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July  2021, 14(7): 2229-2243. doi: 10.3934/dcdss.2020397

Stochastic dynamics of the FitzHugh-Nagumo neuron model through a modified Van der Pol equation with fractional-order term and Gaussian white noise excitation

1. 

Laboratoire de Biophysique, Département de Physique, Faculté des Sciences, Université de Yaoundé I, B.P. 812 Yaoundé, Cameroun

2. 

Botswana International University of Science and Technology, Private Bag 16 Palapye, Botswana

3. 

Département de Physique, Faculté des Sciences, Université de Maroua, B.P. 46 Maroua, Cameroun

4. 

Laboratoire de Mécanique, Département de Physique, Faculté des Sciences, Université de Yaoundé I, B.P. 812 Yaoundé, Cameroun

* Corresponding author: tabic@biust.ac.bw (C. B. Tabi)

Received  May 2019 Revised  September 2019 Published  July 2021 Early access  June 2020

Fund Project: The work of CBT was supported by the Botswana International University of Science and Technology under the grant DVC/RDI/2/1/16I (25). CBT thanks the Kavli Institute for Theoretical Physics (KITP), University of California Santa Barbara (USA), where this work was supported in part by the National Science Foundation Grant no.NSF PHY-1748958 and NIH Grant no.R25GM067110

The stochastic response of the FitzHugh-Nagumo model is addressed using a modified Van der Pol (VDP) equation with fractional-order derivative and Gaussian white noise excitation. Via the generalized harmonic balance method, the term related to fractional derivative is splitted into the equivalent quasi-linear dissipative force and quasi-linear restoring force, leading to an equivalent VDP equation without fractional derivative. The analytical solutions for the equivalent stochastic equation are then investigated through the stochastic averaging method. This is thereafter compared to numerical solutions, where the stationary probability density function (PDF) of amplitude and joint PDF of displacement and velocity are used to characterized the dynamical behaviors of the system. A satisfactory agreement is found between the two approaches, which confirms the accuracy of the used analytical method. It is also found that changing the fractional-order parameter and the intensity of the Gaussian white noise induces P-bifurcation.

Citation: Boris Anicet Guimfack, Conrad Bertrand Tabi, Alidou Mohamadou, Timoléon Crépin Kofané. Stochastic dynamics of the FitzHugh-Nagumo neuron model through a modified Van der Pol equation with fractional-order term and Gaussian white noise excitation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2229-2243. doi: 10.3934/dcdss.2020397
References:
[1]

R. R. Aliev and A. V. Panfilov, A simple two-variable model of cardiac excitation, Chaos, Solitons & Fractals, 7 (1996), 293-301.  doi: 10.1016/0960-0779(95)00089-5.

[2]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A.

[3]

C. D. K. BansiC. B. TabiG. T. Motsumi and A. Mohamadou, Fractional blood flow in oscillatory arteries with thermal radiation and magnetic field effects, J. Magn. Magn. Mater., 456 (2018), 38-45.  doi: 10.1016/j.jmmm.2018.01.079.

[4]

I. Bashkirtseva and L. Ryashko, Analysis of excitability for the FitzHugh-Nagumo model via a stochastic sensitivity function technique, Phys. Rev. E, 83 (2011), 061109. doi: 10.1103/PhysRevE.83.061109.

[5]

M. Caputo, Linear models of dissipation whose $Q$ is almost frequency independent-Ⅱ, Geophysical Journal International, 13 (1967), 529-539.  doi: 10.1111/j.1365-246X.1967.tb02303.x.

[6]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73. doi: 10.12785/pfda/010201.

[7]

A. CheerJ.-P. VincentR. Nuccitelli and G. Oster, Cortical activity in vertebrate eggs I: The activation waves, J. Theor. Biol., 124 (1987), 377-404.  doi: 10.1016/S0022-5193(87)80217-5.

[8]

L. ChenW. WangZ. Li and W. Zhu, Stationary response of Duffing oscillator with hardening stiffness and fractional derivative, Int. J. Non-Linear Mech., 48 (2013), 44-50.  doi: 10.1016/j.ijnonlinmec.2012.08.001.

