The elastic Bernoulli-Euler equation of the microcantilever motion is: (1) ∂ 2 ∂ x 2 [ E I ( x ) ∂ 2 w ( x , t ) ∂ x 2 ] + m ( x ) ∂ 2 w ( x , t ) ∂ t 2 + c ∂ w ( x , t ) ∂ t = q ( x ) p ( t )Where w(x,t) is the deflection; E is Young's modulus; I(x) is the moment of inertia and is assumed to be constant; m(x) is the mass per unit length with the microcantilever assumed to be homogeneous); c is the damping coefficient, and q(x)p(t) represents the applied force per unit length on the microcantilever. The deflection of the microcantilever is modified from the work of Mindlin and Goodman [22]: (2) w ( x , t ) = ξ ( x , t ) + ∑ i = 1 2 Δ i ( t ) g i ( x )in which ξ(x,t) is the surface homogeneous wave function, and Δ_i(t)g_i(x) is the shift-time function associated for time-dependent boundary conditions. Δ₁(t)g₁(x) is the rigid body motion along the microcantilever that is caused by AFM base; Δ₂(t)g₂(x) represents the static deflection with AFM base at various positions. Substituting Equation (2) into Equation (1) yields: (3) E I ξ I V + m ξ + c ξ = q ( x ) p ( t ) − ∑ i = 1 2 [ E I Δ i g i I V + m Δ ¨ i g i + c Δ ˙ i g i ]

The deflection ξ(x,t) under constant boundary conditions is expressed by: (4) ξ ( x , t ) = ∑ n = 1 ∞ Φ n ( x ) T n ( t )in which Φ_n(x) are the mutually orthogonal mode shape functions; and T_n(t) are the mode time functions. Orthogonality can be used to expand g_i(x), g_i^IV, and q_i(x) in Equation (3) as a series of mode functions: (5) g i ( x ) = ∑ n = 1 ∞ G i , n Φ n ( x ) (6) g i I V ( x ) = ∑ n = 1 ∞ G i , n ∗ Φ n ( x ) (7) q i ( x ) = ∑ n = 1 ∞ Q n Φ n ( x )in which the coefficients (G_i,n, G i , n ∗, Q_n) depend on the boundary conditions of the tip-end. Substituting Equations (5)–(7) into Equation (3), dividing both sides by mΦ_n(x)T_n(t), and setting the equation equal ω n 2 yield the equations: (8) E I Φ n I V − m ω n 2 Φ n = 0 (9) m T ¨ n + c T ˙ n + ω n 2 m T n = p ( t ) Q n − ∑ i = 1 2 [ E I Δ i G i , n ∗ + m Δ ¨ i G i , n + c Δ ˙ i G i , n ]

The solutions to Equations (8) and (9) are: (10) Φ n = C n cos β n x + D n sin β n x + E n cosh β n x + F n sinh β n x (11) T n = e − ζ w n ( t − t 0 ) ( A n ( t 0 ) cos [ ω D n ( t − t 0 ) ] + B n ( t 0 ) + A n ( t 0 ) ζ ω n ω D n sin [ ω D n ( t − t 0 ) ] ) + 1 m ω D n ∫ t 0 t { e − ζ w n ( t − τ ) sin ω D n ( t − τ ) ( Q n p ( t ) − ∑ i = 1 2 [ E I Δ i G n ∗ + m Δ ¨ i G n + c Δ ˙ i G n ) } d τwhere ζ = c/(2mω_n), β n = ω n / E I / m, and ω D n = ω n 1 − ζ 2. The coefficients (C_n, D_n, E_n, F_n) can be obtained by applying boundary conditions at the base-end and the tip-end, and [A_n(t₀), B_n(t₀)] can be obtained by applying the initial conditions in every piecewise linear segment. Note that the integration intervals in Equation (11) should be relatively smaller than the jumping interval when the numerical accuracy is considered.

