Abstract
In this paper, the complex dynamical behaviors in a discrete neural network loop with self-feedback are studied. Specifically, an invariant closed set of the system of neural network loops is built and the subsystem restricted on this invariant closed set is topologically conjugate to a two-sided symbolic dynamical system which has two symbols. In the end, some illustrative numerical examples are given to demonstrate our theoretical results.
1. Introduction
In recent years, researchers have found various chaotic phenomena in the nervous system and that chaotic neural networks play an important role in neural activities. Chaos in neural networks systems have been applied to all kinds of practical problems such as combinatorial optimizations, associative recognition memory, deep learning, and biotechnology (see [1–5]). In fact, some nervous systems consist of large-scale and complex nonlinear dynamics. At present, neuroscience has provided abundant evidence to prove that the central nervous system has complex nonlinear dynamic behavior at all levels [6]. So how to analyze the dynamical behavior of neural networks plays an important role in practical applications. In order to obtain a deep and clear understanding of complex neural networks, there are increasing studies on bifurcations and chaotic behaviors of neural network systems [7].
Recently, Huang and Zou in [8] showed the discrete network system consisting of two identical neurons with a uniform delay demonstrates snapback repeller chaotic behaviors near an equilibrium point. For the Hopfield networks with two different neurons [9–11], the conditions that the systems exhibit chaos are obtained. In [12], Wu et al. analyzed the chaotic behaviors of the parameterized discrete dynamics of recurrent m-neuron networks evoked by external inputs and obtained some conditions which the subsystem is topologically conjugate to symbolic dynamical system. In this paper, we will devote to analysis of the chaotic behaviors of the following discrete neural network loops with multiple delays and self-feedback:where , for , is the internal decay rate of the neurons, is the self-feedback strength or the connection strength of the th neuron to the next neuron, and the transmission delay is a positive integer.
For the case of the neural network with m-identical neurons, Cheng constructed a snapback repeller in [13] and then justified chaos in neural networks. When the discrete neural network with m-different neurons has multiple time delays and self-feedback, it is challenging to rigorously analyze the dynamical behaviors. In this paper, we consider the chaotic behaviors of model (1). To this end, we first rewrite the model (1) as a system of difference equations without delay by a novel way. Especially, this transformation requires a little skill. Then, we find an invariant set for the transformed system by projection and show that the system restricted on this set is topologically conjugate to the full shift map on the symbolic dynamical system. This implies that the system has chaotic behaviors. The obtained results extend the related ones in [10, 11, 13]. Also, we provide some numerical simulations to verify the theoretical results.
2. Invariant Subsystem of Model (1)
Let denote the Banach space of bounded sequences of real numbers with the supremum norm defined on it. The norm is denoted by . Let be shift map defined by , for . That is,
Clearly, the shift map on is continuously invertible, and its inverse is being defined by . The th iterate of , , is denoted as . Let denote a symbolic space with symbols. Endowing it with the metric becomes a compact and totally disconnected metric space. The shift map is defined by . Then, is a two-sided symbolic system. To proceed, let be positive integers.
Lemma 1. Let be a positive integer. are different real numbers with and is a real number with . be a subset of . Then, is topological conjugate to .
Proof. Define by , for . In fact, is defined by deleting the elements whose indexes are congruent modulo in , where . It is not difficult to see that is a homeomorphism. By definition of , we have . So and are topological conjugacy.
Lemma 2 (see [14]). Let and be Banach spaces, is an invertible linear map from to , and is a bounded linear map from to . If , then is an invertible linear map from to .
Lemma 3 (see [15]). Let be a metric space, and be Banach spaces, and be open. Suppose that is a continuous map and that there exists a point with the following conditions:(i).(ii) is continuous at , where is Frchet partial derivative of with respect to .(iii) is an invertible linear map.
Then, there exist open balls and , where such that, for any , the equation has a unique continuous solution with .
For convenience, we set when . Let . Without losing generality, we may suppose that . In the other cases, we can discuss it in a similar way. The activation functions have the following conditions (G1): (G1) For every is a continuously differentiable function from to . has two distinct zero points , satisfying , and has a zero point ,satisfying .
Let and define , where
For any , there exists such that . Then, we transform system (1) into the discrete dynamical system without delays on :where is defined aswhere .
To investigate chaos in System (1), we only consider the chaotic behavior of the system . Next, by the projection approach, we are going to find the invariant set of such that the subsystem has chaotic behavior for being sufficiently large.
We consider a family of maps depending on a parameter , and the class of maps is defined by
It is easy to see that if satisfies , then the sequence with satisfies (1). On the contrary, if the sequence satisfies (1), then with satisfies .
Let
Lemma 4. Under the assumption , if and , then we have the following:(i)There exist positive real numbers and such that, for any and , there exists a unique point , satisfying .(ii)For every , there exists such that, for any and , there is a unique point , satisfying and .
is the open ball in centered at with radius .
Proof. For a given sequence , we have . By the assumption () and the definition of , this can ensure the continuous differentiability of . The Fréchet derivative of with respect to at the point be denoted as which is represented asFirstly, we have to show the invertibility of . We denote that , whereLetIt follows from () that the linear operator is invertible. By directing calculation, the inverse operator isSince , , . This implies thatsoby the fact that and . This shows the invertibility of by Lemma 2.