[9]

L. ChenZ. LiQ. Zhuang and W. Zhu, First-passage failure of single-degree-of-freedom nonlinear oscillators with fractional derivative, J. Vib. Control, 19 (2013), 2154-2163.  doi: 10.1177/1077546312456057.

[10]

J. J. Collins, C. C. Chow and T. T. Imhoff, Aperiodic stochastic resonance in excitable systems, Phys. Rev. E, 52 (1995), R3321(R). doi: 10.1103/PhysRevE.52.R3321.

[11]

K. DiethelmN. J. Ford and A. D. Freed, A Predictor-Corrector approach for the numerical solution of fractional differential equations, Nonl. Dyn., 29 (2002), 3-22.  doi: 10.1023/A:1016592219341.

[12]

E. F. Doungmo Goufo, Application of the Caputo-Fabrizio fractional derivative without singular kernel to Korteweg-de Vries–Burgers equation, Math. Model. Anal., 21 (2016), 188-198.  doi: 10.3846/13926292.2016.1145607.

[13]

E. F. Doungmo Goufo and A. Atangana, Analytical and numerical schemes for a derivative with filtering property and no singular kernel with applications to diffusion, Eur. Phys. J. Plus, 131 (2016), 269. doi: 10.1140/epjp/i2016-16269-1.

[14]

E. F. Doungmo Goufo and C. B. Tabi, On the chaotic pole of attraction for Hindmarsh-Rose neuron dynamics with external current input, Chaos, 29 (2019), 023104, 9pp. doi: 10.1063/1.5083180.

[15]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466.  doi: 10.1016/S0006-3495(61)86902-6.

[16]

R. FitzHugh, Thresholds and plateaus in the Hodgkin-Huxley nerve equations, J. Gen. Physiol., 43 (1960), 867-896.  doi: 10.1085/jgp.43.5.867.

[17]

J. Guckenheimer and C. Kuehn, Homoclinic orbits of the FitzHugh-Nagumo equation: Bifurcations in the full system, SIAM J. Appl. Dyn. Syst., 9 (2010), 138-153.  doi: 10.1137/090758404.

[18]

J. L. Hindmarsh and R. M. Rose, A model of the nerve impulse using two first-order differential equations, Nature, 296 (1982), 162-164.  doi: 10.1038/296162a0.

[19]

J. L. Hindmarsh and R. M. Rose, A model of neuronal bursting using three coupled first order differential equations, Proc. Royal. Soc. B, 221 (1984), 87-102.  doi: 10.1098/rspb.1984.0024.

[20]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. physiol., 117 (1952), 500-544.  doi: 10.1113/jphysiol.1952.sp004764.

[21]

Z. L. Huang and X. L. Jin, Response and stability of a SDOF strongly nonlinear stochastic system with light damping modeled by a fractional derivative, J. Sound Vib., 319 (2009), 1121-1135.  doi: 10.1016/j.jsv.2008.06.026.

[22]

E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, Computational Neuroscience. MIT Press, Cambridge, MA, 2007.

[23] C. Koch, Biophysics of Computation: Information Processing in Single Neurons, Oxford University Press, 1999. 
[24]

M. Kostur, X. Sailer and L. Schimansky-Geier, Stationary probability distributions for FitzHugh-Nagumo systems, Fluct. Noise Lett., 3 (2003), L155–L166. doi: 10.1142/S0219477503001221.

[25]

B. LindnerJ. García-OjalvoA. Neiman and L. Schimansky-Geier, Effects of noise in excitable systems, Phys. Rep., 392 (2004), 321-424.  doi: 10.1016/j.physrep.2003.10.015.

[26]

A. Longtin, Stochastic resonance in neuron models, J. Stat. Phys., 70 (1993), 309-327.  doi: 10.1007/BF01053970.

[27]

J. NagumoS. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2070.  doi: 10.1109/JRPROC.1962.288235.

[28]

X. PeiK. Bachmann and F. Moss, The detection threshold, noise and stochastic resonance in the Fitzhugh-Nagumo neuron model, Phys. Lett. A, 206 (1995), 61-65.  doi: 10.1016/0375-9601(95)00639-K.

