The influence of nose perturbations on the behaviors of asymmetric vortices over a slender body with a three-caliber ogive nose is studied in this paper. The tests of a nose-disturbed slender body with surface pressur...The influence of nose perturbations on the behaviors of asymmetric vortices over a slender body with a three-caliber ogive nose is studied in this paper. The tests of a nose-disturbed slender body with surface pressure measurement were conducted at a low speed wind tunnel with subcritical Reynolds number of 1×105 at angle of attack α=50°. The experiment results show that the behaviors and structure of asymmetric vortices over the slender body are mainly controlled by manual perturbation on the nose of body as compared with geometrical minute irregularities on the test model from the machining tolerances. The effect of the perturbation axial location on asymmetric vortices is the strongest if its location is near the model apex. There are four sensitive circumferential locations of manual perturbation at which bistable vortices over the slender body are switched by the perturbation. The flowfield near the reattachment line of lee side is more sensitive to the perturbation, because the saddle point to saddle point topological structure in this reattachment flowfield is unstable. Various types of perturbation do not change the perturbation effect on the behaviors of bistable asymmetric vortices.展开更多
In subcritical Reyolds number flow region, the repeatable and determinate asymmetric vortices flow at regular state can be obtained by manually setting mini-perturbation on the nose of a pointed ogive-cylinder model a...In subcritical Reyolds number flow region, the repeatable and determinate asymmetric vortices flow at regular state can be obtained by manually setting mini-perturbation on the nose of a pointed ogive-cylinder model at high angle of attack and zero side slip. Test results of this study involve surface pressure distributions, sectional-side-force distributions and flow visualizations. The analyses of these results revealed a complicated multi-vortices system at regular state inwhich asymmetric twin vortices with inception region and fully developed region, asymmetric triple vortices, four vortices region, five vortices region and Karman-vortex-street-like flow region aredeveloped along the slender body. The correlation between multi-vortices system structures and corresponding sectional-side-force distribution is given. The behaviors of multi-vortices flow struc-ture at the peculiar points of sectional-side-force distributions and characteristics of corresponding pressure distribution are analysed. Finally, a physical model of asymmetric multi-vortices flowstructure in regular state over slender body is developed.展开更多
In this paper we present the control and synchronization of a coupled Bragg acousto-optic bistable map system using nonlinear feedback technology. This nonlinear feedback technology is useful to control a temporally c...In this paper we present the control and synchronization of a coupled Bragg acousto-optic bistable map system using nonlinear feedback technology. This nonlinear feedback technology is useful to control a temporally chaotic system as well as a spatiotemporally chaotic system. It can be extended to synchronize the spatiotemporal chaos. It can work in a wide range of the controlled and synchronized signals, so it can decrease the sensitivity down to a noise level. The synchronization can be obtained by the analysis of the largest conditional Lyapunov exponent spectrum, and easily implemented in practical systems just by adjusting the coupled strength without any pre-knowledge of the dynamic system required.展开更多
In this paper, a simplified model of the bistable piezoelectric cantilever beam with magnets is established, and the potential energy of the bistable system is analyzed. We have proposed the bistable L-shaped beam str...In this paper, a simplified model of the bistable piezoelectric cantilever beam with magnets is established, and the potential energy of the bistable system is analyzed. We have proposed the bistable L-shaped beam structure, which has the same geometry dimensions of the bistable straight beam in the first time. The comparative study on power generations and dynamic responses of the bistable straight beam and the bistable L-shaped beam plays an important role in exploring excellent piezoelectric generator.The experiment structure includes the base layer and the piezoelectric layer. The harmonic excitation is given to the system.Theoretical analysis results show that the potential energy function of the system has two obvious steady potential wells. In addition, the depth of the upper potential well is different from that of the lower potential well when the gravity potential energy is considered. Experimental results demonstrate that the power generation for the straight beam is better than that of the horizontally placed L-shaped beam when the excitation amplitude is 450 m V. There is the existence that the energy harvesting capacity of the bistable L-shaped beam is better than that of the bistable straight beam when the excitation amplitude is 400 m V.Furthermore, the power generation of the bistable L-shaped beam with the upper potential well is obviously better than that of the bistable L-shaped beam with the lower potential well. In addition, comparing with the straight beam, the dynamic response of the bistable L-shaped beam is more complex when the external excitation frequency is changed. It is also observed that the distance between the magnets has the obvious influence on the dynamic response of the bistable system. It is very effective to select the appropriate distance between the magnets to improve the power generation of the bistable energy harvester under the fixed excitation conditions.展开更多