The clamped end of the microcantilever (x = 0) is supported by the piezoelectric oscillator, and the boundary conditions are: (12) w ( 0 , t ) = S ( ω f t ) (13) w ′ ( 0 , t ) = 0 (14) w ″ ( l , t ) = 0in which S(ω_ft) is the amplitude of excitation at frequency ω_f. The boundary conditions at the tip-end involve the superficial force F(w): (15) E I w ‴ ( l , t ) + F ( w ) = 0where F(w) is defined by the Lennard Jones (LJ) potential force: (16) F ( w ) = − A 1 R 180 ( Z − w ) 8 + A 2 R 6 ( Z − w ) 2 ,in which Z is the distance between AFM base and the sample; R is the radius of the tip, and A₁ and A₂ are the Hamaker constants. F(w) is discretized into several piecewise linear segments, Fⁱ(w); and accordingly, Fⁱ(w) is expanded at with a slope of k i ∗, and F i ( w ) = F 0 i + k i ∗ ( w − w 0 ) = F 0 i + k i ∗ ζ. Substituting Equation (2) into Equations (12)–(15) yields the boundary conditions of the mode functions: (17) ξ ( 0 , t ) + ∑ i = 1 2 Δ i g i ( 0 ) = S ( ω f t ) (18) ξ ′ ( 0 , t ) + ∑ i = 1 2 Δ i g i ′ ( 0 ) (19) ξ ″ ( l , t ) + ∑ i = 1 2 Δ i g i ″ ( l ) = 0 (20) E I [ ξ ‴ ( l , t ) + ∑ i = 1 2 Δ i g i ″ ( l ) ] + F i ( w ) = 0

Let Δ₁ = S(ω_ft) and Δ 2 = F 0 i l 3 / ( 3 E I ). The boundary conditions can be rewritten: (21) ξ ( 0 , t ) = 0 , ξ ′ ( 0 , t ) = 0 , ξ ″ ( l , t ) = 0 , E I ξ ‴ ( l , t ) + k i ∗ ξ = 0 (22) g 1 ( 0 ) = 1 , g 1 ′ ( 0 ) = 0 , g 1 ″ ( l ) = 0 , g 1 ′ ′ ′ ( l ) = 0 (23) g 2 ( 0 ) = 0 , g 2 ′ ( 0 ) = 0 , g 2 ″ ( l ) = 0 , E I Δ 2 g 2 ′ ′ ′ ( l ) + F 0 i = 0

The polynomials of g₁ and g₂ that satisfy Equations (22) and (23) are obtained by: (24) g 1 ( x ) = 1 (25) g 2 ( x ) = ( 3 l x 2 − x 3 ) / ( 2 l 3 )

Applying Equations (4) and (10) to Equation (21) yields C_n = −E_n and D_n = −F_n; the eigenvalues β_n can be numerically determined by solving: (26) E I β n 3 [ 1 + cos β n l cosh β n l ] + k i ∗ [ sin β n l c o h β n l − cos β n l sinh β n l ] = 0

This equation describes a clamped-free condition when I β n 3 ≫ k i ∗, and a clamped-pinned conditions when E I β n 3 ≪ k i ∗. The eigenvalue β_n, depending on k i ∗, vary among the above piecewise linear segments. Moreover, the orthogonality conditions in this modal analysis can be modified from the author's earlier study [20]: (27) ∫ 0 l m Φ m ( x ) Φ n ( x ) d x = 0 (28) ∫ 0 l [ E I Φ m ″ ( x ) Φ n ″ ( x ) − δ ( x − l ) k ∗ Φ m ( x ) Φ n ( x ) ] d x = 0

When the tip of the microcantilever oscillates from one segment to another one, the displacement and velocity conditions in the former segment are regarded as the initial conditions in the new segment. The orthogonality conditions derived in Equations (27) and (28) are used to transform the dynamics from the previous modal basis to the new modal basis. At the interchange time t₀, the initial conditions w₀(x,t₀) and ẇ₀(x,t₀) those were obtained from previous segment are expanded in new modal basis: (29) w 0 ( x , t 0 ) = ∑ n = 1 ∞ Φ n ( x ) T n ( t 0 ) + ∑ i = 1 2 Δ ( t 0 ) g i ( x ) (30) w ˙ 0 ( x , t 0 ) = ∑ n = 1 ∞ Φ n ( x ) T ˙ n ( t 0 ) + ∑ i = 1 2 Δ ˙ i ( t 0 ) g ( x )

Applying Equations (27) and (28) to Equations (29) and (30) yields the time coefficients at the initial time t₀ of the interchange: (31) T n ( t 0 ) = ∫ 0 l [ w 0 ( x , t 0 ) − ∑ i = 1 2 Δ i ( x ) g i ( t 0 ) Φ n ( x ) d x / ∫ 0 l Φ n ( x ) Φ n ( x ) d x (32) T ˙ n ( t 0 ) = ∫ 0 l [ w ˙ 0 ( x , t 0 ) − ∑ i = 1 2 Δ i ( x ) g i ( t 0 ) Φ n ( x ) d x / ∫ 0 l Φ n ( x ) Φ n ( x ) d x