[29]

I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications, Academic Press, San Diego, CA, 1999.

[30]

Z. Ran-Ran, X. Wei, Y. Gui-Dong and H. Qun, Response of a Duffing-Rayleigh system with a fractional derivative under Gaussian white noise excitation, Chin. Phys. B, 24 (2015), 020204. doi: 10.1088/1674-1056/24/2/020204.

[31]

R. SchererS. L. KallaY. Tang and J. Huang, The Grünwald-Letnikov method for fractional differential equations, Comput. Math. Appl., 62 (2011), 902-917.  doi: 10.1016/j.camwa.2011.03.054.

[32]

Y. Shen, P. Wei, C. Sui and S. Yang, Subharmonic resonance of Van-Der Pol oscillator with fractional-order derivative, Math. Probl. Eng., 2014 (2014), Art. ID 738087, 17 pp. doi: 10.1155/2014/738087.

[33]

Y. ShenP. Wei and S. Yang, Primary resonance of fractional-order van der Pol oscillator, Nonlinear Dyn., 77 (2014), 1629-1642.  doi: 10.1007/s11071-014-1405-2.

[34]

Y. Shen, S. Yang and H. Xing, Dynamical analysis of linear single degree-of-freedom oscillator with fractional-order derivative, Acta Physica Sinica, 61 (2012), 110505-1-6. doi: 10.7498/aps.61.110505.

[35]

Y. ShenS. YangH. Xing and G. Gao, Primary resonance of Duffing oscillator with fractional-order derivative, Commun. Nonl. Sci. Numer. Simul., 17 (2012), 3092-3100.  doi: 10.1016/j.cnsns.2011.11.024.

[36]

J. Sneyd and J. Sherratt, On the propagation of calcium waves in an inhomogeneous medium, SIAM J. Appl. Math., 57 (1997), 73-94.  doi: 10.1137/S0036139995286035.

[37]

P. D. Spanos and B. A. Zeldin, Random vibration of systems with frequency-dependent parameters or fractional derivatives, J. Eng. Mech., 123 (1997), 290. doi: 10.1061/(ASCE)0733-9399(1997)123:3(290).

[38]

C. B. Tabi, Dynamical analysis of the FitzHugh–Nagumo oscillations through a modified Van der Pol equation with fractional-order derivative term, Int. J. Nonl. Mech., 105 (2018), 173-178.  doi: 10.1016/j.ijnonlinmec.2018.05.026.

[39]

C. B. Tabi, Fractional unstable patterns of energy in $\alpha-$helix proteins with long-range interactions, Chaos Sol. Fract., 116 (2018), 386-391.  doi: 10.1016/j.chaos.2018.09.037.

[40]

D. Tatchim BemmoM. Siewe Siewe and C. Tchawoua, Nonlinear oscillations of the FitzHugh-Nagumo equations under combined external and two-frequency parametric excitations, Phys. Lett. A, 375 (2011), 1944-1953.  doi: 10.1016/j.physleta.2011.02.072.

[41]

D. Tatchim BemmoM. Siewe Siewe and C. Tchawoua, Combined effects of correlated bounded noises and weak periodic signal input in the modified FitzHugh-Nagumo neural model, Commun. Nonl. Sci. Numer. Simul., 18 (2013), 1275-1287.  doi: 10.1016/j.cnsns.2012.09.016.

[42]

H. Treutlein and K. Schulten, Noise-induced limit cycles of the Bonhoeffer-Van der Pol model of neural pulses, Phys. Chem., 89 (1985), 710-718.  doi: 10.1002/bbpc.19850890626.

[43]

J. C. Tsai and J. Sneyd, Traveling waves in the buffered FitzHugh-Nagumo model, SIAM J. Appl. Math., 71 (2011), 1606-1636.  doi: 10.1137/110820348.

[44]

K. Wiesenfeld, D. Pierson E. Pantazelou, C. Dames and F. Moss, Stochastic resonance on a circle, Phys. Rev. Lett., 72 (1994), 2125. doi: 10.1103/PhysRevLett.72.2125.

