Coupled Neural Network Loops with Delays (I)
Chih-Wen Shih
∗and Jui-Pin Tseng
Department of Applied Mathematics
National Chiao Tung University
Hsinchu, Taiwan
August 14, 2009
Abstract
We consider a neural network that consists of a pair of sub-networks. Each sub-network is a one-way loop with K neurons and the whole network is of two-way coupling between a single neuron of each loop. Each loop has an internal delay and the connection between the loops has a transmission delay. We study the synchronization and synchronous phase of this network. Both delay-dependent and delay-independent criteria are derived for global synchronization, anti-phase motion, convergence to the equilibrium, as well as bifurcation which yields synchronous periodic phase. They are established from observing the geometric structure of neuronal equation, combined with an iteration argument. Our investigation also provides theoretical support to some numerical findings in the literatures. Several numerical simulations illustrate the present theory.
Mathematics Subject Classifications. 34K20, 92B20, 34K18, 92C20
1
Introduction
There have been intensive and growing research interests on nervous systems and neuronal models in the past few decades. One of the most appealing subjects in these investigations is the collective behaviors of coupled neurons. These collective behaviors are determined by individual neuron properties, synaptic properties of coupling, and network structures. Although the real network architecture can be as complicated as one can imagine, rich dynamics arising from the interaction of simple
network motifs are believed to provide similar sources of activities as in real-life sys-tems. For neural network models, the dynamics are determined by the connection strength, nonlinear coupling functions, and transmission delays. Synchrony is one of the most important collective activities for neurons. The phenomenon of synchro-nization has been observed in various biological and physical systems, for example, in pacemaker cells in the heart [21], and hippocampal networks of neurons [17]. For neural systems, synchronization represents the coherent behaviors of neuronal population as conveying information or performing certain functions.
Networks of coupled neurons can exhibit a variety of interesting behaviors which are very different from their behaviors in isolation qualitatively. For example, oscillations may arise as a result of the coupling between sub-networks in a popula-tion of neurons. On the other hand, for realistic considerapopula-tion, time delay, as occur-ring in transmitting signal among neurons, has been incorporated into the neuronal modelling [2, 6, 23, 24, 31, 20]. Indeed, delay can modify the collective dynamics for neural networks; for example, it can induce oscillation or change the stability of the stationary solution [6]. The effects of delay upon neural network dynamics have been studied extensively, see for example [4, 9, 10, 12, 13, 16, 25, 32]. The phenomena for the coupled network systems with delays we are interested in include global synchro-nization [18], desynchrony [5], synchronous phases [30], delayed-induced oscillation [11], delayed-induced stability [15], and oscillation death [1].
There is a large amount of literatures on synchronization in artificial neural networks. Some of these works consider systems with time delays, cf. [6, 30, 32, 34, 35]. One of the conclusions therein is that if the coupling strength is small enough, the system can achieve global synchronization in spite of delays. However, in such a situation, synchronization may mean that all components asymptotically converge to the same synchronous equilibrium point. Crook et al. [7] studied a continuum model of the cortex, with excitatory coupling and distance dependent delays, and found for small enough delay the synchronous oscillation is stable, but for larger delays this oscillation loses stability to a travelling wave. Therefore, in addition to synchronization, it is also important to investigate the synchronous phases and their transitions. Moreover, as delay may cause a completely different behavior, one needs to be aware of when to take delays into consideration and when to ignore them.
Recently, there were also some investigations on coupled networks which con-sist of sub-networks of neurons. It has been reported that certain sub-networks interactions such as pathological synchronization is related to Parkinsons disease
and epilepsy, see [28] and the references therein. Campbell et al. [3] studied a neu-ral network with two coupled loops, each with three neurons. The model considered therein, while similar to (1.2), allows asymmetric coupling between two loops, but is without internal delay. The investigations in [3] focus on the existence of equilibria and their stability for the isolated loop and coupled loops, as well as bifurcations at the trivial equilibrium. Song et al. [27], studied a neural network which con-sists of two sub-networks each with two neurons. The system is again similar to (1.2), but with internal delay identical to transmission delay. They assert that there exists a Hopf bifurcation in some parameter region and that when the coupling de-lay increases the spatio-temporal patterns of bifurcating periodic solutions alternate from in-phase to anti-phase for positive coupling strength. The results in [3, 27, 28] basically depict local behaviors for the system.
In a population of neurons, internal delays within sub-networks and trans-mission delays among sub-networks may be of different time scale and need to be modelled separately, as remarked in [3, 6]. In this presentation, we consider a neural network that consists of a pair of one-way loops each with K neurons and two-way coupling between a single neuron of each loop. Each loop has the form:
˙
xi(t) = −µixi(t) + αig(xi−1(t − τI)), i = 1, 2, · · · , K (mod K). (1.1)
Herein, xi represents the normalized voltage of neuron i; τI ≥ 0 is internal time
delay; the activation function g is of the following class Class A : g ∈ C
2; lim
t→+∞g(ξ) = v ∈ R, limt→−∞g(ξ) = u ∈ R;
g(0) = 0; g0(0) > g0(ξ) > 0, and g00(ξ) · ξ < 0, for ξ 6= 0.
g(ξ) = tanh(ξ) is a typical example for such functions. Denote by L := g0(0) the gain of the activation function, which is assumed equal for all neurons. We denote the bound for the response function by ρ := max{|u|, |v|}.
The coupled two K-loops mainly considered in this investigation is of the following form: ˙xi(t) = −µixi(t) + αig(xi−1(t − τI)), i = 1, 2 · · · , K − 1 (mod K), ˙xK(t) = −µKxK(t) + αKg(xK−1(t − τI)) + cg(yK(t − τT)), ˙
yi(t) = −µiyi(t) + αig(yi−1(t − τI)), i = 1, 2 · · · , K − 1 (mod K),
˙
yK(t) = −µKyK(t) + αKg(yK−1(t − τI)) + cg(xK(t − τT)),
(1.2)
with initial condition φ(t0) = (φ1(t0), φ2(t0), · · · , φ2K(t0)), where φi(t0) ∈ C([−τmax, 0]; R)
loops. The interaction between two loops is inhibitory if c < 0, and excitatory if c > 0. For simplicity of presentation, we set the parameters µi = 1, αi = 1 for all i
and ρ = 1, and focus on the effect of dynamics for (1.2) from the gain of response function (L), the coupling strength between two loops (c), the internal delay (τI)
and the transmission delay (τT).