Equations (8) and (9) have exact solutions when the piezoelectric oscillator of AFM base is driven by a certain sinusoidal amplitude, S(ω_ft) = A_msin(ω_ft), in which A_m is the amplitude and ω_f is the frequency. The applied forces p(t)Q_n are set zero. Since g_i^W = 0, G i , n ∗ = 0, Δ₁ = A_msin(ω_ft) yields Δ ¨ 1 = − A m ω f 2 sin ( ω f t ), and Δ 2 = F 0 i l 3 / ( 3 E I ) yields Δ̈₂ = 0. Rather, Equation (9) is simplified by: (33) T ¨ n + 2 ς ω n T ˙ n + ω n 2 T n = G 1 , n A m ω f ( w f sin ω f t − 2 ς w n cos ω f t )

Its solution can be obtained: (34) T n ( t ) = e − ς ω n ( t − t 0 ) [ A ¯ n ( t 0 ) cos ω D n ( t − t 0 ) + B ¯ n ( t 0 ) sin ω D n ( t − t 0 ) ] + ( A m G 1 , n ω f 2 [ ( 1 − β n 2 ) sin ω f t − 2 ς β n cos ω f t ] ) / ( ω n 2 [ ( 1 − β n 2 ) 2 + ( 2 ς β n ) 2 ] ) + ( 2 ς A m G 1 , n ω n ω f [ ( 1 − β n 2 ) cos ω f t + 2 ς β n sin ω f t ] ) / ( ω n 2 [ ( 1 − β n 2 ) 2 + ( 2 ς β n ) 2 ] )in which: (35) A ¯ n = w ( t 0 ) − A m G 1 , n ω f 2 [ ( 1 − β n 2 ) sin ω f t 0 − 2 ς β n cos ω f t 0 ] ω n 2 [ ( 1 − β n 2 ) 2 + ( 2 ς β n ) 2 ] − 2 ς A m G 1 , n ω f [ ( 1 − β n 2 ) cos ω f t 0 + 2 ς β n sin ω f t 0 ] ω n 2 [ ( 1 − β n 2 ) 2 + ( 2 ς β n ) 2 ] (36) B ¯ n = w ˙ ( t 0 ) ς ω n A ¯ n w D n − A m G 1 , n ω f 3 [ ( 1 − β n 2 ) sin ω f t 0 + 2 ς β n cos ω f t 0 ] ω D n ω n 2 [ ( 1 − β n 2 ) 2 + ( 2 ς β n ) 2 ] − 2 ς A m G 1 , n ω n ω f 2 [ − ( 1 − β n 2 ) cos ω f t 0 + 2 ς β n sin ω f t 0 ] w D n ω n 2 [ ( 1 − β n 2 ) 2 + ( 2 ς β n ) 2 ] .

Applying the orthogonality conditions to g_i(x) yields the coefficient functions: (37) G i , n = ∫ 0 l g i ( x ) Φ n ( x ) d x / ∫ 0 l Φ n ( x ) Φ n d x .

Coefficients G_1,n and G_2,n can be exactly obtained as: (38) G 1 , n = 2 [ 1 − ( 1 + cos β n l cosh β n l ) / ( cos β n l cosh β n l ) ] / [ β n ∫ 0 l Φ n ( x ) Φ n d x ] , (39) G 2 , n = { [ c n 2 l 3 β n − D n ( 6 β n 4 − 3 l 2 β n 2 ) ] sin β n l + [ − C n ( 6 β n 4 + 3 l 2 β n 2 ) − D n 2 l 2 β n ] cos β n l + [ − C n 2 l 2 β n + D n ( 6 β n 4 − 3 l 2 β n 2 ) ] sinh β n l + [ C n ( 6 β n 4 − 3 l 2 β n 2 ) − 2 l 3 β n D n ] cosh β n l } / [ 2 l 3 ∫ 0 l Φ n ( x ) Φ n ( x ) d x ] ,in which the norm satisfies: (40) ∫ 0 l [ Φ n ( x ) Φ n ( x ) ] d x = [ cosh β n l ( C n 2 + D n 2 ) sinh β n l − 2 sin β n l ) + cos β n l × ( C n 2 − D n 2 ) ( sin β n l − 2 sinh β n l ) + 2 C n D n ( sin β n − sinh β n l ) 2 + 2 C n 2 β n l ] / ( 2 β n )

Substituting Equations (34), (38), and (39) into Equations (2), (4), and (5) yields the exact solutions for every segment.

