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Abstract

With the development of chaos theory, Duffing oscillator has been extensively studied in many fields, especially in electronic signal processing. As a nonlinear oscillator, Duffing oscillator is more complicated in terms of equations or circuit analysis. In order to facilitate the analysis of its characteristics, the study analyzes the circuit from the perspective of vibrational science and energetics. The classic Holmes-Duffing model is first modified to make it more popular and concise, and then the model feasibility is confirmed by a series of rigorous derivations. According to experiments, the influence of driving force amplitude, frequency, and initial value on the system is finally explained by the basic theories of physics. Through this work, people can understand the mechanisms and characteristics of Duffing oscillator more intuitively and comprehensively. It provides a new idea for the study of Duffing oscillators and more.

Keywords

Chaos Duffing oscillator spring model driving force initial value

Introduction

Chaos theory was first proposed by Lorentz, who found the butterfly effect in the simulation of Earth meteorology. After long-term continuous development, chaos theory has been widely used in communication, electronics, and economics. Yao used it for the manufacturing of nanoscale non-smooth fibers, and Chun-Hui used it for water collection.^1,2 As a typical second-order chaotic oscillator, Duffing oscillator has attracted extensive attention for its rich motion, especially its sensitivities to the initial value, coefficients, and inputs. Holmes simplified the parameter and obtained Duffing-Holmes oscillator with complex chaotic behavior.³ He also made a corresponding physical model and confirmed its characteristics. People use this oscillator to detect weak signals.^4–6

However, the analysis of beam structure requires a certain mechanical foundation.^7–11 Since many people are not professional, they often study it only from the perspective of equations or circuits and cannot clearly understand its mechanism. It is necessary to transform it into a popular and concise model.^12,13 On this basis, X. Yao and J. He used the homotopy perturbation method and numerical methods for analysis.^1,14 He CH et al. and He JH et al. studied the low frequency characteristics of a fractal vibration model and the frequency–amplitude characteristics of a fractal oscillation, respectively.^15,16 This study studies the Duffing oscillator from the perspective of mechanics.

Duffing oscillator study from the perspective of mechanics

Due to the special superiority of Duffing oscillator in the detection of weak electric signals, it is studied from the perspective of equations or circuits.^17,18 There are also studies from the point view of fractal theory.^19,20 However, they ignore that the essence of Duffing oscillator is a nonlinear elastic system, and the Duffing oscillator is analyzed from the perspective of mechanics.^21,22

Origin of Holmes-Duffing oscillator

Generally, the restoring force of a one-dimensional elastic system is^23–25

f = - κ x - λ x^{2} - a x^{3} + \dots

(1)

Some systems are not linear, and their elastic potential energies are symmetric

U = \frac{1}{2} κ x^{2} + \frac{1}{4} a x^{4}

(2)

Equation (2) is also the simplest form of symmetric elastic potential energy in all nonlinear elastic systems. It corresponds to the motion equation of the elastic system

x^{″} + κ x + a x^{3} = 0

(3)

Due to the resistance in actual movement, the simplest form of it is proportional to the velocity, namely, $μ x^{″}$ . The Duffing equation is developed as

x^{″} + μ x^{'} + κ x + a x^{3} = 0

(4)

Considering the existence of periodic external force, the forced Duffing oscillator equation is obtained. The dissipative system with extrinsic motivation is as follows

x^{″} + μ x^{'} + κ x + a x^{3} = F_{d} \cos (w t)

(5)

Simplify the above parameters, let $a = - κ = 1$ and detect the weak signal by Holmes-Duffing oscillator

x^{″} + μ x^{'} - x + x^{3} = F_{d} \cos (w t)

(6)

For equation (6), Holmes has given a concrete mechanical model and a detailed proof. As shown in Figure 1, the slender steel is fixed on a bracket and the bottom is located in the middle of two permanent magnets. This system model corresponds to equation (4), and there is an unstable equilibrium in the middle, accurately at $x = 0$ . Moreover, there are also two stable equilibria on both sides of $x = \pm 1$ . When the cycle driving power is added to the left side, the motion equation of the steel corresponds to equation (6).