[45]

Y. YangW. XuX. Gu and Y. Sun, Stochastic response of a class of self-excited systems with Caputo-type fractional derivative driven by Gaussian white noise, Chaos Solit. Frac., 77 (2015), 190-204.  doi: 10.1016/j.chaos.2015.05.029.

[46]

Y. YangW. XuW. Jia and Q. Han, Stationary response of nonlinear system with Caputo-type fractional derivative damping under Gaussian white noise excitation, Nonl. Dyn., 79 (2015), 139-146.  doi: 10.1007/s11071-014-1651-3.

show all references

References:
[1]

R. R. Aliev and A. V. Panfilov, A simple two-variable model of cardiac excitation, Chaos, Solitons & Fractals, 7 (1996), 293-301.  doi: 10.1016/0960-0779(95)00089-5.

[2]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A.

[3]

C. D. K. BansiC. B. TabiG. T. Motsumi and A. Mohamadou, Fractional blood flow in oscillatory arteries with thermal radiation and magnetic field effects, J. Magn. Magn. Mater., 456 (2018), 38-45.  doi: 10.1016/j.jmmm.2018.01.079.

[4]

I. Bashkirtseva and L. Ryashko, Analysis of excitability for the FitzHugh-Nagumo model via a stochastic sensitivity function technique, Phys. Rev. E, 83 (2011), 061109. doi: 10.1103/PhysRevE.83.061109.

[5]

M. Caputo, Linear models of dissipation whose $Q$ is almost frequency independent-Ⅱ, Geophysical Journal International, 13 (1967), 529-539.  doi: 10.1111/j.1365-246X.1967.tb02303.x.

[6]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73. doi: 10.12785/pfda/010201.

[7]

A. CheerJ.-P. VincentR. Nuccitelli and G. Oster, Cortical activity in vertebrate eggs I: The activation waves, J. Theor. Biol., 124 (1987), 377-404.  doi: 10.1016/S0022-5193(87)80217-5.

[8]

L. ChenW. WangZ. Li and W. Zhu, Stationary response of Duffing oscillator with hardening stiffness and fractional derivative, Int. J. Non-Linear Mech., 48 (2013), 44-50.  doi: 10.1016/j.ijnonlinmec.2012.08.001.

[9]

L. ChenZ. LiQ. Zhuang and W. Zhu, First-passage failure of single-degree-of-freedom nonlinear oscillators with fractional derivative, J. Vib. Control, 19 (2013), 2154-2163.  doi: 10.1177/1077546312456057.

[10]

J. J. Collins, C. C. Chow and T. T. Imhoff, Aperiodic stochastic resonance in excitable systems, Phys. Rev. E, 52 (1995), R3321(R). doi: 10.1103/PhysRevE.52.R3321.

[11]

K. DiethelmN. J. Ford and A. D. Freed, A Predictor-Corrector approach for the numerical solution of fractional differential equations, Nonl. Dyn., 29 (2002), 3-22.  doi: 10.1023/A:1016592219341.

[12]

E. F. Doungmo Goufo, Application of the Caputo-Fabrizio fractional derivative without singular kernel to Korteweg-de Vries–Burgers equation, Math. Model. Anal., 21 (2016), 188-198.  doi: 10.3846/13926292.2016.1145607.

[13]

E. F. Doungmo Goufo and A. Atangana, Analytical and numerical schemes for a derivative with filtering property and no singular kernel with applications to diffusion, Eur. Phys. J. Plus, 131 (2016), 269. doi: 10.1140/epjp/i2016-16269-1.

[14]

E. F. Doungmo Goufo and C. B. Tabi, On the chaotic pole of attraction for Hindmarsh-Rose neuron dynamics with external current input, Chaos, 29 (2019), 023104, 9pp. doi: 10.1063/1.5083180.

[15]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466.  doi: 10.1016/S0006-3495(61)86902-6.

[16]

R. FitzHugh, Thresholds and plateaus in the Hodgkin-Huxley nerve equations, J. Gen. Physiol., 43 (1960), 867-896.  doi: 10.1085/jgp.43.5.867.

[17]

J. Guckenheimer and C. Kuehn, Homoclinic orbits of the FitzHugh-Nagumo equation: Bifurcations in the full system, SIAM J. Appl. Dyn. Syst., 9 (2010), 138-153.  doi: 10.1137/090758404.