Early bearing faults can generate a series of weak impacts. All the influence factors in measurement may degrade the vibration signal. Currently, bearing fault enhanced detection method based on stochastic resonance...Early bearing faults can generate a series of weak impacts. All the influence factors in measurement may degrade the vibration signal. Currently, bearing fault enhanced detection method based on stochastic resonance（SR） is implemented by expensive computation and demands high sampling rate, which requires high quality software and hardware for fault diagnosis. In order to extract bearing characteristic frequencies component, SR normalized scale transform procedures are presented and a circuit module is designed based on parameter-tuning bistable SR. In the simulation test, discrete and analog sinusoidal signals under heavy noise are enhanced by SR normalized scale transform and circuit module respectively. Two bearing fault enhanced detection strategies are proposed. One is realized by pure computation with normalized scale transform for sampled vibration signal, and the other is carried out by designed SR hardware with circuit module for analog vibration signal directly. The first strategy is flexible for discrete signal processing, and the second strategy demands much lower sampling frequency and less computational cost. The application results of the two strategies on bearing inner race fault detection of a test rig show that the local signal to noise ratio of the characteristic components obtained by the proposed methods are enhanced by about 50% compared with the band pass envelope analysis for the bearing with weaker fault. In addition, helicopter transmission bearing fault detection validates the effectiveness of the enhanced detection strategy with hardware. The combination of SR normalized scale transform and circuit module can meet the need of different application fields or conditions, thus providing a practical scheme for enhanced detection of bearing fault.展开更多
Stochastic resonance（SR） enhances the nonlinear system behavior with the assistance of noise, including the sensitivity and selectivity of the response to the exterior stimulus. The energy-transfer mechanism makes t...Stochastic resonance（SR） enhances the nonlinear system behavior with the assistance of noise, including the sensitivity and selectivity of the response to the exterior stimulus. The energy-transfer mechanism makes the weak information revealed in the output spectrum, while the time-waveform is distorted. The distortion analysis was made both from the particle＇s dynan-fics and signal processing. The factors causing the deviation in the output are presented and the function of the recovery system is proposed. By the investigation of the particle＇s motion track in the bistable system and the suggested recovery system, the influences of noise and system parameters on the recovery course were discussed. Moreover, the pulse distortion appearing the recovery waveform caused by the particle＇s transitions at the bistable potential＇ inflexions was explained. Due to different characteristics, cascaded-bistable SR or mono-stable SR was introduced to process different types of signals. The final recovery signal is just the suggested recovery system＇s response to the SR output. Meanwhile, the recovery system is optional, as parameter-tuned or parameter-fixed. Since the method requires no average processing, it is applicable to a single sample with limited length. The numerical simulations reveal that the SR recovery method can recover the waveform containing weak information submerged ha noise effectively. The engineering application to the vibration analysis of metal cutting chose the combination of mono-stable SR and the parameter-fixed recovery system. Because the optimal SR state is not required strongly, the system parameters are tuned in a wider range than the traditional SR processing methods.展开更多
This paper presents an experimental study of the broadband energy harvesting and dynamic responses of an L-shaped piezoelectric cantilever beam.Experimental results show that the L-shaped piezoelectric beam generates ...This paper presents an experimental study of the broadband energy harvesting and dynamic responses of an L-shaped piezoelectric cantilever beam.Experimental results show that the L-shaped piezoelectric beam generates two optimal voltage peaks when the horizontal beam size is similar to the vertical beam size.Several optimized L-shaped piezoelectric cantilever beam structures are proposed.Power generation using the inverted bistable L-shaped beam is better.It is observed experimentally that the inverted bistable L-shaped beam structure shows obvious bistable characteristics and hard spring characteristics.Furthermore,the corresponding relationship between the bistable phase portrait and the potential energy curve is found in the experiment.This is the first time that a phase portrait for stiffness hardening of an L-shaped beam has been found experimentally.These results can be applied to analysis of new piezoelectric power generation structures.展开更多
By applying the second order Melnikov function, the chaos behaviors of a bistable piezoelectric cantilever power generation system are analyzed. Firstly, the conditions for emerging chaos of the system are derived by ...By applying the second order Melnikov function, the chaos behaviors of a bistable piezoelectric cantilever power generation system are analyzed. Firstly, the conditions for emerging chaos of the system are derived by the second order Melnikov function. Secondly, the effects of each item in chaos threshold expression are analyzed. The excitation frequency and resistance values, which have the most influence on chaos threshold value, are found. The result from the second order Melnikov function is more accurate compared with that from the first order Melnikov function. Finally, the attraction basins of large amplitude motions under different exciting frequency, exciting amplitude, and resistance parameters are given.展开更多