Figure 1.

Holmes-Duffing classic physical model.

Modified model of Holmes-Duffing oscillator

This classic model is in full compliance with Duffing’s equation, and the oscillatory structure of the beam makes it unconcise and unpopular. Therefore, the beam oscillator is converted into a one-dimensional spring oscillator, as shown in Figure 2.

Figure 2.

Improved one-dimensional spring model of Holmes-Duffing.

The spring top is connected to a high fixed point as shown in Figure 2. Its original length is $l$ , the stiffness coefficient is $k$ , and the mass is ignored. An oscillator connected with the bottom carries a positive charge of $q_{1}$ with mass $m$ , and it is set through a fixed, rigid, and thin pole by a hole. The distance between the spring and the high fixed point is $l$ , and the distance that the oscillator deviates from the equilibrium point is set as the generalized coordinates $x$ . There are two point charges with negative charge $q_{2}$ at the origin of the left and right coordinates of $r$ . At a sufficient distance $R$ , a pair of parallel plate capacitors establishes a time-varying uniform electric field through an alternating current connected to one or more ground planes. After the model is established, the theoretical demonstration is carried out. The forces that affect the oscillator motion is the spring force, the Coulomb force between charges, the damping force, and the electric field force exerted on the oscillator by a uniform electric field. They are independent to each other and analyzed, respectively.

The spring restoring force is first analyzed. Lin has analyzed the nonlinear vibration of spring oscillator.^26,27 When the initial state of the oscillator is (0, 0), the kinetic energy of the spring is

T = \frac{1}{2} m {x^{'}}^{2}

(7)

The potential energy is

V = \frac{k}{2} {[\sqrt{l^{2} + x^{2}} - l]}^{2}

(8)

According to Lagrangian function

L = T - V

(9)

Lagrange equation is used as

\frac{d}{d t} \frac{\partial L}{\partial x^{'}} - \frac{\partial L}{\partial x} = 0

(10)

The solution is obtained

m x^{″} = \frac{d}{d t} \frac{\partial L}{\partial x^{'}} = \frac{\partial L}{\partial x} = - k x + \frac{k x}{\sqrt{1 + {(x / l)}^{2}}}

(11)

This is the equation of spring restoring force, and the fraction item is expanded by Taylor series

\frac{1}{\sqrt{1 + {(x / l)}^{2}}} = 1 - \frac{1}{2} {(x / l)}^{2} + \frac{3}{8} {(x / l)}^{4} + O {(x)}^{6}

(12)

Take the first two items as example

- k / 2 l^{2} x^{3} = m x^{″} = F

(13)

2. The Coulomb force is analyzed. Since the three charges are in a straight line, the Coulomb resultant force of the oscillator is affected by two negative charges

F_{e} = k_{e} q_{1} q_{2} [\frac{1}{{(r - x)}^{2}} - \frac{1}{{(r + x)}^{2}}]

(14)

Similarly, expand the fraction items by Taylor series

- \frac{1}{{(r + x)}^{2}} = \frac{1}{r} {[1 - \frac{x}{r} + {(\frac{x}{r})}^{2} - {(\frac{x}{r})}^{3} + O (x^{4})]}^{'}

(15)

- \frac{1}{{(r + x)}^{2}} = \frac{1}{r} [- \frac{1}{r} + \frac{2}{r^{2}} x - \frac{3}{r^{3}} x^{2} + O (x^{3})]

(16)

Namely

\frac{1}{{(r - x)}^{2}} = \frac{1}{r} [\frac{1}{r} + \frac{2}{r^{2}} x + \frac{3}{r^{3}} x^{2} + O (x^{3})]

(17)

Simplify and remove the high-order items

F_{e} = \frac{4 k_{e} q_{1} q_{2}}{r^{3}} x

(18)

3. The simplest is the damping force that positively related to the speed

F_{f} = - μ m x^{'}

(19)

4. For the electric field force

F_{E} = E q_{1} = \frac{U \cos (w t)}{2 R} q_{1}

(20)