In this investigation, we consider global dynamics for the coupled system (1.2). We say that the two coupled loops attain global synchronization (in-phase) if the corresponding components of two loops tend to be identical, namely
xi(t) − yi(t) → 0, as t → ∞, for all i = 1, · · · , K,
for solution (x1(t), · · · , xK(t), y1(t), · · · , yK(t)) of (1.2), starting from arbitrary
ini-tial condition φ ∈ C([−τmax, 0], R2K) at t = t0. We also consider anti-phase for the
two loops, which means
xi(t) + yi(t) → 0, as t → ∞, for all i = 1, · · · , K,
for every solution (x1(t), · · · , xK(t), y1(t), · · · , yK(t)) of (1.2). In addition, we study
the synchronous phase which can be a single equilibrium, multiple equilibria, or a synchronous periodic orbit, for system (1.2). Moreover, we explore the effects from internal delay and transmission delay upon the the global dynamics of (1.2). We shall present a methodology by combining geometric structures of the equations with an iteration scheme to study the global dynamics. Based on this approach, we also provide theoretical support for some numerical findings in [3].
The presentation is organized as follows. In section 2, we study a scalar equa-tion with time-dependent input, which provides a basis for investigating the global dynamics of the coupled system (1.2). We arrange the proofs for the lemmas, propo-sitions, and theorems in section 2.2. Global synchronization, convergence to trivial equilibrium, and anti-phase motion are presented in sections 3.1, 3.2, and 3.3 re-spectively. Hopf bifurcation induced by the internal delay at the trivial equilibrium is studied in section 4. We present some numerical illustrations in section 5. In section 6, we summarize the dynamic scenarios corresponding to various coupling strength and delays.
2
Scalar equation with time-dependent input
This section is a preparation for the main theory of section 3. Let x(t) and y(t) be continuous functions which are eventually attracted by some closed and bounded
interval Q; namely, x(t) and y(t) remain in Q, for all time after some ˜t0 ≥ t0,
and w(t) is a bounded continuous function defined for t ≥ t0. Assume that z(t) =
x(t) − y(t) satisfies the following scalar function:
˙z(t) = −z(t) − β[g(x(t − τ )) − g(y(t − τ ))] + w(t), t ≥ t0, (2.1)
where β ∈ R, τ ≥ 0, and g is of class A. We set ˜
L := min{g0(ξ) : ξ ∈ Q}. (2.2) With the properties of z(t) derived in this section, we shall proceed to analyze the synchronous behavior of system (1.2) in section 3.
2.1
Formulations and properties
The main results in this section assert that there exist certain τ -dependent bounded and closed intervals containing zero to which every solution of (2.1) converges, under certain τ dependent conditions. A variant of this formulation also leads to τ -independent intervals under certain τ --independent condition. First, let us introduce the τ -dependent result. Below, we shall, iteratively, define two kinds of scalar func-tions which depict the upper and lower bounds for the dynamics of (2.1) respectively as time proceeds. Such iterative construction aims at capturing the asymptotical behavior for every solution of (2.1). The construction of upper and lower bounds functions depends on the sign of β. We shall demonstrate the formulation for the τ -dependent result and β > 0 case to present our main idea, and provide the results for the other cases without detailed proofs.
For T ≥ t0, we denote |w|max(T ) := sup{|w(t)| : t ≥ T }. We then define ˆ h(ξ) := −ξ + 2β + |w| max(t 0) if ξ ≥ 0, −(1 + βL)ξ + 2β + |w|max(t 0) if ξ < 0, ˇ h(ξ) := −(1 + βL)ξ − 2β − |w| max(t 0) if ξ ≥ 0, −ξ − 2β − |w|max(t 0) if ξ < 0.
It can be seen that ˆh(ξ) > ˇh(ξ) and ˆh(ξ) = −ˇh(−ξ). The decreasing and piecewise linear functions ˆh and ˇh have unique zeros at ˆAh and ˇAh respectively, where ˆAh =
2β + |w|max(t
0) ≥ 0 and ˇAh = − ˆAh ≤ 0, cf figure 1. Notably, ˆh and ˇh depict
preliminary upper and lower bounds respectively, for the dynamics of (2.1) with β > 0. That is,
ˇ
h(z(t)) < ˙z(t) < ˆh(z(t)), for all t ≥ t0, (2.3)
for arbitrary solution z(t) of (2.1). This leads to the following proposition. Herein, Q and ˜t0 have to be provided a priori as introducing the scalar equation (2.1).
Proposition 2.1. Assume that β > 0. (2.3) holds for arbitrary solution z(t) of (2.1). Subsequently, given any initial value φ ∈ C([−τ, 0], R), there exists some Tφ≥ ˜t0+ τ such that the corresponding z(t) := z(t; t0; φ) belongs to [ ˇAh, ˆAh] for all
t ≥ Tφ− τ . Moreover,
ˇ
h( ˆAh) < ˙z(t) < ˆh( ˇAh), for all t ≥ Tφ− τ.
Remark 2.1. Note that if β > 0, then 0 ≥ ˇh( ˆAh) = −(2+βL)(2β +|w|max(t
0)), (2+
βL)(2β+|w|max(t
0)) = ˆh( ˇAh) ≥ 0; in addition, x(t) and y(t) lie in Q for all t ≥ Tφ−τ ,
where Tφ is given in Proposition 2.1.
In the sequel, we denote by z(t) := z(t; t0; φ) the solution of (2.1) starting
from arbitrary φ ∈ C([−τ, 0], R]), at t = t0. The time Tφ which was introduced in
Proposition 2.1 will be used throughout the presentation.
Now, for each T ≥ t0, we introduce the following functions:
ˆ f(0)(ξ, T ) = −(1 + β ˜L)ξ + τ βLˆh( ˇA h) + |w|max(T ), for ξ ≥ 0, −(1 + βL)ξ + τ βLˆh( ˇAh) + |w|max(T ), for ξ < 0, ˇ f(0)(ξ, T ) = −(1 + βL)ξ + τ βLˇh( ˆA h) − |w|max(T ), for ξ ≥ 0, −(1 + β ˜L)ξ + τ βLˇh( ˆAh) − |w|max(T ), for ξ < 0,
where ˜L is defined in (2.2). The idea for formulation of ˇf(0)(·, T ) and ˆf(0)(·, T ) will be revealed in the following discussions. Notably, ˇf(0)(ξ, T ) = − ˆf(0)(−ξ, T ). For
convenience of later uses, we denote ˆ
f (ξ) := ˆf(0)(ξ, t0), ˇf (ξ) := ˇf(0)(ξ, t0).
We consider the following condition for (2.1).
Condition (H1): β > 0 and τ < 2/[L(2 + βL)(2β + |w|max(t 0))].
Under condition (H1), a direct computation yields that, for T ≥ t0,
ˇ
h(ξ) < ˇf (ξ) ≤ ˇf(0)(ξ, T ) < ˆf(0)(ξ, T ) ≤ ˆf (ξ) < ˆh(ξ), for all ξ ∈ R. (2.4) Herein, ˇf(0)(·, T ) and ˆf(0)(·, T ) depict the lower and upper bounds for the dynamics of (2.1), which are more precise than ˇh(·) and ˆh(·) respectively as time gets larger. Let ˇa(0)(T ) (respectively, ˆa(0)(T )) be the unique solution of ˇf(0)(·, T ) = 0
(respec-tively, ˆf(0)(·, T ) = 0) lying in interval [ ˇAh, ˇAh], cf figure 1. Notably, ˇa(0)(T ) =
−ˆa(0)(T ) ≤ 0.
