Saddle-Node Bifurcation and Homoclinic Persistence in AFMs with Periodic Forcing
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.
Atomic force microscopes (AFMs) were developed in 1986 by Bining and coworkers . 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 iswhere , and a are positive constants and f is a continuous T-periodic function with zero average; that is, .where 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:where are positive integers with , also known as the Lennard-Jones force . On the contrary, the dissipative term of (1)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).
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 . 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 . 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 , (1) is rewritten aswhere 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 to 0 will facilitate the study of the bifurcation diagram for equilibria in the conservative system (). 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 :
Moreover, m has only one positive root, and a direct analysis provides a critical valuesuch that(i)If , then is decreasing(ii)If , then is nonincreasing and has an inflection point in (iii)Finally, if , then has a local maximum (resp., minimum) in (resp., ) and
Therefore, the equilibrium set 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 , , and a.
The proof of Theorem 1 will be made by establishing the equilibria for system (5). Let us define the energy function be
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 , since the Hessian matrix A is such that , , but . In this case, Andronov et al.  showed that the system can be written in the “normal” form:where , and are analytic in a neighborhood of the equilibrium point , , , and . Thus, the degenerate critical point 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 , Theorem 2, pp. 151; Theorem 3, pp. 151).
Theorem 1. The equilibrium solutions of the conservative system associated with (5) are classified as follows:(1)A nonlinear center if eitherand orand(2)Two nonlinear centers and a saddle ifand(3)A nonlinear center and a cusp if eitherandor
Proof. We present here the main steps of the argument:(1)Note that has a unique element if either and or and , and the equilibrium is a nonlinear center since E reaches a local minimum at that point. For the case , is degenerate, and using the expansion given in (9), we have and Therefore, from Theorem 2 (pp. 151) of , this completes (1).(2)Under the hypothesis made, the set 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.(3)In this case, 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 and in (9). Consequently, Perko (, 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:where . 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 , the parameter a does not modify the dynamics of the system as it does when . In fact, there exists numerical evidence [10, 14], which shows that the points , with , , are bifurcation points. In the following theorem, it will be formally shown that those points are saddle-node bifurcation points.
Theorem 2. If , then the points , , 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 , Theorem 3.1, pp 84:Indeed, we have (resp., ) because m has relative minimum (resp., maximum) in (resp., ) and .
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 and , 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 , 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 , , there is a saddle-node bifurcation.
3. Homoclinic Persistence
The discussion in this section is limited to the case and . The objective is to apply Melnikov’s method to (5); when , 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 formwhere f is a vector field Hamiltonian in , , , , and . Now, supposing in an unperturbed system, i.e., in (13), the existence of a family of periodic orbits is given bysuch that approaches a center as and to an invariant curve denoted by as . When is bounded, it is a homoclinic loop consisting of a saddle and a connection. We want to know if persists when (13), where , that is, if is a homoclinic of (13) that is generated by . The first approximation of is given by the zeros of Melnikov’s function which is defined as
Theorem 3 (, Theorem 6.4). Suppose and .(1)If , then there are no limit cycles near for which is sufficiently small(2)If is a simple zero, then there is exactly one limit cycle for which is sufficiently small that approaches when
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, . In particular, when (resp., ), 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 obtainFrom Theorem 1, we have that if and , the unperturbed system has three equilibria from which one is a saddle, denoted by . The function’s energy associated with the conservative system is given by (8) and homoclinic loops, denoted by and , and .
When calculating Melnikov’s function along the separatrix on the right , the computation along is identical; that is,Note thatbecause is an odd function. Consequently,By definingwe prove that and are bounded. Indeed, and in , where are consecutive zeros of . Now if , thenHence,On the contrary,Finally, Melnikov’s function is rewritten as
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
Example 1. For illustrative purposes, we have taken from  the realistic values of the physical parameters in Table 1. The values in Table 1 are related to the following adimensionalized values , and a:For instance, fix and , and Theorem 4 guarantees that if , then the homoclinic persists.
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.
The authors have been financially supported by the Convocatoria Interna UTP 2016 (project CIE 3-17-4).
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