From the above, there is

m x^{″} = - \frac{k}{2 l^{2}} x^{3} + \frac{4 k_{e} q_{1} q_{2}}{r^{3}} x - μ x^{'} + \frac{U \cos (w t)}{2 R} q_{1}

(21)

x^{″} + μ x^{'} - \frac{4 k_{e} q_{1} q_{2}}{m r^{3}} x + \frac{k}{2 m l^{2}} x^{3} = \frac{U q_{1}}{2 m R} \cos (w t)

(22)

Equation (22) can be transformed into equation (6) by choosing appropriate parameters, which proves the effectiveness of the above model.

Influence of system parameters

After the beam oscillator is converted into an easy-to-understand spring oscillator, the amplitude, frequency, and initial phase of the driving force have a more obvious impact on the entire system. Based on Figure 2 and related experimental data, the initial value of the system is (0, 0), the drive frequency is 1, and the initial phase is 0.

Effect of driving force amplitude on the system

$a = - κ = 1$ is assigned into equation (2), and the corresponding change of elastic potential energy is shown in Figure 3. A local maximum can be observed between two local minima; thus, two potential wells separated by a potential barrier are formed. According to Figure 3, the driving force amplitude $F_{d}$ from 0 to infinity is analyzed.

Figure 3.

Duffing equation double potential well.

When $F_{d} = 0$ , the electric field force had no effect. The spring oscillator is affected by the disturbance of $x = 0$ . As a result, the balance force acting on the oscillator is destroyed, and the oscillator moves in a more powerful direction (assuming the x-axis is positive). Finally, as the kinetic energy consumed by the damping increases, the oscillator stops at the minimum potential energy of $x = 1$ . The final trajectory is shown in Figure 4.

Figure 4.

Different motions corresponding to different amplitudes. (From the left to end, the corresponding Fd are 0, 0.1, 0.343, and 0.4, respectively)

$F_{d}$ is the main reason for the voltage increase of the parallel plate capacitor. The oscillator is affected by the positive resultant force in the positive direction and falls into the potential well at $x = 1$ , as shown in Figure 3. Finally, through the damping self-regulation, the energy consumed by the damping is equal to the energy input by the electric field. Therefore, the oscillator will make periodic motions with a certain amplitude near $x = 1$ . For example, the trajectory of the oscillator is shown in Figure 4 when $F_{d} = 0.1$ .

Compared with the internal force of the system, the continuous increase of $F_{d}$ will have a great impact on the oscillatory motion. When the amplitude of the oscillator increases to a certain level (for example, $F_{d} = 0.343$ ), the self-regulation of the internal force cannot guarantee the single-periodical motion of the oscillator. The double amplitude is clearly visible as shown in Figure 4.

Then the system enters the period-doubling bifurcation state. With a further increase of $F_{d}$ , the period-doubling bifurcation state intensifies and enters chaos. At the same time, the oscillator has enough energy to span the potential barrier. The oscillator would make chaotic motions between two potential wells, as shown in Figure 4.

Continually increase $F_{d}$ until an oscillation movement is achieved. When $F_{d} = 0.825$ , the electric field gradually drives the oscillator to build a new period orbit on the two potential wells, called a large period orbit. Here, the temporal and phase printings are both regular, as shown in Figure 5. The input of the electric field is sufficient to provide the energy consumed by the periodic motion of the system. The system will still be responsible for considerable periodic motions when $F_{d}$ changes to a certain range.

Figure 5.

Temporal and phase printing of large period.

When $x$ is too large, the high-order factors in equations (12)–(17) cannot be neglected. Thereby, the model is not valid when $F_{d}$ is too large.

The results show that the input energy is the decisive factor that determines the final state of the oscillator, which is determined by the external driving force $F_{d}$ .