Using the dynamical properties of the polarization bistability that depends on the detuning of the injected light,we propose a novel approach to implement reliable all-optical stochastic logic gates in the cascaded ve...Using the dynamical properties of the polarization bistability that depends on the detuning of the injected light,we propose a novel approach to implement reliable all-optical stochastic logic gates in the cascaded vertical cavity surface emitting lasers（VCSELs） with optical-injection.Here,two logic inputs are encoded in the detuning of the injected light from a tunable CW laser.The logic outputs are decoded from the two orthogonal polarization lights emitted from the optically injected VCSELs.For the same logic inputs,under electro-optic modulation,we perform various digital signal processing（NOT,AND,NAND,XOR,XNOR,OR,NOR） in the all-optical domain by controlling the logic operation of the applied electric field.Also we explore their delay storages by using the mechanism of the generalized chaotic synchronization.To quantify the reliabilities of these logic gates,we further demonstrate their success probabilities.展开更多
Using the principle of minimal energy and S-W model,the exchange bias for ferromagnetic/antiferro-magnetic bilayers has been investigated when the uniaxial anisotropy is misaligned with the exchange anisotropy. Accord...Using the principle of minimal energy and S-W model,the exchange bias for ferromagnetic/antiferro-magnetic bilayers has been investigated when the uniaxial anisotropy is misaligned with the exchange anisotropy. According to the relation between the energy of the bilayer and the orientation of ferro-magnetic magnetization,it is found that the bilayer will be in the monostable state or bistable state when the external field is absent in the initial magnetization state. The monostable state or bistable state of the bilayer,which determines the angular dependence of exchange bias directly,is controlled by the competition between the exchange anisotropy and uniaxial anisotropy. When the applied field is parallel to the intrinsic easy axes and intrinsic hard axes,one of the switching fields of the hysteresis loop shows an abrupt change,while the other keep continuous by analyzing the magnetization reversal processes. Consequently,the exchange bias field and the coercivity will show a jump phenomenon. The numerical calculations indicate that both the magnitude and direction of the exchange anisotropy will significantly affect the angular dependence of exchange bias. The jump phenomenon of exchange bias is an intrinsic property of the bilayer,which is dependent on the interfacial exchange-coupling constant,the orientation of the exchange anisotropy,the thickness and uniaxial anisotropy constant of the ferromagnetic layer.展开更多
Stochastic perturbations and periodic excitations are generally regarded as sources to induce critical transitions in complex systems. However, we find that they are also able to slow down an imminent critical transit...Stochastic perturbations and periodic excitations are generally regarded as sources to induce critical transitions in complex systems. However, we find that they are also able to slow down an imminent critical transition. To illustrate this phenomenon, a periodically driven bistable eutrophication model with Gaussian white noise is introduced as a prototype class of real systems.The residence probability(RP) is presented to measure the possibility that the given system stays in the oligotrophic state versus Gaussian white noise and periodic force. Variations in the mean first passage time(MFPT) and the mean velocity(MV) of the first right-crossing process are also calculated respectively. We show that the frequency of the periodic force can increase the MFPT while reduce the MV under different control parameters. Nevertheless, the noise intensity or the amplitude may result in an increase of the RP only in the case of control parameters approaching the critical values. Furthermore, for an impending critical transition, an increase of the RP appears with the interaction between the amplitude and noise intensity or the combination of the noise intensity and frequency, while the interaction of the frequency and amplitude leads to an extension of the MFPT or a decrease of the MV. As a result, an increase of the RP and MFPT, and a decrease of the MVobtained from our results claim that it is possible to slow down an imminent critical transition via Gaussian white noise and periodic force.展开更多
基金The project supported by the National Natural Science Foundation of China (10172017)the Foundation of National Key Laboratory of Aerodynamic Design and Research (00JS51.3.2 HK01)
文摘The influence of nose perturbations on the behaviors of asymmetric vortices over a slender body with a three-caliber ogive nose is studied in this paper. The tests of a nose-disturbed slender body with surface pressure measurement were conducted at a low speed wind tunnel with subcritical Reynolds number of 1×105 at angle of attack α=50°. The experiment results show that the behaviors and structure of asymmetric vortices over the slender body are mainly controlled by manual perturbation on the nose of body as compared with geometrical minute irregularities on the test model from the machining tolerances. The effect of the perturbation axial location on asymmetric vortices is the strongest if its location is near the model apex. There are four sensitive circumferential locations of manual perturbation at which bistable vortices over the slender body are switched by the perturbation. The flowfield near the reattachment line of lee side is more sensitive to the perturbation, because the saddle point to saddle point topological structure in this reattachment flowfield is unstable. Various types of perturbation do not change the perturbation effect on the behaviors of bistable asymmetric vortices.
