∈ℝ2,where f is a vector field Hamiltonian in ℝ2, gi∈C∞ℝ2×ℝ/Tℤ, i=1,2, g=g1,g2T, and ε≥0. Now, supposing in an unperturbed system, i.e., ε=0in (13), the existence of a family of periodic orbits is given by(14)γe=x1,x2:Ex1,x2=e,e∈α,β,such that γe approaches a center as e⟶α and to an invariant curve denoted by γβ as e⟶β. When γβ is bounded, it is a homoclinic loop consisting of a saddle and a connection. We want to know if γβ persists when (13), where 0<ε≪1, that is, if γβt,ε is a homoclinic of (13) that is generated by γβ. The first approximation of γet,ε is given by the zeros of Melnikov’s function Met which is defined as(15)Met≔∫Ex1,x2=eg2dx1−g1dx2.
Therefore, it is necessary to know the number of zeros of (7). For our purposes, the following theorem, which is an adaptation of [22], will be useful.
If Me0t0≠0, then there are no limit cycles near γe0 for ε+t0+t which is sufficiently small
If Me0t=0 is a simple zero, then there is exactly one limit cycle γe0t0,ε for ε+t0+t which is sufficiently small that approaches γe0 when t,ε⟶t0,0
Remark 1.
Melnikov’s function can be interpreted as the first approximation in ε of the distance between the stable and unstable manifold, measured along the direction perpendicular to the unperturbed connection; that is, dε≔εMβt0/fγβ+Oε2. In particular, when Mβt0>0 (resp., <0), the unstable manifold is above (resp., below) the stable manifold (see [20, 23] for a detail discussion).
Rewriting (5) as a system of the form (13), we obtain(16)fx1,x2=x2mx1+a,gx1,x2,t=0BcosΩt−Cx13x2.
From Theorem 1, we have that if b1<b1∗ and a∈−mxr,−mxl, the unperturbed system has three equilibria from which one is a saddle, denoted by xsa,0. The function’s energy associated with the conservative system is given by (8) and homoclinic loops, denoted by Γl and Γr, and Ex1,x2=Exsa,0=β.
When calculating Melnikov’s function along the separatrix on the right Γr, the computation along Γl is identical; that is,(17)Mβt0=∫Γrg2dx1−g1dx2=∮γβrEx2g1+Ex1g2dt=∫−∞∞x2tBcosΩt+t0−Cx13tx2tdt=BcosΩt0∫−∞∞cosΩtx2tdt−B senΩt0∫−∞∞senΩtx2tdt−C∫−∞∞x22tx13tdt=−2B senΩt0∫0∞senΩtx2tdt−C∫−∞∞x22tx13tdt.
Note that(18)∫−∞∞cosΩtx2tdt=0,because cosΩtx2t is an odd function. Consequently,(19)Mβt0=−2B senΩt0∫0∞senΩtx2tdt−C∫−∞∞x22tx13tdt.
By defining(20)ξ1=−2∫0∞senΩtx2tdt,ξ2=−∫−∞∞x22tx13tdt,we prove that ξ1 and ξ2 are bounded. Indeed, dt=dx1/x1=dx1/x2 and xsa<x1<x¯ in Γr, where xsa and x¯ are consecutive zeros of Ex1,0−β. Now if Ex1,x2=β, then(21)x22=2β+ax1+b2x1−b17x17−x122.
Hence,(22)ξ1≤2∫0∞x2tdt=2∫xsax¯dx1=2x¯−xsa.
On the contrary,(23)ξ2≤2C∫xsax¯x2x13dx1=2C∫xsax¯2β+ax1+b2/x1−b1/7x17−x12/2x13dx1<∞.
Finally, Melnikov’s function is rewritten as(24)Mβt0=Bξ1 senΩt0+Cξ2.
Theorem 4.
Under the conditions of item 2 of Theorem 1, we have that the homoclinic orbits of (5) persist as long as ε is sufficiently small and(25)BC>ξ2ξ1.
Proof.
Condition (25) implies that Melnikov’s function (24) has a simple zero. Consequently, Theorem 3 reaches the desired conclusion.
Example 1.
For illustrative purposes, we have taken from [24] the realistic values of the physical parameters in Table 1. The values in Table 1 are related to the following adimensionalized values b1,b2, and a:(26)b1=11387610000000,b2=1481481000000,a=1.07468,ξ1=0.290315,ξ2=0.382056.
For instance, fix C=1 and Ω=1, and Theorem 4 guarantees that if B>1.316, then the homoclinic persists.
Properties of the case study of the AFM cantilever of Rützel et al. [24].
Symbol
Value
A1
0.001×10−70Jm6
A2
2.96×10−19 J
R
10 nm
K
0.87N/m
Z0
1.68108 nm
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors have been financially supported by the Convocatoria Interna UTP 2016 (project CIE 3-17-4).
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