Abstract
In this paper, we investigate the periodic dynamical behaviors for a class of general Cohen–Grossberg neural networks with discontinuous right-hand sides and mixed time delays involving both time-varying delays and distributed delays. In view of functional differential inclusions theory, we obtain the existence of global solutions. By means of functional differential inclusions theory and fixed-point theorem of multi-valued maps, the existence of one and multiple positive periodic solutions for the neural networks is given. It is worthy to point out that, without assuming the boundedness or under linear growth condition of the discontinuous neuron activation functions, our results on the existence of one and multiple positive periodic solutions will also be valid. We derive some sufficient conditions for the global exponential stability and convergence of the discontinuous neural networks, in terms of non-smooth analysis theory with generalized Lyapunov approach. Finally, we give some numerical examples to show the applicability and effectiveness of our main results.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Forti, M., Nistri, P.: Global convergence of neural networks with discontinuous neuron activations. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 50, 1421–1435 (2003)
Forti, M., Nistri, P., Papini, D.: Global exponential stability and global convergence in finite time of delayed neural networks with infinite gain. IEEE Trans. Neural Netw. 16, 1449–1463 (2005)
Cortés, J.: Discontinuous dynamical systems. IEEE Control Syst. Mag. 28, 36–73 (2008)
Liu, X., Chen, T., Cao, J., Lu, W.: Dissipativity and quasi-synchronization for neural networks with discontinuous activations and parameter mismatches. Neural Netw. 24, 1013–1021 (2011)
Liu, X., Cao, J.: Robust state estimations for neural networks with discontinuous activations. IEEE Trans. Syst., Man, Cybern. Part B 40, 1425–1437 (2010)
Wang, Y., Zuo, Y., Huang, L., Li, C.: Global robust stability of delayed neural networks with discontinuous activation functions. IET Control Theory Appl. 2, 543–553 (2008)
Lu, W., Chen, T.: Dynamical behaviors of delayed neural networks systems with discontinuous activation functions. Neural Comput. 18, 683–708 (2006)
Hopfield, J., Tank, D.: Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Nat. Acad. Sci. 79, 3088–3092 (1984)
Papini, D., Taddei, V.: Global exponential stability of the periodic solution of a delayed neural networks with discontinuous activations. Phys. Lett. A 343, 117–128 (2005)
Liu, X., Cao, J.: On periodic solutions of neural networks via differential inclusions. Neural Netw. 22, 329–334 (2009)
Cai, Z., Huang, L.: Existence and global asymptotic stability of periodic solution for discrete and distributed time-varying delayed neural networks with discontinuous activations. Neurocomputing 74, 3170–3179 (2011)
Huang, L., Guo, Z.: Global convergence of periodic solution of neural networks with discontinuous activation functions. Chaos, Solitons and Fractals 42, 2351–2356 (2009)
Huang, L., Wang, J., Zhou, X.: Existence and global asymptotic stability of periodic solutions for Hopfield neural networks with discontinuous activations. Nonlinear Anal. RWA 10, 1651–1661 (2009)
Xiao, J., Zeng, Z., Shen, W.: Global asymptotic stability of delayed neural networks with discontinuous neuron activations. Neurocomputing 118, 322–328 (2013)
Cai, Z., Huang, L., Guo, Z., Chen, X.: On the periodic dynamics of a class of time-varying delayed neural networks via differential inclusions. Neural Netw. 33, 97–113 (2012)