[18]

J. L. Hindmarsh and R. M. Rose, A model of the nerve impulse using two first-order differential equations, Nature, 296 (1982), 162-164.  doi: 10.1038/296162a0.

[19]

J. L. Hindmarsh and R. M. Rose, A model of neuronal bursting using three coupled first order differential equations, Proc. Royal. Soc. B, 221 (1984), 87-102.  doi: 10.1098/rspb.1984.0024.

[20]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. physiol., 117 (1952), 500-544.  doi: 10.1113/jphysiol.1952.sp004764.

[21]

Z. L. Huang and X. L. Jin, Response and stability of a SDOF strongly nonlinear stochastic system with light damping modeled by a fractional derivative, J. Sound Vib., 319 (2009), 1121-1135.  doi: 10.1016/j.jsv.2008.06.026.

[22]

E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, Computational Neuroscience. MIT Press, Cambridge, MA, 2007.

[23] C. Koch, Biophysics of Computation: Information Processing in Single Neurons, Oxford University Press, 1999. 
[24]

M. Kostur, X. Sailer and L. Schimansky-Geier, Stationary probability distributions for FitzHugh-Nagumo systems, Fluct. Noise Lett., 3 (2003), L155–L166. doi: 10.1142/S0219477503001221.

[25]

B. LindnerJ. García-OjalvoA. Neiman and L. Schimansky-Geier, Effects of noise in excitable systems, Phys. Rep., 392 (2004), 321-424.  doi: 10.1016/j.physrep.2003.10.015.

[26]

A. Longtin, Stochastic resonance in neuron models, J. Stat. Phys., 70 (1993), 309-327.  doi: 10.1007/BF01053970.

[27]

J. NagumoS. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2070.  doi: 10.1109/JRPROC.1962.288235.

[28]

X. PeiK. Bachmann and F. Moss, The detection threshold, noise and stochastic resonance in the Fitzhugh-Nagumo neuron model, Phys. Lett. A, 206 (1995), 61-65.  doi: 10.1016/0375-9601(95)00639-K.

[29]

I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications, Academic Press, San Diego, CA, 1999.

[30]

Z. Ran-Ran, X. Wei, Y. Gui-Dong and H. Qun, Response of a Duffing-Rayleigh system with a fractional derivative under Gaussian white noise excitation, Chin. Phys. B, 24 (2015), 020204. doi: 10.1088/1674-1056/24/2/020204.

[31]

R. SchererS. L. KallaY. Tang and J. Huang, The Grünwald-Letnikov method for fractional differential equations, Comput. Math. Appl., 62 (2011), 902-917.  doi: 10.1016/j.camwa.2011.03.054.

[32]

Y. Shen, P. Wei, C. Sui and S. Yang, Subharmonic resonance of Van-Der Pol oscillator with fractional-order derivative, Math. Probl. Eng., 2014 (2014), Art. ID 738087, 17 pp. doi: 10.1155/2014/738087.

[33]

Y. ShenP. Wei and S. Yang, Primary resonance of fractional-order van der Pol oscillator, Nonlinear Dyn., 77 (2014), 1629-1642.  doi: 10.1007/s11071-014-1405-2.

[34]

Y. Shen, S. Yang and H. Xing, Dynamical analysis of linear single degree-of-freedom oscillator with fractional-order derivative, Acta Physica Sinica, 61 (2012), 110505-1-6. doi: 10.7498/aps.61.110505.

[35]

Y. ShenS. YangH. Xing and G. Gao, Primary resonance of Duffing oscillator with fractional-order derivative, Commun. Nonl. Sci. Numer. Simul., 17 (2012), 3092-3100.  doi: 10.1016/j.cnsns.2011.11.024.

[36]

J. Sneyd and J. Sherratt, On the propagation of calcium waves in an inhomogeneous medium, SIAM J. Appl. Math., 57 (1997), 73-94.  doi: 10.1137/S0036139995286035.

[37]

P. D. Spanos and B. A. Zeldin, Random vibration of systems with frequency-dependent parameters or fractional derivatives, J. Eng. Mech., 123 (1997), 290. doi: 10.1061/(ASCE)0733-9399(1997)123:3(290).