Effect of driving force frequency on the system

The frequency of the driving force indirectly affects the amplitude or energy of the oscillator. In a one-dimensional linear spring oscillator, when the ratio of two frequencies (the driving forces frequency and the natural frequency of the system) is high, the amplitude of the oscillator will be affected. This is because the driving force changes too fast and the corresponding displacement do not have enough time to change under the inertia.⁵ Since the nonlinear oscillator does not have a certain inherent frequency, it has a familiar characteristic. The strength is constant, and the oscillator amplitude decreases to the fast frequency of the driving force. To avoid the interference of frequency and amplitude, factor w is used to modify the equation.

The modified equation

x^{″} + μ w x^{'} - w^{2} x + w^{2} x^{3} = w^{2} F_{d} \cos (w t)

(23)

The corresponding state of the equation

{\begin{cases} x^{'} = w y \\ y^{'} = w (- μ y + x - x^{3} + F_{d} \cos (w t)) \end{cases}

(24)

Effect of system initial value on the system

The driving force directly determines the energy input and motion state of the oscillator. The initial value also has some influence on the oscillator. When the driving force is weak, the initial value will affect the potential well of the oscillator. When the driving force becomes strong, the oscillator supports the chaotic motion between two potential wells. The initial value corresponds to a trajectory and contains the main characteristics of chaotic motion. If the driving force is very strong, the initial value will affect the transition time. For example, when $F_{d} = 0.826$ , different transition times for different initial values are shown in Figure 6, and there is a certain lag between the forced motion of oscillator and driving force. Continuous adjustment of an abnormal initial value is necessary to adapt to the entire system, so that the steady state appears directly.

Figure 6.

Different transition time under different initial values. The first figure corresponds to the initial value [0, 0], and the second corresponds to [−0213, 1.155]

Classification and characteristics of system state

Generally, the solutions of the system are divided into four categories: constant solutions, periodic solutions, quasi-periodic solutions, and chaotic solutions. The focus of this study is to determine whether the system is in a periodic state, and constant and quasi-periodic solutions are not involved. Therefore, these two categories are not discussed in detail. In addition to the study of the periodic state and the chaotic state, a special alternating state between the periodic and chaotic states is also found in the double-excited Duffing oscillator. The alternating state is called the intermittent periodic state (also known as intermittent chaotic state).

The dynamic behavior analysis explains the above states as follows:

The system in a periodic motion state: The total input energy is sufficient to dissipate the energy consumption of the periodic motion of the system. Therefore, regardless of the initial value of the system, the state can always be adjusted to the periodic state under the total driving force. At this point, the energy dissipated by the damping will be equal to the total input energy, and the system will maintain the periodic motion.

The system in a chaotic motion state: The input energy is insufficient to meet the periodic motion of the system or exceeds the adjustable tolerance range of the system. Regardless of the initial value, due to imbalanced energy input and output, the system cannot stably move periodically, leading to disordered motion of the system, that is, the chaos.

The system in an intermittent chaotic state: The input energy is sometimes sufficient to meet the periodic motion of the system, and sometimes not. For example, when the input is a weak periodic signal and a periodic driving force, there is a fixed frequency difference between them, so that the total input amplitude changes periodically. The output of the system will change intermittently from periodic motion to chaotic motion.

From the classic criterion of chaos, Poincare section¹²: When the system is in the periodic motion state, it will always traverse each point on the trajectory in one cycle, and the phase corresponding to each point is fixed. Therefore, the driving force period is regarded as the sampling period, and the Poincare section is obtained. A series of points on the Poincare section should overlap on the same point or be distributed in a small neighborhood. When the system is in chaotic motion state and the sampling period is the same, the Poincare section points are considered to be widely distributed throughout the attraction basin even though the system no longer moves periodically.