It follows from Proposition 2.1 and Remark 2.1 that if β > 0, for each T ≥ Tφ,
ˇ
f(0)(z(t), T ) < ˙z(t) < ˆf(0)(z(t), T ), for all t ≥ T. (2.5) Let us verify (2.5) and explain the formulation of ˇf(0), ˆf(0). First, x(t − τ ) and y(t − τ ) ∈ Q for all t ≥ T ≥ Tφ; therefore, for t ≥ T ≥ Tφ, ˙z(t) = −z(t) −
βg0(ζ)z(t − τ ) + w(t) = −z(t) − βg0(ζ)[z(t) − ˙z(s)τ ] + w(t), for some ζ ∈ Q and s ≥ t−τ ≥ Tφ−τ . Notice that ˇh( ˆAh) < ˙z(s) < ˆh( ˇAh) and ˇh( ˆAh) ≤ 0 and ˆh( ˇAh) ≥ 0.
If z(t) ≥ 0, ˙z(t) < −z(t) − β ˜Lz(t) + τ βLˆh( ˇAh) + |w|max(T ) =: ˆf(0)(z(t), T ). If
z(t) < 0, ˙z(t) < −z(t) − βLz(t) + τ βLˆh( ˇAh) + |w|max(T ) =: ˆf(0)(z(t), T ). Hence, the
right-hand inequality of (2.5) is verified. The left-hand inequality can be treated similarly. We thus derive the following proposition.
Proposition 2.2. Inequality (2.5) holds under condition (H1). Consequently, z(t) eventually enters and stays afterward in [ˇa(0)(T ), ˆa(0)(T )] = [−ˆa(0)(T ), ˆa(0)(T )].
Similar to the construction of ˇf(0)(·, T ) and ˆf(0)(·, T ), we shall iteratively define the following functions. For k ∈ N and T ≥ t0
ˆ f(k)(ξ, T ) := −(1 + β ˜L)ξ + τ βL ˆf k−1(ˇa(k−1)(T ), T ) + |w|max(T ), for ξ ≥ 0, −(1 + βL)ξ + τ βL ˆfk−1(ˇa(k−1)(T ), T ) + |w|max(T ), for ξ < 0, ˇ f(k)(ξ, T ) := −(1 + βL)ξ + τ βL ˇf k−1(ˇa(k−1)(T ), T ) − |w|max(T ), for ξ ≥ 0, −(1 + β ˜L)ξ + τ βL ˇfk−1(ˇa(k−1)(T ), T ) − |w|max(T ), for ξ < 0, where ˇa(k)(T ) (respectively, ˆa(k)(T )) is the unique solution of ˇf(k)(·, T ) = 0 (re-spectively, ˆf(k)(·, T ) = 0). Notice that ˇf(k)(ξ, T ) = − ˆf(k)(−ξ, T ) and ˇa(k)(T ) = −ˆa(k)(T ) ≤ 0. Let us define
|w|max(∞) := lim T →∞|w|
It shall be shown that the successively defined ˆf(k) and ˆf(k) control the dynamics
of (2.1) more precisely as k and T increase. We summarize the properties for the above-defined terms in the following lemma and theorems. Their proofs will be deferred until section 2.2.
Lemma 2.3. Assume that condition (H1) holds. Then, for each T ≥ t0, the
sequences {ˇa(k)(T )}
k≥0, {ˆa(k)(T )}k≥0 can be defined iteratively. Moreover, (i) for
any fixed k ∈ N ∪ {0}, ˆa(k)(T ) is decreasing and ˇa(k)(T ) is increasing with respect
to T ≥ t0; (ii) for any T ≥ t0, there exists a(T ) ≥ 0, such that ˆa(k)(T ) → a(T )
decreasingly, and ˇa(k)(T ) → −a(T ) increasingly, as k → ∞; (iii) there exists a p ≥ 0,
such that a(T ) → ap decreasingly, as T → ∞; (iv) 0 ≤ a(T ) = |w|max(T )/[(1 +
β ˜L) − τ βL(2 + βL + β ˜L)], for any T ≥ t0; (v) ∩T ≥t0[−a(T ), a(T )] = [−ap, ap], and
0 ≤ ap ≤
|w|max(∞)
(1 + β ˜L) − τ βL(2 + βL + β ˜L). (2.6) Theorem 2.4. Every solution of (2.1) converges to interval [−ap, ap], as t → ∞,
under condition (H1).
For the case of β < 0, we can also derive analogous result.
Theorem 2.5. Every solution of (2.1) converges to an interval [−aq, aq] under
condition (H2): −1/L < β < 0, τ < 2(1 + βL)/[L(2 + βL)(2|β| + |w|max(t 0))];
moreover,
0 ≤ aq ≤ |w|max(∞)/{(1 + βL) + τ βL(2 + βL + β ˜L)}.
The assumptions and conclusions in the previous two theorems are both τ -dependent. Indeed, via similar arguments, we can derive the τ -independent conclu-sion as follows:
Theorem 2.6. Every solution of (2.1) converges to an interval [−ar, ar], as t → ∞,
under condition (H3): |β| < 1/L − |w|max(t
0)/2; moreover,
0 ≤ ar ≤ |w|max(∞)/(1 − |β|L).
All results in this section can be extended to the following more general scalar equation:
where v is a continuous function with v(t) → 0, as t → ∞. The form of (2.7) includes the following special one
˙x(t) = −x(t) − βg(x(t − τ )) + w(t) + v(t), (2.8) where x(t) remains in Q, for all time after some ˜t0 ≥ t0. Notably, (2.8) is like y = 0
in (2.7). Since the results in Theorems 2.4-2.6 concern the asymptotical behavior of all solutions to the equation, it is straightforward to conclude the following corollary. Corollary 2.7. Every solution of (2.7) or (2.8) converges to [−ap, ap] (respectively,
[−aq, aq], [−ar, ar]) under condition (H1) (respectively, (H2), (H3)).
Let us also consider the following equation:
˙x(t) = −x(t) + w(t), (2.9) which is a special form of (2.8) with β = 0, and v(t) = 0, for t ≥ t0.
Corollary 2.8. Every solution of (2.9) converges to an interval [−as, as]; moreover,
0 ≤ as ≤ |w|max(∞).
2.2
Proofs of lemmas, propositions and theorems
We provide the proofs for Lemma 2.3, Theorems 2.4-2.6 in this section.
Proof of Lemma 2.3. The labelling in the proof corresponds to the one in the statement of Lemma 2.3.