Figure 2(a,b) shows force-displacement relations of the restoring force of the microcantilever and the LJ potential force. The LJ potential curve is plotted upside-down and with the restoring force curve of the microcantilever, that is modified from Ashhab et al. [23]. The points where these two curves cross represent the equilibrium positions of the tip. Accordingly, the LJ potential curve that is related to a restoring force curve of a stiff microcantilever is compared with that related to a restoring force curve of a soft microcantilever in Figure 2(c,d). The stiff microcantilever is associated with one cross point when AFM base moves toward sample. However, the soft microcantilever case exhibits one, two, or three cross points.

In Figure 3, the cross points A and C are stable points, and the cross point B is a saddle point. When a small ξ(l,t) disturbs the tip to the right of Point B, the restoring force, which exceeds the attracting force, allows the tip to move upward and then to oscillate around Point A. When ξ(l,t) disturbs the tip to the left of Point B, the attracting force causes the tip to move downward and then to oscillate around Point C. As a result, the tip of the soft microcantilever oscillates naturally around its equilibrium position. No jump occurs in this free vibration.

In Figure 4, as the AFM base approached the sample that Δ₁(t) increased, the critical positions were utilized to determine where the tip-jump began and ended. Once, the line of the restoring force became a tangent to the potential curve. Points A and B in Figure 3 degenerated to Point D, and Point C in Figure 3 was replaced by Point E. Then AFM base continued to move toward the sample, and the unbalanced attractive force allowed the tip to jump from Point D to Point E. Then the released attractive energy caused the tip oscillate around Point E in Region III. Next, when AFM base was retracted, the unbalanced restoring force caused the tip to jump from Points F to G and then to oscillate in Region I. Figure 4 is easy to be understood since it bases on force equilibrium not on the numerical results when it is compared with other method.

To estimate the displacements and the velocities of the microcantilever after jump occurred, a model was developed by using the conservation of energy. The tip-jump from Point D to Point E was considered as an example, and that from Point F to Point G was similar. The tip-jump was assumed to occurred immediately after leaving Point D or F, and the tip-jump concerned only on the static attractive deflection term Δ 2 E ( t ), the concept was modified from [24]. Before jump, the velocity of the microcantilever is: (41) w ˙ D ( x ) = ξ ˙ 0 ( x , t ) + Δ ˙ 1 ( t ) g 1 ( x ) .

After jump, the velocity of the microcantilever is: (42) w ˙ E ( x ) = ξ ˙ 0 ( x , t ) + Δ ˙ 1 ( t ) g 1 ( x ) + Δ ˙ 2 E ( t ) g 2 ( x ) .

Consider the variation of the kinetic energy of the microcantilever: (43) Δ E k = ∫ 0 l m ( w ˙ E ) 2 − m ( w ˙ D ) 2 2 d x = ∫ 0 l m 2 { ( Δ ˙ 2 E ) 2 g 2 2 + 2 g 2 Δ ˙ 2 E ( ξ ˙ 0 + Δ ˙ 1 g 1 ) } d x .

The total potential energy between Points D and E is: (44) Δ U = ∫ W D W E F ( w ) d w − ( w E − w D ) ( F E + F D ) 2 .

Finally, the total potential energy equals the kinetic energy ΔE_k = ΔU. The unknown Δ ˙ 2 E, could be obtained numerically. Accordingly, w^E(x) and ẇ^E(x) are the initial conditions for the next segment of oscillation in Region III.

The eigenvalues of the microcantilever with respect to the tip-sample distance were obtained numerically by using Equation (26). Figure 5 displays the first eigenvalues sharply decreasing from right to left in Region I and remaining at zero in Region II. Then the first eigenvalue rises from right to left and then remains constant in Region III. The region in which first eigenvalue is zero is the area of disequilibrium where the jump occurs. The mode shape functions were obtained using Equation (10). Figure 6 plots the first mode shape functions for five tip-sample distances (Z-w), denoted ‘a, b,…,d’, corresponding to the positions ‘a, b,…,d’ in Figure 5. As (Z-w) decreases, the mode shape function changes from a cantilever-type ‘a,’ through an unbalance ‘c,’ to a clamped-pinned beam-type ‘d.’ The above result indicates a significant characteristic of AFM.