Quantitative determination of chaotic state

Deviation mean square judgment method based on Poincare section

The above discussion indicates that the quality of the system state directly determines the quality of detection. Thus, an efficient and accurate determination method is important. Poincare sections of the system under different states have great differences in the dispersion of sample points. Therefore, the mean square of variance of the sample points can be used to determine the different states of the system. Due to periodicity, in the periodic state, the distances between points are extremely small, and the mean square of deviation is also extremely small, close to 0. In contrast, the sampling points in the chaotic state are relatively scattered, so the mean square of deviation will be large, which is several orders of magnitude different from the former. Theoretically, the chaotic points in the intermittent periodic state should be half of the chaotic state, and their values should be approximately equal to the arithmetic mean of the other two states. In terms of numerical characteristics, the logarithm of chaotic and intermittent periodic states is close to each other, and it is significantly different from the logarithm of periodic state. Therefore, the method of determining the periodic state of the system is highly accurate. When a weak signal is detected, the system should be in a critical period state firstly. If a weak signal is input inversely and the resultant force is reduced, the system is unable to maintain a large periodic state. When the weak signal and driving force signal are at the same frequency, the system will be chaotic. However, if there is a small frequency difference between them, the system will enter an intermittent periodic state. The mean value of the deviation will rapidly increase under the two states to achieve signal detection.

Decision process

The initial value of the system is set to ( $x_{0}, y_{0}$ ), and the experimental time length is set to $T_{s}$ . The fourth-order Runge–Kutta algorithm with a fixed step $τ$ is used to solve the numeric equation, and then the corresponding time and phase diagrams are obtained.

The phase diagram adopts a fixed-period stroboscopic sampling method, and the same sampling period is the same as the system input strategy period, that is

T = \frac{2 π}{w}

(25)

3. The number of fixed sampling points is $N$ , and the system output is sampled in $T_{s}$ . In order to avoid interrupting the initial value, the first $N + 1$ values are discarded. The value is obtained from the total $N$ samples ( $x_{i}, y_{i}$ ) after the $N + 1 th$ to the $2 N th$ . Set $ξ$ , there is

ξ = \frac{1}{N} \sum_{i = 1}^{N} {| l_{i} |}^{2}

(26)

Where

l_{i}

is the directed segment from the center point

O

(

\bar{x}, \bar{y}

) to each sample (

x_{i}, y_{i}

), and

\begin{array}{l} \bar{x} = \frac{1}{N} \sum_{i = 1}^{N} x_{i} \\ \bar{y} = \frac{1}{N} \sum_{i = 1}^{N} y_{i} \end{array}

(27)

4. Through experiment, the critical point of different states of $ξ$ is obtained and set as the threshold $ξ_{m}$ .

When $ξ < ξ_{m}$ , the determined result is periodic.

When $ξ \geq ξ_{m}$ , the determined result is non-periodic.

Selection of decision points

The selection of decision points is vital when the intensity of sampling points is calculated. For decision points of different lengths, the computational complexity and processing efficiency are different. Moreover, the selection of decision points will directly affect the accuracy of the final decision. In principle, only two points are needed to obtain the result, that is, the two points are the same as the periodic state and different in the chaotic state. In order to improve the accuracy, the selection of N points is discussed here.

The periodic and chaotic states have obvious characteristics. The intermittent chaotic state is periodic and chaotic, and it is easily misjudged. Therefore, the selected points should be at least equal to the change of intermittent period $T_{c} = 2 π / Δ w$ . The proposed model is based on the principle of decision points determination. $Δ w$ is the fixed frequency difference between the driving force and weak signal.²⁸ Theoretically, when $Δ w < 0.03 w$ , the intermittent chaos is observed.²⁹

Calculation algorithm and threshold

In view of the nonlinearity and computational complexity of Duffing equation, simulation experiments are necessary to obtain numerical solutions. The fourth-order Runge–Kutta method is thus used in this experiment. Considering that the computer may lose some special points when the step length is uncertain and point control is difficult, the experiment adopts fixed step method for sampling. The sampling frequency is set to an integral multiple of the driving force frequency to ensure that each sampling point has a calculated value.

Different systems correspond to different fixed thresholds. In order to reduce the accidental errors and increase the accuracy, multiple experiments are conducted prior to the formal experiment to determine the threshold.

The driving force amplitude of a specific detection system is first adjusted to $N = 10$ times near the critical point. The adjustment step is taken as the detection accuracy to make it in a different state, and $N$ different $ξ$ values are obtained. The maximum value of the periodic state and chaotic state are taken as $ξ_{p}$ and $ξ_{c}$ , respectively. The decision threshold $ξ_{m}$ is obtained from the geometric mean $\sqrt{ξ_{c} ξ_{p}}$ .

