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The homoclinic and heteroclinic chaos in nonlinear systems subjected to trichotomous noise excitation are studied. The Duffing system and the Josephson-junction system are taken for example to calculate the corresponding amplitude thresholds for the onset of chaos on the basis of the stochastic Melnikov process with the mean-square criterion. It is shown that the amplitude threshold for the onset of chaos can be adjusted by changing the internal parameters of trichotomous noise, thereby inducing or suppressing chaotic behaviors in the two systems driven by trichotomous noise. The effects of trichotomous noise on the systems are verified by vanishing the mean largest Lyapunov exponent and demonstrated by phase diagrams and time histories.
The chaotic motion is a kind of irregular and random movement due to sensitive dependence on initial conditions, and it has been studied by many methods such as the largest Lyapunov exponent, Poincare map, power spectrum, fractal dimension, and Melnikov method. Outstanding among them is the Melnikov technique, which renders an analytical method available for the detection of chaotic motions. It was used by Koch and Leven[1] to identify boundaries of subharmonic and homoclinic bifurcations depending on the parameters of a forced pendulum system. With this method, Ariaratnam et al.[2] discussed homoclinic chaos prediction subjected to an axial excitation for the transverse vibration of a buckled column. Shaw and Rand[3] studied the chaotic motions in an inverted pendulum with rigid barriers subjected to periodic excitations. Xie[4] modified the Melnikov method and used it to explore the effect of Gaussian white noise excitation on the chaotic motion of a buckled column. Further, Lin and Yim[5] defined a stochastic Melnikov process and developed the stochastic Melnikov method of determining the threshold for noisy chaos. They observed that external noise has significant effect on the transition from noisy periodic attractor to chaos. Awrejcewicz et al.[6–8] predicted the existence of chaos in a four-dimensional self-excited system and a class of lumped mechanical system with Coulomb-like friction and dry friction, respectively. The stochastic Melnikov technique was applied by Lei[9] to an averaged oscillator in order to determine the threshold of random excitation amplitude for the onset of chaos. Zhang et al.[10] utilized an extended Melnikov method to investigate the multi-pulse global bifurcations and chaotic dynamics of the nonlinear cantilever beam oscillations driven by harmonic axial transverse excitations at the free end in the resonant case. The Melnikov method was also employed by Yang et al.[11] to obtain the chaotic threshold for a nonlinear system under the perturbation of external forcing.
Trichotomous noise, a particular case of non-Gaussian colored noise,[12] can be regarded as a stochastic telegraph process like dichotomous noise, whose effect on dynamic systems has been analyzed by many researchers.[13–18] However, trichotomous noise can represent real noise better than dichotomous noise in actual applications because the former can degenerate into the latter and is more flexible to model random fluctuations in nature.[19] Besides, trichotomous noise can be directly expressed as a physical condition such as the heat transfer between three states or configurations. It has been developed to mimic the fluctuations of the oscillator frequency and mass.[20–23] The influence of variable environment on the population carrying capacity was also modified as the colored three-level Markovian noise.[24] Most of the past research on trichotomous noise focused on the cases of stochastic resonance. For example, Mankin et al. investigated trichotomous-noise-induced transitions[20] and explored the stochastic resonance phenomenon in some linear systems subjected to trichotomous noise.[25–27] The cases of stochastic resonance of a linear oscillator with random damping parameter,[28] several fractional oscillators,[29–31] the coupled underdamped bistable systems[32] and the FitzHugh–Nagumo neuron model[33] driven by trichotomous noise have also been analyzed. Zhong et al.[30] studied two different forms of the generalized stochastic resonance phenomena versus trichotomous noise intensity. For the stochastic chaos, several studies have focused on systems induced by Gaussian noise [34,35] or Poisson noise.[36,37] Lei et al.[17] studied the chaos in a generalized Duffing-type oscillator with a fractional-order deflection subjected to dichotomous noise excitation. They showed that for such an oscillator, the threshold of chaos can be enhanced and then chaotic behaviors can be controlled while appropriately changing the transition rate of dichotomous noise. However, chaos in the system driven by trichotomous noise has not been studied so far.
In this study, the effects of trichotomous noise on chaos in the Duffing system[38–41] and the Josephson–junction system[42–44] are discussed, respectively, with the stochastic Melnikov method. The rest of the paper is organized as follows. Trichotomous noise is generated numerically, and then its spectral density is derived and depicted in Section
One can extend the notion of dichotomous noise further to trichotomous noise denoted by
![]() | (1) |
The transition probabilities[20] between three states
![]() | (2) |
![]() | (3) |
The generation of trichotomous noise is completely determined by Eqs. (
Figures
We calculate the mean value
![]() | (4) |
![]() | (5) |
Then the spectral density of trichotomous noise
![]() | (6) |
Here, we fix a = 1 to show the spectral density of trichotomous noise for several values of parameters τ, q in Fig.
The Melnikov method, developed by Wiggins,[45] of measuring the distance between stable and unstable manifolds, is aimed at predicting whether chaos occurs.