When the AFM base was located at a particular elevation, the microcantilever bent and the tip moved to the equilibrium point (Point A or C in Figure 3). Once the microcantilever underwent a minor disturbance, a free vibration occurred and the tip oscillated around its equilibrium point until the energy was dissipated.

In the following numerical discussion, 30 piecewise linear segments were considered within a certain region (11 nm) from the sample to AFM base. Note that the interval in each piecewise segment is setup to be around 1/3 times small than the jumping interval to ensure the simulation accuracy. In this case, the base was arbitrarily set up at Δ₁ = 8.084 nm, and the equilibrium at Points A and C were obtained at 7.2 and 1.935 nm. Figure 7 presents phase portraits of free vibrations under various initial conditions. The numerical analysis considered the first three modes. IC₁ or IC₂ represents an initial disturbance of displacement and velocity of the tip. In these two cases, motion begins in Region II, and the tip travels into Region I or Region III, oscillates around the equilibrium points.

A demarcation line (D-line) in Region II is introduced to distinguish the oscillation vibration around Point A from that around Point C. When the tip is initially located above the D-line, the tip finally oscillates around Point C; otherwise, the tip finally oscillates around Point A. The D-line can be obtained by finding the points whose potential energy equal to the potential energy at Point B, where Point B is the cross-points as discussed in Figure 3. The two 0%-damped orbits show the oscillations that included jumps and periodically circled around Point A and Point C. The two 5%-damped orbits examples of dissipation orbits and finally sink to Points A and C.

Unlike damped driven spring systems that have been described elsewhere, this system exhibited a jump region that separated AFM dynamics into two oscillation systems. One oscillation system was in Region I and exhibited noncontact oscillation, and its behavior was simple. The other oscillation system was in Region III and involved noncontact and contact states; the oscillating behavior in contact state was complicated and was the most likely cause of bifurcation and chaos.

When the AFM base was set at a certain height elevation, two forced vibrations were driven at two equilibrium points with various amplitudes. In the cases considered herein, area A_ab was set to exceed area A_bc, as shown in Figure 3, since the tip always approached the sample from far away. Accordingly, the oscillation around A was more stable than that oscillation around C. Figures 8 and 9 the tip phase portraits under sinusoidal forced excitation with various amplitudes, A_m = 0.05, 0.09, 0.2, and 0.4 nm. The first 20 modes were considered in the simulation. Two excitation frequencies were set to values determined by the first eigenvalues when the tip was located at Points A (Z-w = 7.2 nm) and Point C (Z-w = 1.935 nm). They were ω f A = 6.5575 × 10 4 rad s − 1 in Figure 8 and ω f C = 2.12447 × 10 5 rad s − 1 in Figure 9. The orbit a's (red) and the orbit c's (blue) represent oscillations triggered from A and C, respectively; the steady state orbits were acquired to demonstrate the final periodicity.

Figure 8(a,b) presents the phase portraits around the equilibrium points with amplitudes A_m = 0.05 and 0.09 nm. Period doubling is observed in the orbits marked by ‘c’, and the number of period doublings seems to depend on the amplitudes of oscillation and the sample-contact condition. In Figure 8(c) with A_m = 0.2 nm, the orbit c initially circles around C; then snaps to Region I, and finally circles around A. The tip does not return to Region III because its amplitude is too low to leave in Region I. In Figure 8(d), with the excitation amplitude A_m = 0.4 nm, both orbits oscillate through three regions and exhibit hysteresis, and 14 period doublings occur. Unlike a damped driven spring system, which exhibits contact and noncontact behaviors, AFM oscillates in Region I, jumps in Region II, and exhibits either noncontact or contact oscillation in Region III. The results also indicate bistability when the amplitude is sufficiently large, period doubling when the vibrating tip is shown in Region III, and snapping when the amplitude is too small to exhibit hysteresis.

The excitation frequency was set to ω f C in Figure 9, where ω f C was the natural frequency when the tip was located at Point C. The excitation amplitude was set to A_m = 0.05, 0.09, 0.2, and 0.4 nm in Figure 9(a–d), respectively. In Figure 9(a), the amplitude of oscillation of the orbit a (red) is less than that in Figure 8(a), but that of the orbit c (blue) is greater. In Figure 9(b–d), applying the same excitation frequency but different excitation amplitudes yields the orbit c that initially oscillates in Region III; jumps to Region I, and then remains in Region I, oscillating with small amplitude. The orbits marked by ‘a’ remain stable in Region I. Hence, the excitation frequency ω f C, obtained from Region III, affects large oscillation amplitude in Region III, causing it to snap into Region I. The excitation frequency ω f C produces extremely stable oscillation in Region I.