Algorithm complexity analysis

Since the algorithm only takes the points from the $N + 1 th$ to the $2 N th$ in the second period, the solution only needs to be calculated in two periods, which can greatly reduce the calculation time. When the sampling points are processed, equation (26) can be simplified

ξ = \frac{1}{N} \sum_{i = 1}^{N} (x_{i}^{2} + y_{i}^{2}) - {(\frac{1}{N} \sum_{i = 1}^{N} x_{i})}^{2} - {(\frac{1}{N} \sum_{i = 1}^{N} y_{i})}^{2}

(28)

It can be seen from equation (28) that this algorithm only uses $2 N + 5$ multiplication and $3 N + 2$ additions. The complexity of the algorithm is thus low.

Intelligent detector for weak signal

Experimental design and premise

Li et al. showed that when the detection threshold was 0.825, the detection accuracy is 0.001.³⁰ Therefore, the driving force amplitude is set to 0.825. Without affecting the experimental conclusions, the initial phase of the driving force is set to be 0, $μ = 0.5$ , and $w = 2 π$ . The sampling period is $T = 1$ and the number of sample points is $N = \frac{2 π}{0.03 w} = 34$ . In order to verify the feasibility and effectiveness of the proposed model and make the experiment more obvious, the sample points are increased to 100. The entire experimental time is $T_{s} = 2 * N * T = 200$ , and the initial value of the system ( $x_{0}, y_{0}$ ) is set to (−0.213, 1.155).

Before formal experiment, a prior test is conducted to determine the decision threshold

ξ_{m}

. Near the critical point of 0.825, let

F_{d}

take a different value and keep the remaining parameters unchanged. The experiment is repeated several times, and Table 1 is obtained.

Table 1.

System states and values of $ξ$ corresponding to different F_d.

$F_{d}$	$ξ$	State
0.829	3.40 × 10⁻⁶	Periodic state
0.828	3.77 × 10⁻⁶
0.827	5.75 × 10⁻⁶
0.826	4.66 × 10⁻⁶
0.825	4.73 × 10⁻⁶
0.824	0.134411657	Chaotic state
0.823	0.131836814
0.822	0.159901807
0.821	0.172665167
0.820	0.155147772

Table 1 shows the change of $ξ$ and the corresponding system states when the driving force amplitude $F_{d}$ changes near the critical point of 0.825. In the periodic and chaotic states, $ξ$ is less than $1 \times 10^{- 5}$ and greater than 0.1. The geometric mean $1 \times 10^{- 3}$ of $ξ$ is taken as the resolution threshold.

The Duffing oscillator has strong immunity to zero-mean noise.¹³ Under most natural conditions, the system noise is zero-mean white noise. The noise is thus temporarily ignored in the experiment, and only the influence of a single-frequency strategy on the system is considered.

Intelligent weak signal detection model

According to the above design, the corresponding intelligent weak signal detector is shown in Figure 7.

Figure 7.

Proposed intelligent weak signal detection model.

The detection process in Figure 7 is divided into three parts:

Once the input signal is completely copied, it is split into two parts. One part is directly added to the driving force and then enters the chaotic system. The other part moves through $π$ phases, added the driving force and then sends it to the chaotic system. It is aimed to reduce the dead zone caused by the phase difference between the input signal and driving force signals to a small phase neighborhood $π / 2$ , so that the weak signal can be detected under any phase.

After a series of calculation modules, $x$ and $\dot{x}$ are taken and their $ξ$ values are calculated, respectively. This process completes the automatic chaotic processing of the signal, calculates the mean square value of the deviation, and obtains the quantitative index.

If one of $ξ$ is greater than the threshold $ξ_{m}$ and the other is less than it, there will be a signal to be tested; otherwise, there is no signal to be tested.

Experimental verification

In this section, the single variable method is used to verify the effects of weak signal amplitude, initial phase, and frequency on the detection.

