TY  - JOUR
AU  - Bingshuo, Wang
AU  - Li, Wei
AU  - Zhao, Junfeng
AU  - Trišović, Nataša
PY  - 2024
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/7738
AB  - In this paper, we take the resting T cells into account and interpret the progression and
regression of tumors by a predator-prey like tumor-immune system. First, we construct an appropriate
Lyapunov function to prove the existence and uniqueness of the global positive solution to the system.
Then, by utilizing the stochastic comparison theorem, we prove the moment boundedness of tumor
cells and two types of T cells. Furthermore, we analyze the impact of stochastic perturbations on the
extinction and persistence of tumor cells and obtain the stationary probability density of the tumor cells
in the persistent state. The results indicate that when the noise intensity of tumor perturbation is low,
tumor cells remain in a persistent state. As this intensity gradually increases, the population of tumors
moves towards a lower level, and the stochastic bifurcation phenomena occurs. When it reaches a
certain threshold, instead the number of tumor cells eventually enter into an extinct state, and further
increasing of the noise intensity will accelerate this process.
PB  - AIMS Press Journals
T2  - Mathematical Biosciences and Engineering, Special Issue: Modelling and investigations of predator-prey dynamics
T1  - Longtime evolution and stationary response of a stochastic tumor-immune system with resting T cells
EP  - 2834
IS  - 2
SP  - 2813
VL  - 21
DO  - 10.3934/mbe.2024125
ER  - 

@article{
author = "Bingshuo, Wang and Li, Wei and Zhao, Junfeng and Trišović, Nataša",
year = "2024",
abstract = "In this paper, we take the resting T cells into account and interpret the progression and
regression of tumors by a predator-prey like tumor-immune system. First, we construct an appropriate
Lyapunov function to prove the existence and uniqueness of the global positive solution to the system.
Then, by utilizing the stochastic comparison theorem, we prove the moment boundedness of tumor
cells and two types of T cells. Furthermore, we analyze the impact of stochastic perturbations on the
extinction and persistence of tumor cells and obtain the stationary probability density of the tumor cells
in the persistent state. The results indicate that when the noise intensity of tumor perturbation is low,
tumor cells remain in a persistent state. As this intensity gradually increases, the population of tumors
moves towards a lower level, and the stochastic bifurcation phenomena occurs. When it reaches a
certain threshold, instead the number of tumor cells eventually enter into an extinct state, and further
increasing of the noise intensity will accelerate this process.",
publisher = "AIMS Press Journals",
journal = "Mathematical Biosciences and Engineering, Special Issue: Modelling and investigations of predator-prey dynamics",
title = "Longtime evolution and stationary response of a stochastic tumor-immune system with resting T cells",
pages = "2834-2813",
number = "2",
volume = "21",
doi = "10.3934/mbe.2024125"
}

Bingshuo, W., Li, W., Zhao, J.,& Trišović, N.. (2024). Longtime evolution and stationary response of a stochastic tumor-immune system with resting T cells. in Mathematical Biosciences and Engineering, Special Issue: Modelling and investigations of predator-prey dynamics
AIMS Press Journals., 21(2), 2813-2834.
https://doi.org/10.3934/mbe.2024125

Bingshuo W, Li W, Zhao J, Trišović N. Longtime evolution and stationary response of a stochastic tumor-immune system with resting T cells. in Mathematical Biosciences and Engineering, Special Issue: Modelling and investigations of predator-prey dynamics. 2024;21(2):2813-2834.
doi:10.3934/mbe.2024125 .

Bingshuo, Wang, Li, Wei, Zhao, Junfeng, Trišović, Nataša, "Longtime evolution and stationary response of a stochastic tumor-immune system with resting T cells" in Mathematical Biosciences and Engineering, Special Issue: Modelling and investigations of predator-prey dynamics, 21, no. 2 (2024):2813-2834,
https://doi.org/10.3934/mbe.2024125 . .