Figure 10(a–d) shows time histories that correspond to the cases in Figures 8(c), 8(d), 9(b), and 9(d), respectively. In Figure 10(a,b), the time histories marked by ‘a’ (red) are associated with the oscillation that is driven at initial Point A with ω f A, and the time histories marked by ‘c’ (blue) are associated with the oscillation that is driven at initial Point C with ω f C. All time histories are run from transient states to steady states, and jumps are explicitly indicated. The orbit c in Figure 10(a) presents the snapping phenomenon. In Figure 10(b), the orbit a and the orbit c exhibit hysteresis. Figure 10(c,d) displays the time histories that are associated with the cases in Figure 9(b,d). Clearly, the orbit c's finally converge to the orbit a's, so their phase portraits overlap, as displayed in Figure 9(b,d). These figures also reveal that the time histories have smaller amplitudes and are relatively stable in Region I since they are not driven at their dominate frequency. The tip might jump to the other region, while the driving amplitudes are sufficiently large enough to pass Region II. As a result, the acquired kinetic energy exceeded the superficial potential energy is an important criterion for snapping and intermittency.

With a constant driving amplitude of A_m = 0.4 nm, the forced vibrations were induced at five elevations of AFM base. The associated tip positions were initially in Region I and moved to Region III. Therefore, the tip-sample distances were 8.019, 6.694, 5.368, 4.706, and 2.010 nm, associated with one, three, three, two, and one equilibrium points as presented in Figure 11. The frequencies, ω f A = 6.5575 × 10 4 rad s − 1 and ω f C = 2.12447 × 10 5 rad s − 1, were set. In Figure 11(a–e), the orbits marked by ‘a’ (blue) represent the oscillation that is driven at the frequency ω f A, that is the traditional AFM case which drives the microcantilever at dominate frequency. The orbits marked by ‘c’ (red) represent that is driven at the frequency ω f C.

In Figure 11(b), the case with a single-equilibrium point involves stable oscillation, and the orbit a is amplified when the tip triggers at Point A. In Figure 11(c), the orbit c remains stable in Region I, but the orbit a that is amplified in Region I and enters Region III, finally becoming chaos. Clearly, AFM can easily detect the signals from the amplitude, and that is the way the traditional AFM used. Moreover, in Figure 11(d), the orbit a snaps to oscillate around Point C since the oscillation is stable in Region III, and the orbit c remains stable in Region I. In Figure 11(e), the tip is initially driven at the equilibrium point A, at which is very close to Region II. The orbit a finally converges in Region III, and the orbit c is amplified because the frequency ω f C increase the amplitude of oscillation in Region III. In Figure 11(f), the oscillations are driven at the equilibrium point in Region III, and the orbits are amplified to run in three regions and come into contact with the samples. The orbit c is distinctly larger than the orbit a and it is chaotic, because it is driven at ω f C and let the microcantilever resonate in Region III.

Figure 12(a–d) display detailed time histories with various tip-sample distances associated with Figure 11(b–e), respectively. The time history marked by ‘a’ (blue) represents the oscillation that is driven at ω f A, and the time history marked by ‘c’ (red) represents oscillation that is driven at ω f C. In Figure 12(a), intermittency is evident in the time history a, which ends in chaos. In Figure 12(b), the time history a snaps to Region III, and the time history c remains stable in Region I. In Figure 12(c,d), the orbit c's exhibit intermittency and then become chaotic, but the orbit a's remain periodic and exhibit hysteresis.

Comparing Figure 11 with Figure 12 reveals that the oscillations driven at ω f A exhibit period doubling, snapping, intermittency, and chaos. However, the oscillations driven at ω f C are stable until the tip touches into Region II. Some early experimental works [15,16] indicated that microcantilever driven at the second eigenvalue improved the sensitivity of an AFM and could avoid chaotic. Indeed, the second eigenvalue was close to the value of the first eigenvalue in Region III, as shown in Figure 5. Therefore, the oscillation driven at ω f A leads large amplitude and results unstable. However, the oscillation driven at ω f C has stable amplitudes before the tip comes into Region II, and it provides an obvious amplitude difference after the tip came into Region III. As a result, this specific excitation frequency at ω f C increased the sensitivity of AFM measurement by enabling the jumps to be recognized and eliminating the uncertainties.