Lyapunov Functionals and Stability of Stochastic Functional Differential Equations pp 113-152 | Cite as

# Stability of Systems with Nonlinearities

Chapter

## Abstract

In Chap. 5 stability of the zero solution and positive equilibrium points for nonlinear systems is studied. In particular, differential equations with nonlinearities in deterministic and stochastic parts are considered, differential equation with fractional nonlinearity. It is shown that investigation of stability in probability for nonlinear systems with the level of nonlinearity higher than one can be reduced to investigation of asymptotic mean square stability of the linear part of the considered nonlinear system. The obtained results are illustrated by 18 figures.

## Keywords

Equilibrium Point Trivial Solution Nonlinear Differential Equation Initial Function Stochastic Perturbation
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## References

- 19.Bandyopadhyay M, Chattopadhyay J (2005) Ratio dependent predator–prey model: effect of environmental fluctuation and stability. Nonlinearity 18:913–936 MathSciNetMATHCrossRefGoogle Scholar
- 27.Beretta E, Kolmanovskii V, Shaikhet L (1998) Stability of epidemic model with time delays influenced by stochastic perturbations. Math Comput Simul 45(3–4):269–277 (Special Issue “Delay Systems”) MathSciNetMATHCrossRefGoogle Scholar
- 38.Bradul N, Shaikhet L (2007) Stability of the positive point of equilibrium of Nicholson’s blowflies equation with stochastic perturbations: numerical analysis. Discrete Dyn Nat Soc 2007:92959. doi: 10.1155/2007/92959. 25 pages MathSciNetCrossRefGoogle Scholar
- 39.Bradul N, Shaikhet L (2009) Stability of difference analogue of mathematical predator–prey model by stochastic perturbations. Vest Odesskogo Naz Univ Mat Mekh 14(20):7–23 (in Russian) Google Scholar
- 49.Carletti M (2002) On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment. Math Biosci 175:117–131 MathSciNetMATHCrossRefGoogle Scholar
- 95.Grove EA, Ladas G, McGrath LC, Teixeira CT (2001) Existence and behavior of solutions of a rational system. Commun Appl Nonlinear Anal 8(1):1–25 MathSciNetMATHGoogle Scholar
- 99.Gurney WSC, Blythe SP, Nisbet RM (1980) Nicholson’s blowflies revisited. Nature 287:17–21 CrossRefGoogle Scholar
- 101.Gyori I, Trofimchuk S (1999) Global attractivity in \(\dot{x}(t)=-\delta x(t)+pf(x(t-\tau))\). Dyn Syst Appl 8:197–210 MathSciNetGoogle Scholar
- 102.Gyori I, Trofimchuck SI (2002) On the existence of rapidly oscillatory solutions in the Nicholson’s blowflies equation. Nonlinear Anal 48:1033–1042 MathSciNetCrossRefGoogle Scholar
- 117.Jovanovic M, Krstic M (2012) Stochastically perturbed vector-borne disease models with direct transmission. Appl Math Model 36:5214–5228 MathSciNetMATHCrossRefGoogle Scholar
- 162.Kulenovic MRS, Nurkanovic M (2002) Asymptotic behavior of a two dimensional linear fractional system of difference equations. Rad Mat 11(1):59–78 MathSciNetMATHGoogle Scholar
- 163.Kulenovic MRS, Nurkanovic M (2005) Asymptotic behavior of a system of linear fractional difference equations. Arch Inequal Appl 2005(2):127–143 MathSciNetMATHGoogle Scholar
- 164.Kulenovic MRS, Nurkanovic M (2006) Asymptotic behavior of a competitive system of linear fractional difference equations. Adv Differ Equ 2006(5):19756. doi: 10.1155/ADE/2006/19756. 13 pages MathSciNetGoogle Scholar
- 183.Liz E, Röst G (2009) On global attractor of delay differential equations with unimodal feedback. Discrete Contin Dyn Syst 24(4):1215–1224 MathSciNetMATHCrossRefGoogle Scholar
- 190.Mackey MC, Glass L (1997) Oscillation and chaos in physiological control system. Science 197:287–289 CrossRefGoogle Scholar
- 214.Mukhopadhyay B, Bhattacharyya R (2009) A nonlinear mathematical model of virus-tumor-immune system interaction: deterministic and stochastic analysis. Stoch Anal Appl 27:409–429 MathSciNetMATHCrossRefGoogle Scholar
- 224.Nicholson AJ (1954) An outline of the dynamics of animal populations. Aust J Zool 2:9–65 CrossRefGoogle Scholar
- 232.Paternoster B, Shaikhet L (2008) Stability of equilibrium points of fractional difference equations with stochastic perturbations. Adv Differ Equ 2008:718408. doi: 10.1155/2008/718408, 21 pages MathSciNetCrossRefGoogle Scholar
- 248.Röst G, Wu J (2007) Domain decomposition method for the global dynamics of delay differential equations with unimodal feedback. Proc R Soc Lond Ser A, Math Phys Sci 463:2655–2669 MATHCrossRefGoogle Scholar
- 257.Sarkar RR, Banerjee S (2005) Cancer self remission and tumor stability—a stochastic approach. Math Biosci 196:65–81 MathSciNetMATHCrossRefGoogle Scholar
- 261.Shaikhet L (1975) Stability investigation of stochastic systems with delay by Lyapunov functionals method. Probl Pereda Inf 11(4):70–76 (in Russian) Google Scholar
- 267.Shaikhet L (1998) Stability of predator–prey model with aftereffect by stochastic perturbations. Stab Control: Theory Appl 1(1):3–13 MathSciNetGoogle Scholar
- 278.Shaikhet L (2011) Lyapunov functionals and stability of stochastic difference equations. Springer, London MATHCrossRefGoogle Scholar

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