基金This work was supported by the National Natural Science Foundation of China(Grant No.10172017)Aeronautical Science Foundation ofChina(02A51048)Foundation of National Key Laboratory ofAerodynamic Design and Research(00JS51.3.2HK01).
文摘In subcritical Reyolds number flow region, the repeatable and determinate asymmetric vortices flow at regular state can be obtained by manually setting mini-perturbation on the nose of a pointed ogive-cylinder model at high angle of attack and zero side slip. Test results of this study involve surface pressure distributions, sectional-side-force distributions and flow visualizations. The analyses of these results revealed a complicated multi-vortices system at regular state inwhich asymmetric twin vortices with inception region and fully developed region, asymmetric triple vortices, four vortices region, five vortices region and Karman-vortex-street-like flow region aredeveloped along the slender body. The correlation between multi-vortices system structures and corresponding sectional-side-force distribution is given. The behaviors of multi-vortices flow struc-ture at the peculiar points of sectional-side-force distributions and characteristics of corresponding pressure distribution are analysed. Finally, a physical model of asymmetric multi-vortices flowstructure in regular state over slender body is developed.
文摘In this paper we present the control and synchronization of a coupled Bragg acousto-optic bistable map system using nonlinear feedback technology. This nonlinear feedback technology is useful to control a temporally chaotic system as well as a spatiotemporally chaotic system. It can be extended to synchronize the spatiotemporal chaos. It can work in a wide range of the controlled and synchronized signals, so it can decrease the sensitivity down to a noise level. The synchronization can be obtained by the analysis of the largest conditional Lyapunov exponent spectrum, and easily implemented in practical systems just by adjusting the coupled strength without any pre-knowledge of the dynamic system required.
基金supported by the National Natural Science Foundation of China(Grant Nos.11772008,11172009,11372015,11232009,10872010,11290152&10732020)
文摘In this paper, a simplified model of the bistable piezoelectric cantilever beam with magnets is established, and the potential energy of the bistable system is analyzed. We have proposed the bistable L-shaped beam structure, which has the same geometry dimensions of the bistable straight beam in the first time. The comparative study on power generations and dynamic responses of the bistable straight beam and the bistable L-shaped beam plays an important role in exploring excellent piezoelectric generator.The experiment structure includes the base layer and the piezoelectric layer. The harmonic excitation is given to the system.Theoretical analysis results show that the potential energy function of the system has two obvious steady potential wells. In addition, the depth of the upper potential well is different from that of the lower potential well when the gravity potential energy is considered. Experimental results demonstrate that the power generation for the straight beam is better than that of the horizontally placed L-shaped beam when the excitation amplitude is 450 m V. There is the existence that the energy harvesting capacity of the bistable L-shaped beam is better than that of the bistable straight beam when the excitation amplitude is 400 m V.Furthermore, the power generation of the bistable L-shaped beam with the upper potential well is obviously better than that of the bistable L-shaped beam with the lower potential well. In addition, comparing with the straight beam, the dynamic response of the bistable L-shaped beam is more complex when the external excitation frequency is changed. It is also observed that the distance between the magnets has the obvious influence on the dynamic response of the bistable system. It is very effective to select the appropriate distance between the magnets to improve the power generation of the bistable energy harvester under the fixed excitation conditions.