[38]

C. B. Tabi, Dynamical analysis of the FitzHugh–Nagumo oscillations through a modified Van der Pol equation with fractional-order derivative term, Int. J. Nonl. Mech., 105 (2018), 173-178.  doi: 10.1016/j.ijnonlinmec.2018.05.026.

[39]

C. B. Tabi, Fractional unstable patterns of energy in $\alpha-$helix proteins with long-range interactions, Chaos Sol. Fract., 116 (2018), 386-391.  doi: 10.1016/j.chaos.2018.09.037.

[40]

D. Tatchim BemmoM. Siewe Siewe and C. Tchawoua, Nonlinear oscillations of the FitzHugh-Nagumo equations under combined external and two-frequency parametric excitations, Phys. Lett. A, 375 (2011), 1944-1953.  doi: 10.1016/j.physleta.2011.02.072.

[41]

D. Tatchim BemmoM. Siewe Siewe and C. Tchawoua, Combined effects of correlated bounded noises and weak periodic signal input in the modified FitzHugh-Nagumo neural model, Commun. Nonl. Sci. Numer. Simul., 18 (2013), 1275-1287.  doi: 10.1016/j.cnsns.2012.09.016.

[42]

H. Treutlein and K. Schulten, Noise-induced limit cycles of the Bonhoeffer-Van der Pol model of neural pulses, Phys. Chem., 89 (1985), 710-718.  doi: 10.1002/bbpc.19850890626.

[43]

J. C. Tsai and J. Sneyd, Traveling waves in the buffered FitzHugh-Nagumo model, SIAM J. Appl. Math., 71 (2011), 1606-1636.  doi: 10.1137/110820348.

[44]

K. Wiesenfeld, D. Pierson E. Pantazelou, C. Dames and F. Moss, Stochastic resonance on a circle, Phys. Rev. Lett., 72 (1994), 2125. doi: 10.1103/PhysRevLett.72.2125.

[45]

Y. YangW. XuX. Gu and Y. Sun, Stochastic response of a class of self-excited systems with Caputo-type fractional derivative driven by Gaussian white noise, Chaos Solit. Frac., 77 (2015), 190-204.  doi: 10.1016/j.chaos.2015.05.029.

[46]

Y. YangW. XuW. Jia and Q. Han, Stationary response of nonlinear system with Caputo-type fractional derivative damping under Gaussian white noise excitation, Nonl. Dyn., 79 (2015), 139-146.  doi: 10.1007/s11071-014-1651-3.

Figure 1.  The panel shows the stationary probability density function (PDF) of the amplitude for different values of the fractional-order parameter $ \alpha $. Solid lines correspond to analytical results, while symbol ($ \triangle $) corresponds to results from numerical calculations
Figure 2.  The panels show the joint PDF of the displacement $ X $ and velocity $ Y $, corresponding to Fig. 1, for different values of the fractional-order parameter $ \alpha $. Panels (aj)$ _{j = 1, 2, 3} $ correspond to our analytical calculations, while their corresponding panels (bj)$ _{j = 1, 2, 3} $ are obtained from numerical simulations: (a1)-(b1) $ \alpha = 0.8 $, (a2)-(b2) $ \alpha = 0.6 $ and (a3)-(b3) $ \alpha = 0.3 $
Figure 3.  The panel shows the stationary probability density function (PDF) of the amplitude for different values of the fractional-order parameter $ \alpha $. Solid lines correspond to analytical results, while symbol ($ \triangle $) corresponds to results from numerical calculations
Figure 4.  The panels show numerical results for the joint PDF displacement $ X $ and velocity $ Y $. The lines form top to bottom correspond to different values of $ D $, the Gaussian white noise intensity: (aj)$ _{j = 1, 2, 3} $ $ D = 0.1 $, (bj)$ _{j = 1, 2, 3} $ $ D = 0.05 $ and (cj)$ _{j = 1, 2, 3} $ $ D = 0.01 $. The columns from left to right respectively correspond to $ \alpha = 0.8 $, $ \alpha = 0.6 $ and $ \alpha = 0.3 $
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