TY  - JOUR
AU  - Li, Wei
AU  - Huang, Dongmei
AU  - Zhang, Meiting
AU  - Trišović, Nataša
AU  - Zhao, Junfeng
PY  - 2019
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/3145
AB  - Fractional-order PID (FOPID) controller, as the results of recent development of fractional calculus, is becoming wide-used in many deterministic dynamical systems, but not in stochastic dynamical systems. This paper explores stochastic bifurcation of a generalized Van del Pol (VDP) system under the control of FOPID controller. Firstly, introducing the transformation between fast-varying and slow-varying variables of the system response, and utilizing the properties of fractional calculus, we obtain a new expression in the form of slow-varying variables for FOPID controller. Based on this work, the stochastic averaging method is applied to obtain the Fokker-Planck-Kolmogorov (FPK) equation and the stationary probability density function (PDF) of the amplitude response. Then a new numerical algorithm is proposed to testify the analytical results in the case of the coexistence of fractional integral and fractional derivative. After that, stochastic bifurcations induced by the order of the fractional integral, the order of the fractional derivative and the coefficient in FOPID controller are investigated in detail. The agreement between analytical and numerical results verifies the correctness and effectiveness of our proposed methods.
PB  - Pergamon-Elsevier Science Ltd, Oxford
T2  - Chaos Solitons & Fractals
T1  - Bifurcation control of a generalized VDP system driven by color-noise excitation via FOPID controller
EP  - 38
SP  - 30
VL  - 121
DO  - 10.1016/j.chaos.2019.01.026
ER  - 

@article{
author = "Li, Wei and Huang, Dongmei and Zhang, Meiting and Trišović, Nataša and Zhao, Junfeng",
year = "2019",
abstract = "Fractional-order PID (FOPID) controller, as the results of recent development of fractional calculus, is becoming wide-used in many deterministic dynamical systems, but not in stochastic dynamical systems. This paper explores stochastic bifurcation of a generalized Van del Pol (VDP) system under the control of FOPID controller. Firstly, introducing the transformation between fast-varying and slow-varying variables of the system response, and utilizing the properties of fractional calculus, we obtain a new expression in the form of slow-varying variables for FOPID controller. Based on this work, the stochastic averaging method is applied to obtain the Fokker-Planck-Kolmogorov (FPK) equation and the stationary probability density function (PDF) of the amplitude response. Then a new numerical algorithm is proposed to testify the analytical results in the case of the coexistence of fractional integral and fractional derivative. After that, stochastic bifurcations induced by the order of the fractional integral, the order of the fractional derivative and the coefficient in FOPID controller are investigated in detail. The agreement between analytical and numerical results verifies the correctness and effectiveness of our proposed methods.",
publisher = "Pergamon-Elsevier Science Ltd, Oxford",
journal = "Chaos Solitons & Fractals",
title = "Bifurcation control of a generalized VDP system driven by color-noise excitation via FOPID controller",
pages = "38-30",
volume = "121",
doi = "10.1016/j.chaos.2019.01.026"
}

Li, W., Huang, D., Zhang, M., Trišović, N.,& Zhao, J.. (2019). Bifurcation control of a generalized VDP system driven by color-noise excitation via FOPID controller. in Chaos Solitons & Fractals
Pergamon-Elsevier Science Ltd, Oxford., 121, 30-38.
https://doi.org/10.1016/j.chaos.2019.01.026

Li W, Huang D, Zhang M, Trišović N, Zhao J. Bifurcation control of a generalized VDP system driven by color-noise excitation via FOPID controller. in Chaos Solitons & Fractals. 2019;121:30-38.
doi:10.1016/j.chaos.2019.01.026 .

Li, Wei, Huang, Dongmei, Zhang, Meiting, Trišović, Nataša, Zhao, Junfeng, "Bifurcation control of a generalized VDP system driven by color-noise excitation via FOPID controller" in Chaos Solitons & Fractals, 121 (2019):30-38,
https://doi.org/10.1016/j.chaos.2019.01.026 . .

TY  - JOUR
AU  - Li, Wei
AU  - Chen, Lincong
AU  - Zhao, Junfeng
AU  - Trišović, Nataša
PY  - 2018
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/2768
AB  - In this paper, the reliability of stochastic dynamical systems under Gaussian white noise excitations with fractional order proportional-inegral-derivative (FOPID) controller is estimated. First, the FOPID controller is approximated by a set of combination of displacement and velocity based on the generalized van der Pol transformation. Then, the stochastic averaging method of energy envelope is applied to obtain a diffusive differential equation, from which the Backward Kolmogorov equation, governing the conditional reliability function, and the Generalized Pontryagin equation, governing the statistical moments of first-passage time, are derived from the averaged equation and solved numerically. Finally, in the two examples, the critical parameters in the FOPID controller are shown to be capable of improving the reliability of the stochastic dynamical system apparently, and all numerical results are verified to be efficient and correct by the Monte Carlo simulation.
PB  - World Scientific Publ Co Pte Ltd, Singapore
T2  - International Journal of Structural Stability and Dynamics
T1  - Reliability Estimation of Stochastic Dynamical Systems with Fractional Order PID Controller
IS  - 6
VL  - 18
DO  - 10.1142/S0219455418500839
ER  - 