Consider the two-dimensional perturbed Hamiltonian system as follows:
![]() | (7) |
![](cpb_26_3_030502/cpb_26_3_030502_eqn011.gif)
![]() | (8) |
![]() |
![]() | (9) |
If system (
![]() | (10) |
![](cpb_26_3_030502/cpb_26_3_030502_eqn017.gif)
![](cpb_26_3_030502/cpb_26_3_030502_eqn018.gif)
![](cpb_26_3_030502/cpb_26_3_030502_eqn019.gif)
![]() | (11) |
The deterministic portion of the stochastic Melnikov process is
![]() | (12) |
![]() | (13) |
The mean of the stochastic Melnikov process
![]() |
![]() | (14) |
Thus, the second moment of the stochastic portion can be derived through the convolution integral as a filtering process in the frequency domain as follows:
![]() | (15) |
From the viewpoint of the Smale–Birkhoff theorem, if the Melnikov function has a simple zero point, the transversal intersection between the stable and unstable manifolds will occur. Once the two manifolds intersect, they will intersect infinite times, indicating the appearance of chaos. However, the mean of the stochastic Melnikov process is usually negative and not zero since the average ignores the effects of stochastic excitations. However, the standard Melnikov method only provides a necessary condition for the occurrence of chaos in the sense of Smale’s horseshoes. Accordingly, the criterion for chaotic behavior in stochastic systems can be described with a mean-square representation in view of energy.[5] That is, the stochastic Melnikov process has a simple zero point associated with the mean-square criterion when
As mentioned above, we discuss the possible domain for the onset of chaos in a nonlinear system with two illustrative examples.
Consider the Duffing system as
![]() | (16) |
By variable substitution
![]() | (17) |
The Hamiltonian function of the unperturbed system relating to system (
![]() | (18) |
We obtain the homoclinic orbits as follows:
![]() | (19) |
The stochastic Melnikov function for system (
![]() | (20) |
![]() |
Based on the above derivation, the impulse response function is h(t) = y0(t), which correlates with the frequency response function as follows:
![]() | (21) |
Thus, the second moment of the stochastic part of the Melnikov process can be obtained in the frequency domain as follows:
![]() | (22) |
As mentioned above, when
![]() | (23) |
We consider the Josephson-junction system perturbed by trichotomous noise excitation as follows:
![]() | (24) |
With the transformation θ = x,
![]() | (25) |
The Hamiltonian function of the unperturbed system associated with system (
![]() | (26) |
The analytical solutions of two heteroclinic orbits can be calculated from
![]() | (27) |
In a similar way, we derive the stochastic Melnikov function of system (
![]() | (28) |
![]() |
As in the case of the Duffing system, the corresponding frequency response function is
![]() | (29) |
Thus, the second moment of the stochastic part
![]() | (30) |
Then, the conditions of occurrence of chaos for the stochastic Melnikov process in system (
![]() | (31) |
Hence, the threshold for chaotic behaviors in system (
![]() | (32) |
Equations (
Since the result obtained by the stochastic Melnikov method together with the mean-square criterion is not a necessary and sufficient condition for the occurrence of chaos, it is desirable for us to further account for the analytical result by using numerical simulations.
For the existence of chaos, the system can be judged intuitively from the largest Lyapunov exponent which characterizes the exponential rate of divergence or convergence of nearby orbits. A positive largest Lyapunov exponent means that in phase space of the system, no matter how close the two initial trajectories are, the difference between them will increase exponentially with time evolution so that the long-term behavior of the system cannot be predicted. We utilize Wolf’s algorithm[46] to calculate the largest Lyapunov exponent. The basic idea of Wolf’s algorithm is to study the long-term evolution of infinitesimal n-dimensional sphere under the initial condition. The Lyapunov exponent is defined as
![]() |
However, for the stochastic system, we can only determine the speed of divergence or convergence of a sample orbit. Thus, in this work for minimizing the effect of random orbits,[47] we utilize the mean largest Lyapunov exponent for investigating whether the system with a stochastic excitation is chaotic through averaging different sample orbits.
For the nonlinear system with given initial value under the trichotomous noise excitation, if an external noise orbit is selected as input, then a sample orbit of the stochastic system will be obtained as output correspondingly, whose processing method can be similar to that of the deterministic system,[48] by using the variation equation to obtain the linearized system along the fiducial sample orbit. The average divergence or convergence degree of sample orbits can be described through the linearized system. Then, we select another sample orbit to repeat the calculation of the largest Lyapunov exponent
![]() | Fig. 4. (color online) Plots of mean largest Lyapunov exponent L versus amplitude μ in system ( |
To better reveal the dependence of the threshold on the correlation time and stationary probability, the plots of function of the amplitude threshold versus the correlation time for various values of probability are obtained by selecting the critical value of the mean largest Lyapunov exponent (from non-chaos to chaos) as shown in Figs.
We utilize phase diagrams and time histories to numerically investigate dynamical behaviors. Figures
In the present paper, the effects of trichotomous noise excitation on the chaotic behaviors of the Duffing system and the Josephson-junction system are discussed, analytically and numerically. The stochastic Melnikov process in the mean-square sense is used to obtain the threshold of trichotomous noise amplitude for the onset of chaos. When the stationary probability of trichotomous noise is fixed and the correlation time increases, the threshold of noise amplitude for the onset of chaos increases after decreasing first in the Duffing system, but always decreases continuously in the Josephson-junction system. However, in reverse, with the increasing of the stationary probability of trichotomous noise whose correlation time is fixed, the threshold decreases continually in both systems. The extensive numerical simulations including the mean largest Lyapunov exponent, phase diagrams and time histories are carried out to verify the analytical results, and we can see that both results are in agreement with each other. We conclude that the correlation time and the stationary probability of trichotomous noise play significant roles in inducing and suppressing chaos in both the Duffing system and the Josephson-junction system.
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