Wu, H.: Stability analysis for periodic solution of neural networks with discontinuous neuron activations. Nonlinear Anal. RWA 10, 1717–1729 (2009)
Wang, J., Huang, L., Guo, Z.: Global asymptotic stability of neural networks with discontinuous activations. Neural Netw. 22, 931–937 (2009)
Li, Y., Wu, H.: Global stability analysis for periodic solution in discontinuous neural networks with nonlinear growth activations. Adv. Differ. Equ. 2009, 1–4 (2009). doi:10.1155/2009/798685
Gonzalez, S.: Neural Networks for Macroeconomic Forecasting: A Complementary Approach to Linear Regression Models. Department of Finance, Canada (2000)
Cheng, C., Lin, K., Shi, C.: Multistability an convergence in delayed neural networks. Phys. D 225, 61–64 (2007)
Yi, Z., Tan, K., Lee, T.: Multistability analysis for recurrent neural networks with unsaturating piecewise linear transfer functions. Neural Comput. 15, 639–662 (2003)
Lu, W., Wang, L., Chen, T.: On attracting basins of multiple equilibria of a class of cellular neural networks. IEEE Trans. Neural Netw. 22, 381–394 (2011)
Zeng, Z., Wang, J.: Multiperiodicity and exponential attractivity evoked by periodic external inputs in delayed cellular neural networks. Neural Comput. 18, 848–870 (2006)
Cao, J., Feng, G., Wang, Y.: Multistability and multiperiodicity of delayed Cohen-Grossberg Neural networks with a general class of activation functions. Phys. D 237, 1734–1749 (2008)
Zeng, Z., Huang, T., Zheng, W.: Multistability of recurrent neural networks with time-varying delays and the piecewise linear activation function. IEEE Trans. Neural Netw. 21, 1371–1377 (2010)
Macro, M., Forti, M., Grazzini, M., Pancioni, L.: Limit set dichotomy and multistability for a class of cooperative neural networks with delays. IEEE Trans. Neural Netw. Learn. Syst. 23, 1473–1485 (2012)
Zhou, T., Wang, M., Long, M.: Existence and exponential stability of multiple periodic solutions for a multidirectional associative memory neural network. Neural Process Lett. 35, 187–202 (2012)
Nie, X., Huang, Z.: Multistability and multiperiodicity of high-order competitive neural networks with a general class of activation functions. Neurocomputing 82, 1–13 (2012)
Huang, Y., Zhang, H., Wang, Z.: Multistability and multiperiodicity of delayed bidirectional associative memory networks with discontinuous activation functions. Appl. Math. Comput. 219, 899–910 (2012)
Huang, Y., Zhang, H., Wang, Z.: Dynamical stability analysis of multiple equilibrium points in time-varying delayed recurrent neural networks with discontinuous activation functions. Neurocomputing 91, 21–28 (2012)
Hou, C., Qian, J.: Stability analysis for neural dynamics with time-varying delays. IEEE Trans. Neural Netw. 9, 221–223 (2008)
Huang, H., Ho, D., Lam, J.: Stochastic stability analysis of fuzzy Hopfield neural networks with time-varying delays. IEEE Trans. Circuits Syst. II(52), 251–255 (2005)
Gopalsamy, K., He, X.: Delay-independent stability in bidirectional associative memory network. IEEE Trans. Neural Netw. 5, 998–1002 (1994)
Zheng, C., Gong, C., Wang, Z.: Stability criteria for Cohen-Grossberg neural networks with mixed delays and inverse Lipschitz neuron activations. J. Frankl. Inst. 349, 2903–2924 (2012)
Cohen, M., Grossberg, S.: Absolute stability and global pattern formation and parallel memory storage by competitive neural networks. IEEE Trans. Syst. Man Cybern. 13, 815–826 (1983)