基金supported by National Natural Science Foundation of China(Grant Nos. 51075391, 51105366)
文摘Early bearing faults can generate a series of weak impacts. All the influence factors in measurement may degrade the vibration signal. Currently, bearing fault enhanced detection method based on stochastic resonance（SR） is implemented by expensive computation and demands high sampling rate, which requires high quality software and hardware for fault diagnosis. In order to extract bearing characteristic frequencies component, SR normalized scale transform procedures are presented and a circuit module is designed based on parameter-tuning bistable SR. In the simulation test, discrete and analog sinusoidal signals under heavy noise are enhanced by SR normalized scale transform and circuit module respectively. Two bearing fault enhanced detection strategies are proposed. One is realized by pure computation with normalized scale transform for sampled vibration signal, and the other is carried out by designed SR hardware with circuit module for analog vibration signal directly. The first strategy is flexible for discrete signal processing, and the second strategy demands much lower sampling frequency and less computational cost. The application results of the two strategies on bearing inner race fault detection of a test rig show that the local signal to noise ratio of the characteristic components obtained by the proposed methods are enhanced by about 50% compared with the band pass envelope analysis for the bearing with weaker fault. In addition, helicopter transmission bearing fault detection validates the effectiveness of the enhanced detection strategy with hardware. The combination of SR normalized scale transform and circuit module can meet the need of different application fields or conditions, thus providing a practical scheme for enhanced detection of bearing fault.
基金supported by National Hi-tech Research and Development Program of China (863 Program, Grant No. 2007AA04Z414)National Natural Science Foundation of China (Grant No. 50675153)+1 种基金Tianjin Municipal Natural Science Foundation of China (Grant No. 07JCYBJC04600)Specialized Research Fund for Doctoral Program of Higher Education of China (Grant No. 20060056016)
文摘Stochastic resonance（SR） enhances the nonlinear system behavior with the assistance of noise, including the sensitivity and selectivity of the response to the exterior stimulus. The energy-transfer mechanism makes the weak information revealed in the output spectrum, while the time-waveform is distorted. The distortion analysis was made both from the particle＇s dynan-fics and signal processing. The factors causing the deviation in the output are presented and the function of the recovery system is proposed. By the investigation of the particle＇s motion track in the bistable system and the suggested recovery system, the influences of noise and system parameters on the recovery course were discussed. Moreover, the pulse distortion appearing the recovery waveform caused by the particle＇s transitions at the bistable potential＇ inflexions was explained. Due to different characteristics, cascaded-bistable SR or mono-stable SR was introduced to process different types of signals. The final recovery signal is just the suggested recovery system＇s response to the SR output. Meanwhile, the recovery system is optional, as parameter-tuned or parameter-fixed. Since the method requires no average processing, it is applicable to a single sample with limited length. The numerical simulations reveal that the SR recovery method can recover the waveform containing weak information submerged ha noise effectively. The engineering application to the vibration analysis of metal cutting chose the combination of mono-stable SR and the parameter-fixed recovery system. Because the optimal SR state is not required strongly, the system parameters are tuned in a wider range than the traditional SR processing methods.
基金supported by the National Natural Science Foundation of China(Grants 11772008,11172009,11372015,11232009,10872010,11290152,10732020)the Tianjin Natural Science Foundation(Grant 19JCZDJC32300).
文摘This paper presents an experimental study of the broadband energy harvesting and dynamic responses of an L-shaped piezoelectric cantilever beam.Experimental results show that the L-shaped piezoelectric beam generates two optimal voltage peaks when the horizontal beam size is similar to the vertical beam size.Several optimized L-shaped piezoelectric cantilever beam structures are proposed.Power generation using the inverted bistable L-shaped beam is better.It is observed experimentally that the inverted bistable L-shaped beam structure shows obvious bistable characteristics and hard spring characteristics.Furthermore,the corresponding relationship between the bistable phase portrait and the potential energy curve is found in the experiment.This is the first time that a phase portrait for stiffness hardening of an L-shaped beam has been found experimentally.These results can be applied to analysis of new piezoelectric power generation structures.