@article{
author = "Li, Wei and Chen, Lincong and Zhao, Junfeng and Trišović, Nataša",
year = "2018",
abstract = "In this paper, the reliability of stochastic dynamical systems under Gaussian white noise excitations with fractional order proportional-inegral-derivative (FOPID) controller is estimated. First, the FOPID controller is approximated by a set of combination of displacement and velocity based on the generalized van der Pol transformation. Then, the stochastic averaging method of energy envelope is applied to obtain a diffusive differential equation, from which the Backward Kolmogorov equation, governing the conditional reliability function, and the Generalized Pontryagin equation, governing the statistical moments of first-passage time, are derived from the averaged equation and solved numerically. Finally, in the two examples, the critical parameters in the FOPID controller are shown to be capable of improving the reliability of the stochastic dynamical system apparently, and all numerical results are verified to be efficient and correct by the Monte Carlo simulation.",
publisher = "World Scientific Publ Co Pte Ltd, Singapore",
journal = "International Journal of Structural Stability and Dynamics",
title = "Reliability Estimation of Stochastic Dynamical Systems with Fractional Order PID Controller",
number = "6",
volume = "18",
doi = "10.1142/S0219455418500839"
}

Li, W., Chen, L., Zhao, J.,& Trišović, N.. (2018). Reliability Estimation of Stochastic Dynamical Systems with Fractional Order PID Controller. in International Journal of Structural Stability and Dynamics
World Scientific Publ Co Pte Ltd, Singapore., 18(6).
https://doi.org/10.1142/S0219455418500839

Li W, Chen L, Zhao J, Trišović N. Reliability Estimation of Stochastic Dynamical Systems with Fractional Order PID Controller. in International Journal of Structural Stability and Dynamics. 2018;18(6).
doi:10.1142/S0219455418500839 .

Li, Wei, Chen, Lincong, Zhao, Junfeng, Trišović, Nataša, "Reliability Estimation of Stochastic Dynamical Systems with Fractional Order PID Controller" in International Journal of Structural Stability and Dynamics, 18, no. 6 (2018),
https://doi.org/10.1142/S0219455418500839 . .

TY  - JOUR
AU  - Li, Wei
AU  - Chen, Lincong
AU  - Trišović, Nataša
AU  - Cvetković, Aleksandar
AU  - Zhao, Junfeng
PY  - 2015
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/3932
AB  - In this paper, the first-passage failure of stochastic dynamical systems with fractional derivative and power-form restoring force subjected to Gaussian white-noise excitation is investigated. With application of the stochastic averaging method of quasi-Hamiltonian system, the system energy process will converge wealdy to an Ito differential equation. After that, Backward Kolmogorov (BK) equation associated with conditional reliability function and Generalized Pontryagin (GP) equation associated with statistical moments of first-passage time are constructed and solved. Particularly, the influence on reliability caused by the order of fractional derivative and the power of restoring force are also examined, respectively. Numerical results show that reliability function is decreased with respect to the time. Lower power of restoring force may lead the system to more unstable evolution and lead first passage easy to happen. Besides, more viscous material described by fractional derivative may have higher reliability. Moreover, the analytical results are all in good agreement with Monte-Carlo data.
PB  - Pergamon-Elsevier Science Ltd, Oxford
T2  - International Journal of Non-Linear Mechanics
T1  - First passage of stochastic fractional derivative systems with power-form restoring force
EP  - 88
SP  - 83
VL  - 71
DO  - 10.1016/j.ijnonlinmec.2015.02.002
ER  - 