Balasubramaniam, P., Vembarasan, V.: Robust stability of uncertain fuzzy BAM neural networks of neutral-type with Markovian jumping parameters and impulses. Comput. Math. Appl. 62, 1838–1861 (2011)
Balasubramaniam, P., Vembarasan, V.: Asymptotic stability of BAM neural networks of neutral-type with impulsive effects and time delay in the leakage term. Int. J. Comput. Math. 88, 3271–3291 (2011)
Sathy, R., Balasubramaniam, P.: Stability analysis of fuzzy Markovian jumping Cohen–Grossberg BAM neural networks with mixed time-varying delays. Commun. Nonlinear Sci. Numer. Simul. 16, 2054–2064 (2011)
Lu, W., Chen, T.: Dynamical behaviors of Cohen–Grossberg neural networks with discontinuous activation functions. Neural Netw. 18, 231–242 (2005)
Meng, Y., Huang, L., Guo, Z., Hu, Q.: Stability analysis of Cohen–Grossberg neural networks with discontinuous neuron activations. Appl. Math. Model. 34, 358–365 (2010)
He, X., Lu, W., Chen, T.: Nonnegative periodic dynamics of delayed Cohen–Grossberg neural networks with discontinuous activations. Neurocomputing 73, 2765–2772 (2010)
Wang, D., Huang, L., Cai, Z.: On the periodic dynamics of a general Cohen–Grossberg BAM neural networks via differential inclusions. Neurocomputing 118, 203–214 (2013)
Wang, D., Huang, L.: Periodicity and global exponential stability of generalized Cohen–Grossberg neural networks with discontinuous activations and mixed delays. Neural Netw. 51, 80–95 (2014)
Chen, X., Song, Q.: Global exponential stability of the periodic solution of delayed Cohen–Grossberg neural networks with discontinuous activations. Neurocomputing 73, 3097–3104 (2010)
Wang, D., Huang, L.: Almost periodic dynamical behaviors for generalized Cohen–Grossberg neural networks with discontinuous activations via differential inclusions. Commun. Nonlinear Sci. Numer. Simul. 19, 3857–3879 (2014)
Filippov, A.: Differential Equations with Discontinuous Right-hand Side. Mathematics and Its Applications (Soviet Series). Kluwer Academic, Boston (1988)
Huang, L., Guo, Z., Wang, J.: Theory and Applications of Differential Equations with Discontinuous Right-Hand Sides. Science Press, Beijing (2011). (In Chinese)
Aubin, J., Cellina, A.: Differential Inclusions. Springer, Berlin (1984)
Aubin, J., Frankowska, H.: Set-Valued Analysis. Birkhauser, Boston (1990)
Agarwal, R., O’Regan, D.: A note on the existence of multiple fixed points for multivalued maps with applications. J. Differ. Equ. 160, 389–403 (2000)
Hong, S.: Multiple positive solutions for a class of integral inclusions. J. Comput. Appl. Math. 214, 19–29 (2008)
Benchohra, M., Hamani, S., Henderson, J.: Functional differential inclusions with integral boundary conditions. Electron. J. Qual. Theory Differ. Equ. 15, 1–13 (2007)
Zecca, P., Zezza, P.: Nonlinear boundary value problems in Banach spaces for multivalued differential equations in noncompact intervals. Nonlinear Anal. 3, 347–352 (1979)
Balasubramaniam, P., Ntouyas, S.: Controllability for neutral stochastic functional differential inclusions with infinite delay in abstract space. J. Math. Anal. Appl. 324, 161–176 (2006)
Xue, X., Yu, J.: Periodic solutions for semi-linear evolution inclusions. J. Math. Anal. Appl. 331, 1246–1262 (2007)
Clarke, F.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was jointly supported by the National Natural Science Foundation of China (11371127, 11401228, 11501221, 61573004), the Natural Science Foundation of Fujian Province (2015J01584), the Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University (ZQN-PY201, ZQN-YX301).