Amplitude factor: The driving force parameter is first initialized. The initial phase $φ$ of the weak signal is 0, and the frequency of it is the same as that of the driving force. The amplitude $ε$ varies from 0.0 to 0.5, and the detection accuracy of 0.001 as the step size. A total of 501 input detectors are tested, and the resultant positive and negative values $λ$ are shown in Figure 8.

Figure 8.

Different values of $λ$ corresponding to $ε$ varying from 0.0 to 0.5.

In Figure 8, when $ε = 0$ , the weak signal does not exist, and $λ > 0$ . When $0 < ε \leq 0.481$ , the decisions are correct and $λ < 0$ . However, when $ε > 0.481$ , misjudgments appear due to $λ > 0$ . The reason is that the amplitude of weak signal exceeds the holding thickness of the chaotic interval. When the amplitude of the weak signal is greater than 0.481, the system will skip the chaotic state and directly enter another cycle state, leading to the detection loss. Therefore, when weak signals are detected, it is necessary to ensure the signal strength is not too high. The detection accuracy can reach 100% when the amplitude of weak signal is controlled in a certain range.

2. Phase factor: The driving force parameter is initialized and the weak signal amplitude $ε$ is set to 0.001. The frequency of weak signal is the same as that of the driving force. The initial phase $φ$ varies from $π / 6$ to $π$ , and the step length is $π / 6$ , 6 values in all. Input the values to the detector, and the positive and negative values of $λ$ are shown in Figure 9.

Figure 9.

Different $φ$ corresponds to the value of $λ$ .

In Figure 9, the phase is in the detection dead zone only when the phase is $π / 2$ and an error appears. Theoretically, the range of the dead zone is small and can be ignored. Therefore, the detector has a high detection rate for weak signals with different phases.

3. Frequency factor: Firstly, the driving force parameter is initialized and the weak signal amplitude $ε$ is set to 0.001. The initial phase $φ$ is 0, and the frequency varies from $0.97 w$ to $1.03 w$ .The step size is $0.001 w$ and 61 values in all. The 61 values are input to the detector, and the positive and negative values of $λ$ are shown in Figure 10.

Figure 10.

Values of $λ$ corresponding to the frequency from $0.97 ω$ to $1.03 ω$ .

Figure 10 shows the frequency deviation between the weak signal and driving force in a certain range, that is, $| Δ w | \leq 0.03 w$ . The detection rate of the weak signal is 100%.

According to the above experiments, the detector based on Duffing oscillator can effectively detect a wide range of weak signals. Compared with traditional methods, such as wavelet analysis, it is difficult to find the corresponding basis functions for different forms of signals,³¹ or stochastic resonance that not suitable for long datasets.³² Even the homotopy perturbation method is nonlinear, and it also has uncertain convergence.³³ The Duffing oscillator detector thus has high reliability and accuracy.

Conclusion

In view of the development of chaos theory, the classical Duffing oscillator in chaos is studied.

The basic principle of Holmes-Duffing oscillator is first studied. Based on the analysis of the original Holmes-Duffing oscillator model and in view of its shortcomings, the beam model is changed to a spring model, which makes the Holmes-Duffing equation more concise in calculation and analysis. The validity of the new model is proved by strict theoretical deductions.

Based on the modified model, the characteristics of Duffing oscillator are analyzed in detail. The effects of driving force amplitude and the initial value of frequency on the oscillator are studied, respectively, which reveals the change law of Duffing vibrator under different states.

In addition, through a new method and the Deviation Mean Square Judgment Method based on Poincare section, the quantitative judgment of the chaotic state is realized. This method has low computational complexity and high accuracy.

Finally, based on the above theory, an intelligent weak signal detector based on the Duffing oscillator is designed. The correctness and practicability of the above-mentioned theory are proved through a series of simulation experiments.

Through this work, people can comprehend the mechanisms and characteristics of the Duffing oscillator more intuitively and comprehensively. It provides a new idea for the study of Duffing oscillators and more.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Science Foundation in China (Grant No. 61701534).

ORCID iD

Xiangyu Cao

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