@article{
author = "Li, Wei and Chen, Lincong and Trišović, Nataša and Cvetković, Aleksandar and Zhao, Junfeng",
year = "2015",
abstract = "In this paper, the first-passage failure of stochastic dynamical systems with fractional derivative and power-form restoring force subjected to Gaussian white-noise excitation is investigated. With application of the stochastic averaging method of quasi-Hamiltonian system, the system energy process will converge wealdy to an Ito differential equation. After that, Backward Kolmogorov (BK) equation associated with conditional reliability function and Generalized Pontryagin (GP) equation associated with statistical moments of first-passage time are constructed and solved. Particularly, the influence on reliability caused by the order of fractional derivative and the power of restoring force are also examined, respectively. Numerical results show that reliability function is decreased with respect to the time. Lower power of restoring force may lead the system to more unstable evolution and lead first passage easy to happen. Besides, more viscous material described by fractional derivative may have higher reliability. Moreover, the analytical results are all in good agreement with Monte-Carlo data.",
publisher = "Pergamon-Elsevier Science Ltd, Oxford",
journal = "International Journal of Non-Linear Mechanics",
title = "First passage of stochastic fractional derivative systems with power-form restoring force",
pages = "88-83",
volume = "71",
doi = "10.1016/j.ijnonlinmec.2015.02.002"
}

Li, W., Chen, L., Trišović, N., Cvetković, A.,& Zhao, J.. (2015). First passage of stochastic fractional derivative systems with power-form restoring force. in International Journal of Non-Linear Mechanics
Pergamon-Elsevier Science Ltd, Oxford., 71, 83-88.
https://doi.org/10.1016/j.ijnonlinmec.2015.02.002

Li W, Chen L, Trišović N, Cvetković A, Zhao J. First passage of stochastic fractional derivative systems with power-form restoring force. in International Journal of Non-Linear Mechanics. 2015;71:83-88.
doi:10.1016/j.ijnonlinmec.2015.02.002 .

Li, Wei, Chen, Lincong, Trišović, Nataša, Cvetković, Aleksandar, Zhao, Junfeng, "First passage of stochastic fractional derivative systems with power-form restoring force" in International Journal of Non-Linear Mechanics, 71 (2015):83-88,
https://doi.org/10.1016/j.ijnonlinmec.2015.02.002 . .

TY  - JOUR
AU  - Li, Wei
AU  - Chen, Lincong
AU  - Trišović, Nataša
AU  - Cvetković, Aleksandar
AU  - Zhao, Junfeng
PY  - 2015
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/2197
AB  - In this paper, the first-passage failure of stochastic dynamical systems with fractional derivative and power-form restoring force subjected to Gaussian white-noise excitation is investigated. With application of the stochastic averaging method of quasi-Hamiltonian system, the system energy process will converge wealdy to an Ito differential equation. After that, Backward Kolmogorov (BK) equation associated with conditional reliability function and Generalized Pontryagin (GP) equation associated with statistical moments of first-passage time are constructed and solved. Particularly, the influence on reliability caused by the order of fractional derivative and the power of restoring force are also examined, respectively. Numerical results show that reliability function is decreased with respect to the time. Lower power of restoring force may lead the system to more unstable evolution and lead first passage easy to happen. Besides, more viscous material described by fractional derivative may have higher reliability. Moreover, the analytical results are all in good agreement with Monte-Carlo data.
PB  - Pergamon-Elsevier Science Ltd, Oxford
T2  - International Journal of Non-Linear Mechanics
T1  - First passage of stochastic fractional derivative systems with power-form restoring force
EP  - 88
SP  - 83
VL  - 71
DO  - 10.1016/j.ijnonlinmec.2015.02.002
ER  - 

@article{
author = "Li, Wei and Chen, Lincong and Trišović, Nataša and Cvetković, Aleksandar and Zhao, Junfeng",
year = "2015",
abstract = "In this paper, the first-passage failure of stochastic dynamical systems with fractional derivative and power-form restoring force subjected to Gaussian white-noise excitation is investigated. With application of the stochastic averaging method of quasi-Hamiltonian system, the system energy process will converge wealdy to an Ito differential equation. After that, Backward Kolmogorov (BK) equation associated with conditional reliability function and Generalized Pontryagin (GP) equation associated with statistical moments of first-passage time are constructed and solved. Particularly, the influence on reliability caused by the order of fractional derivative and the power of restoring force are also examined, respectively. Numerical results show that reliability function is decreased with respect to the time. Lower power of restoring force may lead the system to more unstable evolution and lead first passage easy to happen. Besides, more viscous material described by fractional derivative may have higher reliability. Moreover, the analytical results are all in good agreement with Monte-Carlo data.",
publisher = "Pergamon-Elsevier Science Ltd, Oxford",
journal = "International Journal of Non-Linear Mechanics",
title = "First passage of stochastic fractional derivative systems with power-form restoring force",
pages = "88-83",
volume = "71",
doi = "10.1016/j.ijnonlinmec.2015.02.002"
}