Appendix
Appendix
1.1 Appendix A. Proof of Proposition 2.1
Proof
It is easy to see that the multi-valued map
where
is upper semi-continuous with non-empty compact convex values, the local existence of a solution \(x(t)=(x_1(t),x_2(t),\ldots ,x_n(t))^T\) of (2.2) can be guaranteed [46, 47]. That is, the IVP of system (2.2) has at least a solution \(x(t)=(x_1(t),x_2(t),\ldots ,x_n(t))^T\) on \([0,\mathcal {T})\) for some \(\mathcal {T}\in (0,+\infty ]\) and the derivative of x(t) is a measurable selection from \(Q(x(t))\mathcal {F}(t,f)\). For a real matrix \(\Lambda =(\lambda _{ij})_{m\times n}\), denote \(||\Lambda ||_{\mathcal {M}}=\max _{1\leqslant i\leqslant n}\sum _{j=1}^m|\lambda _{ij}|\). From the Continuation Theorem (Theorem 2, P78, [46]), we have that either \(\mathcal {T}=+\infty \), or \(\mathcal {T}<+\infty \) and \(\lim _{t\rightarrow T^{-}}||x(t)||_{\mathcal {M}}=+\infty \). In the following, we will prove that \(\lim _{t\rightarrow T^{-}}||x(t)||_{\mathcal {M}}<+\infty \) if \(\mathcal {T}<+\infty \), which means that the maximal existing interval of x(t) can be extended to \(+\infty \).
First, we’ll show that x(t) is uniformly bounded about \(t\in [0,\mathcal {T})\).
Let \(r_0=||x(0)||_{\mathcal {M}}\), \(r_s=\max _{0\leqslant t\leqslant s}||x(t)||_{\mathcal {M}}\). And the interval \([r_0, r_s]\) can be divided as follows
Denote \(t_l^*=\min \{t\in [0,s]\mid ||x(t)||_{\mathcal {M}}=r_l\}, l=0,1,2,\ldots ,m;\) and \(t_l^{**}=\max \{t\in [0,t_{l+1}^*]\mid ||x(t)||_{\mathcal {M}}=r_l\}, l=0,1,2,\ldots ,m-1.\) Note that \(||x(t)||_{\mathcal {M}}\) is a continuous function, we have
and
Thus, we have
Since x(t) is a solution of the differential inclusions (2.2) with the initial condition \([\phi ,\psi ]\), we can obtain
where
and \(q^M=\max _{1\leqslant i\leqslant n}\{q_i^M\}\), \(||\psi ||_{\mathcal {M},\infty }=\max _{1{\leqslant } i{\leqslant } n}\{||\psi _i||_{\infty }\}{=}\max _{1{\leqslant } i{\leqslant } n}\{\hbox {ess}\sup _{s\in ({-}\infty ,0]}|\psi _i(s)|\}\).
It follows from (7.1) and (7.2) that
where \(||x(\eta _l)||_{\mathcal {M}}+W(||x(\eta _l)||_{\mathcal {M}})=\max _{t\in [t_l^{**},t_{l+1}^*]}\{||x(t)||_{\mathcal {M}}+W(||x(t)||_{\mathcal {M}})\}\). From (7.3), we have
Summing on both sides of (7.4) from 0 to \(m-1\) with respect to l, we can derive
Since the division of segment \([r_0,r_s]\) is arbitrary, and from the definition of integration, we have
which, together with the condition (H3) imply that \(r_s\) is uniformly bounded about \(s\in [0,\mathcal {T})\). Hence, x(t) is uniformly bounded about \(t\in [0,\mathcal {T})\).
Next, we will show that \(\lim _{t\rightarrow T^-}||x(t)||_{\mathcal {M}}<+\infty \). There exists \(M_2\,{>}\,0\), such that \(||x(t)||_{\mathcal {M}}\leqslant M_2\), \(\forall t\in [0,\mathcal {T})\), since x(t) is uniformly bounded about \(t\in [0,\mathcal {T})\). For \(0\leqslant t_1<t_2<\mathcal {T}\), we have
which implies that \(\lim _{t\rightarrow T^-}x(t)\) exist, i.e., \(\lim _{t\rightarrow T^-}||x(t)||_{\mathcal {M}}<+\infty \). The proof is completed. \(\square \)
1.2 Appendix B. Proof of Lemma 3.1
Suppose that \(x(t)=(x_1(t),x_2(t),\ldots ,x_n(t))^T\) is a \(\omega \)-periodic solution of system (1.1) in the sense of Filippov, in view of Definition 2.1, we can obtain from (2.2) that
Thus
Note that the periodicity of x(t), by integrating both sides of differential inclusion (7.5) over the interval \([t, t+\omega ](0\leqslant t\leqslant \omega )\), we get the following integral inclusions
That is, x(t) is a \(\omega \)-periodic solution of integral inclusions (3.1).