(i) Let us show that for any T ≥ t0, ˇa(k)(T ) and ˆa(k)(T ) are well-defined for all
k ∈ N ∪ {0}. First, let us claim that the following inequalities ˇ
f(k−1)(ξ, T ) ≤ ˇf(k)(ξ, T ) ≤ ˆf(k)(ξ, T ) ≤ ˆf(k−1)(ξ, T ), (2.10) hold for all ξ ∈ R, k ∈ N. To justify that (2.10) holds as k = 1, by the definitions of ˇf(0), ˇf(1), ˆf(1) and ˆf(0), it suffices to verify the following three inequalities:
ˆ h( ˇAh) ≥ ˆf(0)(ˇa(0)(T ), T ), ˆ f(0)(ˇa(0)(T ), T ) ≥ ˇf(0)(ˆa(0)(T ), T ), ˇ f(0)(ˆa(0)(T ), T ) ≥ ˇh( ˆAh).
It is obvious that all these three inequalities hold under condition (H1), cf figure 1. Assume that (2.10) holds for k = 1, · · · , j − 1, where j ∈ N. To show that it also
holds for k = j, by definitions of ˇf(j−1), ˇf(j), ˆf(j) and ˆf(j−1), it suffices to justify the
following three inequalities: ˆ f(j−2)(ˇa(j−2)(T ), T ) ≥ ˆf(j−1)(ˇa(j−1)(T ), T ), ˆ f(j−1)(ˇa(j−1)(T ), T ) ≥ ˇf(j−1)(ˆa(j−1)(T ), T ), ˇ f(j−1)(ˆa(j−1)(T ), T ) ≥ ˇf(j−2)(ˆa(j−2)(T ), T ).
Note that ˇf(j−2)(ξ, T ) ≤ ˇf(j−1)(ξ, T ) ≤ ˆf(j−1)(ξ, T ) ≤ ˆf(j−2)(ξ, T ) for ξ ∈ R, since (2.10) holds for k = j −1. It is obvious that all these three inequalities hold, cf figure 2. Hence (2.10) hold for all k ∈ N; subsequently, ˇf (ξ) ≤ ˇf(k)(ξ, T ) ≤ ˆf(k)(ξ, T ) ≤
ˆ
f (ξ) for all ξ ∈ R, and k ∈ N ∪ {0}, owing to (2.4). Note that ˇf(k)(ξ, T ) and
ˆ
f(k)(ξ, T ) are both vertical shift of ˇf (ξ) and ˆf (ξ). Accordingly, both ˇa(k)(T ) and
ˆ
a(k)(T ) are well defined for all k ∈ N ∪ {0} under condition (H1). Moreover, it
is a straightforward result that ˆa(j)(T
1) ≤ ˆa(j)(T2) and ˇa(j)(T1) ≥ ˇa(j)(T2), for any
T1 > T2 ≥ t0, since that ˆf(j)(·, T1) ≤ ˆf(j)(·, T2) and ˇf(j)(·, T1) ≥ ˇf(j)(·, T2). Thus,
for each k ∈ N ∪ {0}, ˇa(k)(T ) increases and ˆa(k)(T ) decreases, with respect to T .
(ii) By (2.10), it can be shown that, for each T ≥ t0,
ˆ
a(k+1)(T ) ≤ ˆa(k)(T ), ˇa(k+1)(T ) ≥ ˇa(k)(T ), for all k ≥ 0.
Moreover, ˆa(k)(T ) ≥ 0 and ˇa(k)(T ) = −ˆa(k)(T ), for i ∈ N. It follows that, for any
T ≥ t0, there exist some a(T ) ≥ 0 such that limk→∞ˆa(k)(T ) = a(T ) ≥ 0, and
limk→∞ˇa(k)(T ) = −a(T ) ≤ 0.
(iii) For each k ∈ N ∪ {0}, it has been shown that ˆa(k)(T2) ≥ ˆa(k)(T1), if
T1 > T2 ≥ t0; subsequently, limk→∞ˆa(k)(T2) ≥ limk→∞ˆa(k)(T1), i.e. a(T2) ≥ a(T1).
Therefore, there exists some ap ∈ R such that a(T ) → ap decreasingly as T → ∞,
since a(T ) is bounded below for all T ≥ t0.
(iv) It is obvious that for all T ≥ t0, a(T ) ≥ 0. Next, we justify that a(T ) ≤
|w|max(T )/[(1 + β ˜L) − τ βL(2 + βL + β ˜L)], for any T ≥ t
0. For any fixed T ≥ t0,
it is not difficult to observe that { ˆf(k)(·, T )|
[ ˇAh, ˆAh]}k≥0 are uniformly bounded and
equicontinuous. Moreover ˆf(k)(ξ, T ) decreases with respect to k. By Ascoli-Azela
Theorem, for any fixed T ≥ t0, there exists some continuous function ˆf(∞)(·, T )
defined on [ ˇAh, ˆAh] such that ˆ
f(k)(·, T ) ↓ ˆf(∞)(·, T ) uniformly on [ ˇAh, ˆAh], as k → ∞. Recall that ˇf(k)(ξ, T ) = − ˆf(k)(−ξ, T ). It follows that
ˇ
Figure 2: Configurations for functions ˇf(j−1)(·, T ), ˇf(j−2)(·, T ), ˆf(j−2)(·, T ) and ˆ
f(j−1)(·, T ) for fixed T ≥ t 0.
where ˇf(∞)(ξ, T ) = − ˆf(∞)(−ξ, T ). It is obvious that, for any fixed T ≥ t 0
ˇ
f(∞)(ξ, T ) ≤ ˆf(∞)(ξ, T ), for all ξ ∈ [ ˇAh, ˆAh].
We summarize the properties of ˇf(∞)(·, T ) and ˆf(∞)(·, T ) as follows:
(P1): ˆf(k)(ˆa(k)(T ), T ) → ˆf(∞)(a(T ), T ), ˆf(k)(ˇa(k)(T ), T ) → ˆf(∞)(−a(T ), T ) as k →
∞,
(P2): ˆf(∞)(ξ, T ) = −(1 + β ˜L)ξ + τ βL ˆf(∞)(−a(T ), T ) + |w|max(T ), for ξ ≥ 0,
−(1 + βL)ξ + τ βL ˆf(∞)(−a(T ), T ) + |w|max(T ), for ξ < 0,
(P3): ˆf(∞)(a(T ), T ) = 0.