Li, W., Chen, L., Trišović, N., Cvetković, A.,& Zhao, J.. (2015). First passage of stochastic fractional derivative systems with power-form restoring force. in International Journal of Non-Linear Mechanics
Pergamon-Elsevier Science Ltd, Oxford., 71, 83-88.
https://doi.org/10.1016/j.ijnonlinmec.2015.02.002

Li W, Chen L, Trišović N, Cvetković A, Zhao J. First passage of stochastic fractional derivative systems with power-form restoring force. in International Journal of Non-Linear Mechanics. 2015;71:83-88.
doi:10.1016/j.ijnonlinmec.2015.02.002 .

Li, Wei, Chen, Lincong, Trišović, Nataša, Cvetković, Aleksandar, Zhao, Junfeng, "First passage of stochastic fractional derivative systems with power-form restoring force" in International Journal of Non-Linear Mechanics, 71 (2015):83-88,
https://doi.org/10.1016/j.ijnonlinmec.2015.02.002 . .

TY  - CONF
AU  - Li, Wei
AU  - Zhao, Junfeng
AU  - Trišović, Nataša
AU  - Zhang, Y.
PY  - 2014
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/1981
AB  - In this paper, the Galerkin method is presented to estimate the approximate stationary and non-stationary responses of nonlinear random dynamical systems with fractional derivative damping. Based on equivalent linearization method, the equation of motion is reverted to a linear equation in terms of envelope and phase process at first. After that, stochastic averaging method is used to get an averaged Ito differential equation with fractional derivative approximated by a periodic function. Finally, the approximate non-stationary probability density function solution of the associated Fokker-Planck equation is obtained by applying the Galerkin method. The application on a Rayleigh oscillator shows that the methods we utilized in this paper are efficient and reliable by comparing the stationary solution between the exact and approximated one.
PB  - American Society of Civil Engineers (ASCE)
C3  - Vulnerability, Uncertainty, and Risk: Quantification, Mitigation, and Management - Proceedings of th
T1  - Solutions to Stochastic Dynamical Systems with Fractional Derivative Damping
EP  - 1963
SP  - 1949
DO  - 10.1061/9780784413609.195
ER  - 

@conference{
author = "Li, Wei and Zhao, Junfeng and Trišović, Nataša and Zhang, Y.",
year = "2014",
abstract = "In this paper, the Galerkin method is presented to estimate the approximate stationary and non-stationary responses of nonlinear random dynamical systems with fractional derivative damping. Based on equivalent linearization method, the equation of motion is reverted to a linear equation in terms of envelope and phase process at first. After that, stochastic averaging method is used to get an averaged Ito differential equation with fractional derivative approximated by a periodic function. Finally, the approximate non-stationary probability density function solution of the associated Fokker-Planck equation is obtained by applying the Galerkin method. The application on a Rayleigh oscillator shows that the methods we utilized in this paper are efficient and reliable by comparing the stationary solution between the exact and approximated one.",
publisher = "American Society of Civil Engineers (ASCE)",
journal = "Vulnerability, Uncertainty, and Risk: Quantification, Mitigation, and Management - Proceedings of th",
title = "Solutions to Stochastic Dynamical Systems with Fractional Derivative Damping",
pages = "1963-1949",
doi = "10.1061/9780784413609.195"
}

Li, W., Zhao, J., Trišović, N.,& Zhang, Y.. (2014). Solutions to Stochastic Dynamical Systems with Fractional Derivative Damping. in Vulnerability, Uncertainty, and Risk: Quantification, Mitigation, and Management - Proceedings of th
American Society of Civil Engineers (ASCE)., 1949-1963.
https://doi.org/10.1061/9780784413609.195

Li W, Zhao J, Trišović N, Zhang Y. Solutions to Stochastic Dynamical Systems with Fractional Derivative Damping. in Vulnerability, Uncertainty, and Risk: Quantification, Mitigation, and Management - Proceedings of th. 2014;:1949-1963.
doi:10.1061/9780784413609.195 .

Li, Wei, Zhao, Junfeng, Trišović, Nataša, Zhang, Y., "Solutions to Stochastic Dynamical Systems with Fractional Derivative Damping" in Vulnerability, Uncertainty, and Risk: Quantification, Mitigation, and Management - Proceedings of th (2014):1949-1963,
https://doi.org/10.1061/9780784413609.195 . .