On the other hand, suppose that x(t) is a \(\omega \)-periodic solution of integral inclusions (3.1). By the integral representation theorem [48], there exist a measurable function \(\gamma =(\gamma _1,\gamma _2,\ldots ,\gamma _n)^T:[0,\mathcal {T})\rightarrow \mathbb {R}^n\) such that \(\gamma _i(t)\in \overline{co}[f_i(x_i(t))]\) for a.e. \(t\in [0, \mathcal {T})\) and
Since the right-hand sides of (7.6) is absolutely continuous, deviating the two sides of (7.6) about t, for a.e. \(t\in [0, \mathcal {T})\), we obtain
Note that the periodicity of x(t) and \(\gamma (t)\), we have
In view of Definition 2.1, we know that \(x(t)=(x_1(t),x_2(t),\ldots ,x_n(t))^T\) is a \(\omega \)-periodic solution of system (1.1) in the sense of Filippov.
The proof of Lemma 3.1 is completed.
1.3 Appendix C. Proof of Lemma 3.2
First, for any \(x(t)=(x_1(t),x_2(t),\ldots ,x_n(t))^T\in \overline{\Omega }_R\cap \mathbb {P}\) and \(y(t)=(y_1(t),y_2(t),\ldots ,y_n(t))^T\in \varphi (x)\). There exists a measurable function \(\gamma =(\gamma _1,\gamma _2,\ldots ,\gamma _n)^T:[0,\mathcal {T})\rightarrow \mathbb {R}^n\) such that \(\gamma _i(t)\in \overline{co}[f_i(x_i(t))]\) with \( |\gamma _i(t)|\leqslant \max _{0\leqslant s\leqslant R}\{W_i(s)\} (i=1,2,\ldots ,n)\) for a.e. \(t\in [0, \mathcal {T})\) and
Thus, for \(t\leqslant v\leqslant t+\omega \) and \(x\in \overline{\Omega }_R\cap \mathbb {P}\), we have
which implies
From (7.7), for \(t\leqslant v\leqslant t+\omega \) and \(x\in \overline{\Omega }_R\cap \mathbb {P}\), we also have
Therefore, for any \(x\in \overline{\Omega }_R\cap \mathbb {P}\) and \(y\in \varphi (x)\), we have \(y\in \mathbb {P}\). That is, \(\varphi (x)\in \mathbb {P}\) for every fixed \(x\in \mathbb {P}\), i.e., \(\varphi :\overline{\Omega }_R\cap \mathbb {P}\rightarrow \mathbb {P}\).
Next, we prove that \(\varphi (x)\) is convex for each \(x\in \overline{\Omega }_R\cap \mathbb {P}\). In fact, for any \(x=(x_1,x_2,\ldots ,x_n)^T\), if \(y=(y_1,y_2,\ldots ,y_n)^T, z=(z_1,z_2,\ldots ,z_n)^T\in \varphi (x)\), then there exist \(\gamma =(\gamma _1,\gamma _2,\ldots ,\gamma _n)^T:[0,\mathcal {T})\rightarrow \mathbb {R}^n\) and \(\eta =(\eta _1,\eta _2,\ldots ,\eta _n)^T:[0,\mathcal {T})\rightarrow \mathbb {R}^n\) with \(\gamma _i(t)\in \overline{co}[f_i(x_i(t))]\) and \(\eta _i(t)\in \overline{co}[f_i(x_i(t))]\) for a.e. \(t\in [0,\mathcal { T})\), such that for each \(t\in [0,\mathcal { T})\) we have
and
Let \(0\leqslant \alpha \leqslant 1\). Then for each \(t\in [0,\omega ]\) we have
that is
Hence,
Finally, it is easy to see that \(\varphi (x)\) is closed. The proof of Lemma 3.2 is completed.