Below, let us justify these properties one by one. The first result in (P1) holds since | ˆf(k(ˆa(k)(T ), T ) − ˆf(∞)(a(T ), T )|
≤ | ˆf(k)(ˆa(k)(T ), T ) − ˆf(∞)(ˆa(k)(T ), T )| + | ˆf(∞)(ˆa(k)(T ), T ) − ˆf(∞)(a(T ), T )|, and both | ˆf(k)(ˆa(k)(T ), T ) − ˆf(∞)(ˆa(k)(T ), T )| and | ˆf(∞)(ˆa(k)(T ), T ) − ˆf(∞)(a(T ), T )|
converge to 0 as k → ∞. The remaining part in (P1) can be justified similarly. If ξ ≥ 0, ˆ f(∞)(ξ, T ) = lim k→∞ ˆ f(k)(ξ, T ) = lim k→∞{−(1 + β ˜L)ξ + τ βL ˆf k−1(ˇa(k−1)(T ), T ) + |w|max(T )} = −(1 + β ˜L)ξ + τ βL lim k→∞{ ˆf (k−1)(ˇa(k−1)(T ), T )} + |w|max(T ) = −(1 + β ˜L)ξ + τ βL ˆf(∞)(−a(T ), T ) + |w|max(T ).
Similar argument can be applied to the case ξ < 0. Hence, property (P2) follows. At last, (P3) follows from property (P1) since ˆf(k)(ˆa(k)(T ), T ) = 0. Due to properties
(P2) and (P3), for each T , ˆf(∞)(·, T ) is a strictly decreasing function and has a
unique zero at a(T ). As ˇf(∞)(ξ, T ) = − ˆf(∞)(−ξ, T ), −a(T ) is the unique zero
of ˇf(∞)(·, T ) = 0. Since ˆf(∞)(−a(T ), T ) = (1 + βL)a(T ) + τ βL ˆf(∞)(−a(T ), T ) +
|w|max(T ), we derive 0 ≤ ˆf(∞)(−a(T ), T ) = (1 + βL)a(T ) + |w| max(T ) 1 − τ βL . Consequently, for ξ ≥ 0, ˆ f(∞)(ξ, T ) = −(1 + β ˜L)ξ + τ βL[(1 + βL)a(T ) + |w| max(T )] 1 − τ βL + |w| max (T ). (2.11)
With the help of (2.11) and ˆf(∞)(a(T ), T ) = 0, it can be derived that
a(T ) = |w|max(T )/[(1 + β ˜L) − τ βL(2 + βL + β ˜L)]. (2.12) (v) It is obvious that ∩T ≥t0[−a(T ), a(T )] = [−ap, ap]. In addition, ap ≤ a(T )
for any T ≥ t0. With a(T ) given in (2.12), we thus obtain
0 ≤ ap ≤ |w|max(∞)/{(1 + β ˜L) − τ βL(2 + βL + β ˜L)}.
Proof of Theorem 2.4. Let z(t) be a solution to (2.1). The following claim will lead to that for each T ≥ Tφ, z(t) converges to [−a(T ), a(T )] as t → ∞.
Subsequently z(t) converges to [−a, a] as t → ∞. To complete the proof, it suffices to prove the claim.
Claim: For arbitrarily fixed T ≥ Tφ, and any fixed n ∈ N, there exist some
increasing sequence {Tk}nk=0 with Tk+1 ≥ Tk+ τ , for k = 0, 1, · · · , n − 1 and T0 ≥
T + τ , such that ˇ
f(k)(z(t), T ) < ˙z(t) < ˆf(k)(z(t), T ), for t ≥ Tk+ τ, k = 0, 1, · · · , n − 1;
z(t) ∈ [ˇa(k)(T ), ˆa(k)(T )], for t ≥ Tk+1, k = 0, 1, · · · , n − 1.
(2.13) Let us justify the claim. First, it is easy to show that ˇf(0)(z(t), T ) < ˙z(t) <
ˆ
f(0)(z(t), T ) for all t ≥ T
0+τ := (T +τ )+τ > T due to Proposition 2.2. Accordingly,
there exists some T1 ≥ T0+ τ , such that z(t) ∈ [ˇa(0)(T ), ˆa(0)(T )] for all t ≥ T1. Thus,
(2.13) holds for n = 1. Now, we assume that (2.13) holds for n = ` − 1 ≥ 1. We shall show that (2.13) holds for n = `. For any t ≥ T`−1+ τ , g(x(t − τ )) − g(y(t − τ )) =
g0(ζ)z(t−τ ), for some ζ ∈ Q. z(t−τ ) = z(t)− ˙z(s)τ for some s ∈ (t−τ, t). Therefore, ˙z(t) = −z(t) − βg0(ζ)z(t − τ ) + w(t) = −z(t) − βg0(ζ)[z(t) − ˙z(s)τ ] + w(t). No-tice that ˇf(`−2)(ˆa(`−2)(T ), T ) < ˙z(s) < ˆf(`−2)(ˇa(`−2)(T ), T ); ˇf(`−2)(ˆa(`−2)(T ), T ) ≤ 0
and ˆf(`−2)(ˇa(`−2)(T ), T ) ≥ 0 since that s ≥ T
`−1 ≥ T`−2 + τ . If z(t) ≥ 0, ˙z(t) <
−z(t) − β ˜Lz(t) + τ βL ˆf(`−2)(ˇa(`−2)(T ), T ) + |w|max(T ) = ˆf(`−1)(z(t), T ). If z(t) < 0,
˙z(t) < −z(t) − βLz(t) + τ βL ˆf(`−2)(ˇa(`−2)(T ), T ) + |w|max(T ) = ˆf(`−1)(z(t), T ).
Sim-ilarly, we can show that ˙z(t) > ˇf(`−1)(z(t), T ). Accordingly, there exists some
T` ≥ T`−1 + τ such that z(t) ∈ [ˇa(`−1)(T ), ˆa(`−1)(T )], for t ≥ T`. The claim is
thus justified.
formulation for the case of β > 0 as follows: ˆ h(ξ) := −(1 + βL)ξ + 2|β| + |w| max(t 0), if ξ ≥ 0, −ξ + 2|β| + |w|max(t 0), if ξ < 0; ˇ h(ξ) := −ξ − 2|β| − |w| max(t 0), if ξ ≥ 0, −(1 + βL)ξ − 2|β| − |w|max(t 0), if ξ < 0; ˆ f(0)(ξ, T ) := −(1 + βL)ξ − τ βLˆh( ˇA h) + |w|max(T ), for ξ ≥ 0, −(1 + β ˜L)ξ − τ βLˆh( ˇAh) + |w|max(T ), for ξ < 0; ˇ f(0)(ξ, T ) := −(1 + β ˜L)ξ − τ βLˇh( ˆA h) − |w|max(T ), for ξ ≥ 0, −(1 + βL)ξ − τ βLˇh( ˆAh) − |w|max(T ), for ξ < 0; ˆ f(k)(ξ, T ) := −(1 + βL)ξ − τ βL ˆf k−1(ˇa(k−1)T ), T ) + |w|max(T ), for ξ ≥ 0, −(1 + β ˜L)ξ − τ βL ˆfk−1(ˇa(k−1)(T ), T ) + |w|max(T ), for ξ < 0; ˇ f(k)(ξ, T ) := −(1 + β ˜L)ξ − τ βL ˇf k−1(ˇa(k−1)(T ), T ) − |w|max(T ), for ξ ≥ 0, −(1 + βL)ξ − τ βL ˇfk−1(ˇa(k−1)(T ), T ) − |w|max(T ), for ξ < 0.