Therefore, according the implicit function theorem, there exist positive constants such that, for every , there is a unique point with .
To complete the proof of (i), it only needs to prove that there exist two positive constants which are independent of such that the conclusion is satisfied in (i). From the proof of the implicit function theorem, for the given , the constants and above are chosen such that, for and , we haveHere, is the constant such that .
We now give the above estimates which are independent of . Firstly, we have, for any ,where is given by (8). Secondly, by assumption , there exists such thatfor ,for , andfor . Note thatTaking , where , we have that, for , with and :On the contrary, let , and it follows from the definition of thatwhen .
Finally, take and then the constants and satisfy (i).
For every , (ii) follows by taking and the proof of (i).
3. Chaos in System (1)
In this section, we shall show that the system (1) exists chaotic behaviors. By Lemma 4, for sufficiently large , we define the map from to bywhere is the unique solution of , satisfying . Then, we have the following proposition.
Proposition 1. For sufficiently large , let , then the map and the shift map are commutative, i.e.,Moreover, .
Proof. Note that if is a solution of so is . Thus, for any is a solution of . On the contrary, by Lemma 4, which leads to . Hence, by the uniqueness of in Lemma 4, we have . Note that , it follows that .
For every , we define the projection bywhere for , is given by
Proposition 2. Let , then is invariant for .
Proof. For each , then there exists such that . Therefore,This proves .
On the contrary, by Proposition 1, we have . Thus, there exists such that . Thus,which shows that . Therefore, .
Theorem 1. Under the assumption of , if and , then there exists such that, for any , is topologically conjugate to the full shift map , and therefore, the system is chaotic in the sense of Devaney.
Proof. Note that is an invariant subsystem. By Lemma 1 and Proposition 1, we only need to prove that there is such that, for any , is topological conjugate to .
Let , then is a set in consisting of elements, denoted byLet and be given as in Lemma 4, and let be small enough such that the family of closed balls in is piecewise disjoint.
For the given and any , by (ii) in Lemma 4, there exists an such that, for every , there exists a unique satisfying and . By the definition of the projections and , we have . So we letThe set is a subset of . For every , for all , we setWe claim the following:(1)For every , contains a unique point.(2).In fact, for each , we note thatTherefore, there exists a unique such that, for all , . Then, by the definition of and Lemma 4, there exists a unique , satisfying and . So is a bounded orbit of , that is, . Therefore, , which implies is nonempty.
On the contrary, for any , for all , there are . Thus, is a bounded orbit of . Then, there exists such that . So , and . Again by Lemma 4 (ii), there is , and hence, . Claim (1) holds.
For Claim (2), let . Then, there exists a such that Let be the corresponding sequence of . Similar to the above argument, we have . Therefore,From Claim (1), each contains a unique point which belongs to , so the converse inclusion holds. This proves Claim (2).
For every , define a map by . We claim that is a conjugacy from to . To prove this, we need to show that both and are continuous andBy Claim (2) and the definition of , it is easy see that is surjective. From Claim (1) and Lemma 4, it follows that is injective. Therefore, is bijective. Since is a map from a compact metric space to a Hausdorff space , to prove that h is homeomorphic, we just need to show the continuity of . Let the corresponding subindex sequence of be . It follows from Claim (1) thatwhere denotes the diameter of the set . Thus, for any , there exists a positive integer such that . Take . Then, for any with , it follows that agrees with in those terms with lower indices from to . Let be the symbolic sequences corresponding to and , respectively. We have agrees with in those terms with subscripts from to . Thus, and . This shows the continuity of . Hence, we conclude that is a homeomorphism.
Finally, for any , we haveThus,The Theorem 1 holds.
4. Some Simulations
In this section, we will give some numerical simulation results to verify our theoretical results. We choose , . In this case, system (1) becomes
In Figure 1, for every value, the initial values were reset to , , , and . After time steps being iterated, we plot the data consisting of 500 points for per value. The plotting is for vs the parameter . The bifurcation figures illustrate that the fixed point of loses stability and period bifurcation occurs when , and the fixed point of loses stability and period bifurcation occurs when . Making the bifurcation figures of the vs and the vs similar, they are omitted.
In Figure 2, we show the largest Lyapunov exponent diagram for . For every value, the initial values were the same as Figure 1. From the simulation results in Figure 2, we can find that the largest Lyapunov exponent is negative when and is positive when . Thus, the figures illustrate that the system (38) has chaotic behaviors when is large enough.
In Figure 3, we show the chaotic figures. For each value, after time steps being iterated, plot the 6000 data points. The figure illustrates that there are no chaos for small (e.g., ) and chaotic behavior occurs when is larger (e.g., ). Those numerical simulations support the theoretical results in Section 2.
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5. Conclusion
In this paper, the chaos of a discrete neural network loops with self-feedback is studied. The discrete neural network loops with multiple delays and self-feedback can demonstrate chaotic behavior when the interconnection strengths are large enough. Numerical simulations support the theoretical results. The theoretical results are to provide some new methods for the design of chaotic neural networks.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This study was supported in part by the National Natural Science Foundation of P. R. China (11671410) and Natural Science Foundation of Guangdong Province (2017A030313037).