基金supported by the National Natural Science Foundation of China (Grant 11172199)
文摘By applying the second order Melnikov function, the chaos behaviors of a bistable piezoelectric cantilever power generation system are analyzed. Firstly, the conditions for emerging chaos of the system are derived by the second order Melnikov function. Secondly, the effects of each item in chaos threshold expression are analyzed. The excitation frequency and resistance values, which have the most influence on chaos threshold value, are found. The result from the second order Melnikov function is more accurate compared with that from the first order Melnikov function. Finally, the attraction basins of large amplitude motions under different exciting frequency, exciting amplitude, and resistance parameters are given.
基金Project supported by the National Natural Science Foundation of China(Grant No.61475120)the Innovative Projects in Guangdong Colleges and Universities,China(Grant Nos.2014KTSCX134 and 2015KTSCX146)
文摘Using the dynamical properties of the polarization bistability that depends on the detuning of the injected light,we propose a novel approach to implement reliable all-optical stochastic logic gates in the cascaded vertical cavity surface emitting lasers（VCSELs） with optical-injection.Here,two logic inputs are encoded in the detuning of the injected light from a tunable CW laser.The logic outputs are decoded from the two orthogonal polarization lights emitted from the optically injected VCSELs.For the same logic inputs,under electro-optic modulation,we perform various digital signal processing（NOT,AND,NAND,XOR,XNOR,OR,NOR） in the all-optical domain by controlling the logic operation of the applied electric field.Also we explore their delay storages by using the mechanism of the generalized chaotic synchronization.To quantify the reliabilities of these logic gates,we further demonstrate their success probabilities.
基金Supported by the National Natural Science Foundation of China (Grant No. 10762001)the Program for New Century Excellent Talents in University of China (Grant No. NCET- 05- 0272)the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 200801260003 )
文摘Using the principle of minimal energy and S-W model,the exchange bias for ferromagnetic/antiferro-magnetic bilayers has been investigated when the uniaxial anisotropy is misaligned with the exchange anisotropy. According to the relation between the energy of the bilayer and the orientation of ferro-magnetic magnetization,it is found that the bilayer will be in the monostable state or bistable state when the external field is absent in the initial magnetization state. The monostable state or bistable state of the bilayer,which determines the angular dependence of exchange bias directly,is controlled by the competition between the exchange anisotropy and uniaxial anisotropy. When the applied field is parallel to the intrinsic easy axes and intrinsic hard axes,one of the switching fields of the hysteresis loop shows an abrupt change,while the other keep continuous by analyzing the magnetization reversal processes. Consequently,the exchange bias field and the coercivity will show a jump phenomenon. The numerical calculations indicate that both the magnitude and direction of the exchange anisotropy will significantly affect the angular dependence of exchange bias. The jump phenomenon of exchange bias is an intrinsic property of the bilayer,which is dependent on the interfacial exchange-coupling constant,the orientation of the exchange anisotropy,the thickness and uniaxial anisotropy constant of the ferromagnetic layer.
基金supported by the National Natural Science Foundation of China(Grant Nos.11772255&11872305)the Fundamental Research Funds for the Central Universities+2 种基金Shaanxi Province Project for Distinguished Young ScholarsInnovation Foundation for Doctor Dissertation of Northwestern Polytechnical Universitythe China Postdoctoral Science Foundation
文摘Stochastic perturbations and periodic excitations are generally regarded as sources to induce critical transitions in complex systems. However, we find that they are also able to slow down an imminent critical transition. To illustrate this phenomenon, a periodically driven bistable eutrophication model with Gaussian white noise is introduced as a prototype class of real systems.The residence probability(RP) is presented to measure the possibility that the given system stays in the oligotrophic state versus Gaussian white noise and periodic force. Variations in the mean first passage time(MFPT) and the mean velocity(MV) of the first right-crossing process are also calculated respectively. We show that the frequency of the periodic force can increase the MFPT while reduce the MV under different control parameters. Nevertheless, the noise intensity or the amplitude may result in an increase of the RP only in the case of control parameters approaching the critical values. Furthermore, for an impending critical transition, an increase of the RP appears with the interaction between the amplitude and noise intensity or the combination of the noise intensity and frequency, while the interaction of the frequency and amplitude leads to an extension of the MFPT or a decrease of the MV. As a result, an increase of the RP and MFPT, and a decrease of the MVobtained from our results claim that it is possible to slow down an imminent critical transition via Gaussian white noise and periodic force.