Then the proof of the theorem is similar to the one for Theorem 2.4.
Proof of Theorem 2.6. We recompose the upper and lower functions in the formulation for Theorem 2.4 as follows:
ˆ h(ξ) := −ξ + 2|β| + |w|max(t0), ˇh(ξ) := −ξ − 2|β| − |w|max(t0), ˆ f(0)(ξ, T ) := −ξ + |β|L ˆAh+ |w|max(T ), ˇf(0)(ξ, T ) := −ξ − |β|L ˆAh − |w|max(T ), ˆ f(k)(ξ, T ) := −ξ + |β|Lˆa(k−1)(T ) + |w|max(T ), ˇ f(k)(ξ, T ) := −ξ − |β|Lˆa(k−1)(T ) − |w|max(T ).
Then the proof follows from similar process as Theorem 2.4.
3
The coupled K-loops
We shall study global synchronization (in-phase) for the coupled loops (1.2) in sec-tion 3.1, global convergence to the trivial equilibrium in secsec-tion 3.2, and anti-phase motion in section 3.3. Notably, system (1.2) is a dissipative system, hence solution evolved from any initial condition φ ∈ C([−τmax, 0], R2K) at initial time t0 exists for
3.1
Global synchronization
In this section, we shall derive both transmission τT-dependent and τT-independent
criteria for (1.2) to attain global synchronization; that is
xi(t) − yi(t) → 0, as t → ∞, for all i = 1, · · · , K
for solution (x1(t), · · · , xK(t), y1(t), · · · , yK(t)) of (1.2), starting from arbitrary
ini-tial condition. For this purpose, we shall consider the following difference system of (1.2):
˙zi(t) = −zi(t) + wi(t), i = 1, · · · , K − 1,
˙zK(t) = −zK(t) − c[g(xK(t − τT)) − g(yK(t − τT))] + wK(t).
(3.1) where zi(t) = xi(t) − yi(t), i = 1, · · · , K; wi(t) = g(xi−1(t − τI)) − g(yi−1(t − τI)),
i = 1, · · · , K (mod K). Notice that both xK(t) and yK(t) are eventually attracted
by [−1 − |c|, 1 + |c|], as seen from the equation for xK and yK in (1.2) (with αi, ρ
set to one). We denote ˜
Lc:= min{g
0
(ξ) : ξ ∈ [−1 − |c|, 1 + |c|]}.
Obviously, every component of (3.1) is of the form (2.1). In fact, the first K − 1 components satisfy (2.9) and the K-component satisfies (2.1) with β = c. Now, we introduce the following τT-dependent condition for global synchronization of (1.2):
Condition (S1): c > 0, τT ≤ 1/[L(2 + cL)(1 + c)], and 1 + c ˜Lc− τTcL(2 + cL +
c ˜Lc) > LK.
The second inequality in condition (S1) matches the second inequality in condition (H1) with |wi|max(t0) ≤ 2, for all i. The third inequality in condition (S1) is needed
for contraction of sequence of intervals in the following Theorem 3.3. Obviously, under condition (S1), the Kth component of (3.1) satisfies condition (H1). By Theorem 2.4 and Corollary 2.8, there exist K intervals Ii := [−ai, ai], i = 1, · · · , K,
to which zi(t) converges respectively. Moreover,
ai ≤ |wi|max(∞), i = 1, 2 · · · , K − 1,
aK ≤ |wK|max(∞)/[1 + c ˜Lc− τTcL(2 + cL + c ˜Lc)].
(3.2)
Notably, the magnitude of aidepends on |wi|max(∞) which cannot be measured
which are definite terms, and then derive a rough estimate of ai. From this estimate,
we compute more precise estimates on ai through an iterative process. The idea for
estimating ai is illustrated and implemented in the following proposition.
Proposition 3.2. Assume that condition (S1) holds. Then for any i = 1, · · · , K, there exists a sequence of intervals {a(k)i }∞
k=1 such that for each k, the ith component
zi(t) of every solution to (3.1) converges to I (k) i := [−a (k) i , a (k) i ] as t → ∞, and a (k) i satisfies 0 ≤ a(k)1 = La(k−1)K , 0 ≤ a(k)i = La(k)i−1, i = 2, · · · , K − 1, 0 ≤ a(k)K = La(k)K−1/[1 + c ˜Lc− τTcL(2 + cL + c ˜Lc)], (3.3) where a(0)K := 2/L.
Proof. First, z1(t) converges to interval [−a1, a1], as t → ∞, where
a1 ≤ |w1|max(∞) ≤ 2 =: a(1)1 .
For notational use, we set a(0)K := 2/L so that La(0)K = a(1)1 . We may say that z1(t) converges to a closed and bounded interval I
(1) 1 := [−a (1) 1 , a (1) 1 ] ⊃ I1 with
a(1)1 = La(0)K . Notice that |w2(t)| = |g(x1(t − τI)) − g(y1(t − τI))| ≤ L|x1(t −
τI) − y1(t − τI)| = L|z1(t − τI)|, for all t ≥ t0. Hence, |w2|max(∞) ≤ La (1) 1 . It
follows that a2 ≤ |w2|max(∞) ≤ La (1)
1 . We may say that z2(t) converges to a closed
and bounded interval I2(1) := [−a(1)2 , a(1)2 ] ⊃ I2 and a (1)
2 := La (1)
1 . Iteratively, we
obtain that, for i = 2, · · · , K − 1, zi(t) converges to a closed and bounded interval
Ii(1) := [−a(1)i , a(1)i ] ⊃ Ii with a (1)
i := La (1)
i−1. For i = K, zK(t) converges to interval
[−aK, aK], where
aK ≤ |wK|max(∞)/[1 + c ˜Lc− τTcL(2 + cL + c ˜Lc)],
≤ La(1)K−1/[1 + c ˜Lc− τTcL(2 + cL + c ˜Lc)],
by (3.2) and |wK(t)| = |g(xK−1(t−τI))−g(yK−1(t−τI))| ≤ L|xK−1(t−τI)−yK−1(t−
τI)| = L|zK−1(t − τI)|, for all t ≥ t0. Therefore, by setting a (1)
K = La
(1)
K−1/[1 +
c ˜Lc− τTcL(2 + cL + c ˜Lc)], then zK(t) converges to a closed and bounded interval
IK(1) := [−a(1)K , a(1)K ] ⊃ IK. By continuing the above process and employing the loop
structure, we can derive (3.3). This completes the proof.
So far, we have shown that every component of arbitrary solution to system (3.1) converges to a sequence of closed intervals whose lengths 2a(k)i can be controlled
by iterative formula (3.3). Next, it will be examined that for each i, a(k)i converges to zero as k → ∞. Thus the interval to which each component of the solution converges degenerates into a single point. One can then conclude that system (3.1) achieves global convergence to zero; accordingly system (1.2) attains global synchronization. Such arguments are implemented in the following theorem.