1.4 Appendix D. Proof of Lemma 3.3
It is enough to show that \(\varphi :\overline{\Omega }_R\cap \mathbb {P}\rightarrow P_{kc}(\mathbb {P})\) is a compact map. According to the Ascoli-Arzela Theorem, it suffices to show that \(\varphi (\overline{\Omega }_R\cap \mathbb {P})\) is an uniformly bounded and equi-continuous set. For any \(x=(x_1,x_2,\ldots ,x_n)^T\in \overline{\Omega }_R\cap \mathbb {P}\) and \(y=(y_1,y_2,\ldots ,y_n)^T\in \varphi (x)\). There exists a measurable function \(\gamma =(\gamma _1,\gamma _2,\ldots ,\gamma _n)^T:[0,\mathcal {T})\rightarrow \mathbb {R}^n\) such that \(\gamma _i(t)\in \overline{co}[f_i(x_i(t))]\) with \( |\gamma _i(t)|\leqslant \max _{0\leqslant s\leqslant R}\{W_i(s)\}\ (i=1,2,\ldots ,n)\) for a.e. \(t\in [0, \mathcal {T})\) and
Hence
where \(||\psi ||_{\infty }=\hbox {ess}\sup _{s\in (-\infty ,0]}|\psi (s)|\). Which yields
Thus, \(\varphi (\overline{\Omega }_R\cap \mathbb {P})\) is an uniformly bounded set for all \(x\in \overline{\Omega }_R\cap \mathbb {P}\).
Let \(t_1, t_2\in [0,\omega ]\), then for any \(x\in \overline{\Omega }_R\cap \mathbb {P}\) and each \(i=1,2, \ldots ,n\), we have
According to the mean value theorem of derivations, we obtain
where \(0<\lambda <1\). And
Hence
where \(M_i=G_iq_i^M(q_i^Md_i^M+2)[\sum _{j=1}^n(a_{ij}^M+b_{ij}^M+c_{ij}^M)(\max _{0\leqslant s\leqslant R}\{W_i(s)\}+||\psi ||_{\infty })+I_i^M]\).
As a result we have that
Hence, \(\varphi (\overline{\Omega }_R\cap \mathbb {P})\) is an equi-continuous set in X.
Therefore, the multi-valued map \(\varphi :\overline{\Omega }_R\cap \mathbb {P}\rightarrow P_{cp,cv}(\mathbb {P})\) is a k-set-contractive map with \(k=0\).
1.5 Appendix E. Proof of Lemma 3.4
We will show that \(\varphi \) has closed graph. Denote
Let \(|||\digamma (t,x)|||=\sup \{|u|: u\in \digamma (t,x)\}\) and \(L^1([0,\omega ],\mathbb {R}^n)\) be the Banach space of all functions \(u=(u_1,u_2,\ldots ,u_n)^T:[0,\omega ]\rightarrow \mathbb {R}^n\) which are Lebesgue integrable. Define the multi-valued operator
by letting
It is easy to show that \(\digamma (t,x)\) is a \(L^1\)-Carathéodory map and the set F(x) is non-empty for each fixed \(x\in \overline{\Omega }_R\cap \mathbb {P}\).
Consider the linear continuous operator \(\mathfrak {L}: L^1([0,\omega ], \mathbb {R}^n)\rightarrow C([0,\omega ], \mathbb {R}^n)\),
Hence, it follows from Lemma 2.3 that \(\varphi =\mathfrak {L}\circ F\) is a closed graph operator. We should be point out that USC is equivalent to the condition of being a closed graph multi-valued map when the map has non-empty compact values, that is to say, we have shown that \(\varphi \) is USC.
Rights and permissions
About this article
Cite this article
Wang, D., Huang, L. Periodicity and multi-periodicity of generalized Cohen–Grossberg neural networks via functional differential inclusions. Nonlinear Dyn 85, 67–86 (2016). https://doi.org/10.1007/s11071-016-2667-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-016-2667-7