Theorem 3.3. The coupled K-loops (1.2) attain global synchronization under condition (S1).
Proof. Under condition (S1), a(k)1 are decreasing with respective to k, since a(j+1)1 = La(j)K = L2a(j)
K−1/[1+c ˜Lc−τTcL(2+cL+c ˜Lc)] = LKa (j)
1 /[1+c ˜Lc−τTcL(2+cL+c ˜Lc)] <
a(j)1 . By similar arguments, it can also be shown that a(k)i are decreasing with respective to k for i = 2, · · · , K. Suppose a(k)i → ˜ai, as k → ∞, for i = 1, · · · , K.
By (3.3), it follows that 0 ≤ ˜a1 = L˜aK, 0 ≤ ˜ai = L˜ai−1, i = 2, · · · , K − 1, 0 ≤ ˜aK = L˜aK−1/[1 + c ˜Lc− τTcL(2 + cL + c ˜Lc)]. (3.4)
Subsequently, 0 ≤ ˜a1 = LK˜a1/[1 + c ˜Lc − τTcL(2 + cL + c ˜Lc)], and it yields that
˜
a1 = ˜a2 = · · · = ˜aK = 0 under condition (S1). This completes the proof.
Applying the same treatment as Theorem 3.3 and using Theorems 2.5, 2.6, we can derive criteria which are τT-dependent with c < 0 and τT-independent
respec-tively, for synchronization of (1.2); namely
Condition (S2): −1/L < c < 0, τT ≤ (1 + cL)/[L(2 + cL)(1 + |c|)] and
1 + cL + τTcL(2 + cL + c ˜Lc) > LK.
Condition (S3): 0 < L < 1 and |c| ≤ 1/L − 1.
Theorem 3.4. The coupled K-loops (1.2) attains global synchronization under condition (S2) or (S3).
It is necessary that L < 1 for condition (S2); but not for condition (S1). With this restriction, we may say that positive coupling strength (c > 0, excitatory) is advantageous for (1.2) to be synchronized. On the other hand, in section 3.3, it can be seen that negative coupling strength (c < 0, inhibitory) is advantageous for (1.2) (especially for the case of L > 1) to tend to antis-phase.
Conditions (S1) and (S2) are dependent on τT and favor small τT, but are
independent of τI. In section 4, we shall show that, under condition (S1) and some
extra conditions, there exist nontrivial synchronous periodic solution. In addition, it will be shown that (1.2) actually achieves global convergence to the trivial equilibria under condition (S3)
3.2
Global convergence to trivial equilibrium
In this section, we shall derive both τT-independent and τT-dependent criteria for
system (1.2) to admit global convergence to trivial equilibrium. For this purpose, we rewrite (1.2) as follows: ˙xi(t) = −xi(t) + wi(t), i = 1, 2 · · · , K − 1 (mod K), ˙xK(t) = −xK(t) + cg(xK(t − τT)) + wK(t) + vK(t), ˙ yi(t) = −yi(t) + ˜wi(t), i = 1, 2 · · · , K − 1 (mod K), ˙ yK(t) = −yK(t) + cg(yK(t − τT)) + ˜wK(t) + ˜vK(t). (3.5)
where wi(t) = g(xi−1(t − τI)), ˜wi(t) = g(yi−1(t − τI)), i = 1, 2 · · · , K − 1; vK(t) =
c[g(yK(t − τT)) − g(xK(t − τT))], and ˜vK(t) = c[g(xK(t − τT)) − g(yK(t − τT))]. We
impose the following conditions: Condition (C): 0 < |c| < 1/L, τT ≤ min{1/[L(2 + |c|L)(1 + |c|)], (1 − |c|L)/[(1 + |c|)(2 − |c|L)L]}, min{1 + |c| ˜Lc− τT|c|L(2 + |c|L + |c| ˜Lc), 1 − |c|L − τT|c|L(2 − |c|L − |c| ˜Lc)} > LK.
Theorem 3.5. Every solution of the coupled K-loops (1.2) converges to the trivial equilibrium as t → ∞, under condition (C).
Proof. We merely prove the case of c > 0. The situation for c < 0 can be treated similarly. Notably, the latter two inequalities in condition (C) yield condition (S1). By Theorem 3.3, (1.2) achieves global synchronization under condition (C), hence each of the first K−1 components (respectively, the Kth component) of (3.5) is of the form (2.9) (respectively, (2.8) with β = −c). Moreover, the Kth component satisfies condition (H2) under condition (C). Notice that both xK(t) and yK(t) are eventually
attracted by [−1 − |c|, 1 + |c|]. By Corollary 2.7 and Corollary 2.8, there exist K intervals [−ai, ai], i = 1, · · · , K, to which xi(t) converges respectively. Furthermore,
ai ≤ |wi|max(∞), i = 1, 2 · · · , K − 1,
aK ≤ |wK|max(∞)/[1 − cL − τTcL(2 − cL − c ˜Lc)].
Similar to Proposition 3.2, there exists a sequence of intervals {a(k)i }∞
k=1 such that
for each k, the ith component xi(t) of every solution x(t) to (3.5) converges to
Ii(k):= [−a(k)i , a(k)i ] as t → ∞, and a(k)i satisfies 0 ≤ a(k)1 = La(k−1)K , 0 ≤ a(k)i = La(k)i−1, i = 2, · · · , K − 1, 0 ≤ a(k)K = La(k)K−1/[1 − cL − τTcL(2 − cL − c ˜Lc)], (3.7)
where a(0)K := 2/L. Due to the last inequality in condition (C), {a(k)i } are decreasing with respective to k for i = 1, · · · , K, since a(j+1)i = LKa(j)i /[1 − cL − τTcL(2 − cL −
c ˜Lc)] < a (j)
i . Suppose {a (k)
i } converges to ˜ai, for i = 1, · · · , K. It follows from (3.7)
that 0 ≤ ˜a1 = L˜aK, 0 ≤ ˜ai = L˜ai−1, i = 2, · · · , K − 1, 0 ≤ ˜aK = L˜aK−1/[1 − cL − τTcL(2 − cL − c ˜Lc)].
Subsequently, 0 ≤ ˜a1 = LKa˜1/[1 + c ˜Lc− τTcL(2 + cL + c ˜Lc)]. Hence, it yields that
˜
a1 = ˜a2 = · · · = ˜aK = 0, thanks to the last inequality in condition (C). Therefore,
every xi(t) converges to zero as t → ∞. By similar arguments, we can show that
every yi(t) converges to zero. This completes the proof.
By using the same techniques as Theorem 3.5, we can also derive the following theorem under τT-independent condition (S3).
Theorem 3.6. Every solution of the coupled K-loops (1.2) converges to the trivial equilibrium as t → ∞, under condition (S3).
Notably, in [3], the globally asymptotical stability (global convergence) of the origin for the coupled 3-loops of form (1.2) with K = 3 and without internal delay τI has been addressed under condition independent of τT. In this presentation, we
have derived the same conclusion, under τT-dependent criteria, for system (1.2). On
the other hand, the criteria for system (1.2) to converge to multiple equilibria have been established in [26].
3.3
Anti-phase motion
In this section, we shall consider (1.2) with the activation functions of the following class
We denote system (1.2) with activation functions in class B by (1.2)B. We shall
derive criteria to guarantee (1.2)B to admit global anti-phase motion; that is,
xi(t) + yi(t) → 0, as t → ∞, for all i = 1, · · · , K,
for solution (x1(t), · · · , xK(t), y1(t), · · · , yK(t)) of (1.2), starting from any initial
condition. We set ˜yi(t) = −yi(t), then from (1.2), we obtain
˙xi(t) = −xi(t) + g(xi−1(t − τI)), i = 1, 2 · · · , K − 1 (mod K), ˙xK(t) = −xK(t) + g(xK−1(t − τI)) − cg(˜yK(t − τT)),
˙˜yi(t) = −˜yi(t) + g(˜yi−1(t − τI)), i = 1, 2 · · · , K − 1 (mod K),
˙˜yK(t) = −˜yK(t) + g(˜yK−1(t − τI)) − cg(xK(t − τT)).
(3.8)
Showing that (1.2)B achieves global anti-phase motion amounts to justifying that
(3.8) achieve global synchronization. By employing the same treatment in section 3.1, we can conclude the following result.
Theorem 3.7. The coupled K-loops with activation function of class B, (1.2)B,
attains global anti-phase motion under condition (S3) or (A1) or (A2), where Condition (A1): c < 0, τT ≤ 1/[L(2 − cL)(1 − c)] and 1 − c ˜Lc+ τTcL(2 − cL −
c ˜Lc) > LK,
Condition (A2): 1/L > c > 0, τT ≤ (1 − cL)/[L(2 − cL)(1 + |c|)] and 1 − cL −
τTcL(2 − cL − c ˜Lc) > LK.
4
Hopf Bifurcation
In Theorem 3.3, we have shown that if the transmission delay τT is small enough,
(1.2) can attain global synchronization in spite of the magnitude of internal delay τI. In this section, via bifurcation analysis, we shall show that there exist nontrivial
synchronous periodic solutions for (1.2) induced by internal delay τI. To simplify
the presentation, we consider (1.2) with K = 3 in this section. First, let us consider a circle block matrix
circ(A0, A1, · · · , An−1) = A0 A1 · · · An−1 An−1 A0 · · · An−2 .. . ... ... ... A1 An−1 · · · A0 ,
where Aj, j = 0, 1, · · · , n − 1, is a k × k matrix. Let vj = e
2πj
n i, and define a function
of matrices G(x) = A0+ xA1+ · · · + xn−1An−1. In [33], it is shown that
where Ik is the k × k identity matrix. The linearization of (1.2) at the trivial equilibrium (0, 0, 0, 0, 0, 0) is given by ˙u1(t) = −u1(t) + Lu3(t − τI), ˙u2(t) = −u2(t) + Lu1(t − τI),
˙u3(t) = −u3(t) + Lu2(t − τI) + cLu6(t − τT),
˙u4(t) = −u4(t) + Lu6(t − τI),
˙u5(t) = −u5(t) + Lu4(t − τI),
˙u6(t) = −u6(t) + Lu5(t − τI) + cLu3(t − τT).
(4.2)
The linear system (4.2) is in the form ˙
u(t) = Lut,
which admits the characteristic equation,
∆(λ) := det[λI6− L(eλ·I6)] = 0,
cf [14, 29]. For convenience, we set
d := cL. Then the characteristic equation for (4.2) is
∆(λ) := det 1 + λ 0 −Le−λτI 0 0 0 −Le−λτI 1 + λ 0 0 0 0 0 −Le−λτI 1 + λ 0 0 −de−λτT 0 0 0 1 + λ 0 −Le−λτI 0 0 0 −Le−λτI 1 + λ 0 0 0 −de−λτT 0 −Le−λτI 1 + λ = 0.
Thanks to (4.1), the characteristic equation can be factored as ∆+(λ)∆−(λ) = 0,
∆±(λ) := (1 + λ)2(1 + λ ∓ de−λτT) − L3e−3λτI.
In the following, we shall adopt the delayed Hopf bifurcation theory to seek for nontrivial periodic solution bifurcated from the trivial equilibrium. We substitute λ = iw with w > 0 into ∆±(λ) = 0 and collect the real and imaginary parts to yield
L3sin(3τ
Iw) = (w3 − 3w) ± (w2− 1)d sin(τTw) ± 2d cos(τTw)w,
L3cos(3τ
Iw) = (1 − 3w2) ∓ (1 − w2)d cos(τTw) ∓ 2d sin(τTw)w.
Summing up the square of equations (4.3) gives
Q±(w) = L6, (4.4)
where Q±(w) := Q1(w) ± Q2(w) and Q1(w) := w6 + (3 + d2)w4 + (3 + 2d2)w2 +
d2+ 1 and Q2(w) := d[2 sin(τTw)w5− 2 cos(τTw)w4+ 4 sin(τTw)w3− 4 cos(τTw)w2+
2 sin(τTw)w − 2 cos(τTw)]. Therefore the positive solution of (4.4) corresponds to
the purely imaginary roots of ∆±(w) = 0. Direct computation gives Q
0
1(w) = 6w5+
(12+4d2)w3+(6+4d2)w and Q0
2(w) = d[2τT cos(τTw)w5+(10+2τT) sin(τTw)w4−(8−
4τT) cos(τTw)w3+ (12 + 4τT) sin(τTw)w2− (8 − 2τT) cos(τTw)w + (2 + 2τT) sin(τTw)].
Note that | sin θ| ≤ 1 and | cos θ| ≤ 1, for all θ ∈ R. Therefore, for all w ≥ 0,
Q0±(w) ≥ P (w). (4.5) where P (w) := (6 − 2|d|τT)w5 − |d|(10 + 2τT)w4 + (12 + 4d2 − |d||8 − 4τT|)w3 −
|d|(12 + 4τT)w2+ (6 + 4d2− |d||8 − 2τT|)w − |d|(2 + 2τT). As P is a polynomial, we
set
˜
w := the largest zero of P (w). (4.6) If 6 − 2|d|τT > 0, then P (w) > 0 for all w > ˜w ≥ 0, since P (0) ≤ 0; hence Q±(w)
are increasing, for all w > ˜w. Now, let us introduce the conditions, labelled by “ + ” and “ − ”, to guarantee the existence of purely imaginary roots of ∆+(w) = 0 and
∆−(w) = 0 respectively.
Condition (B1)±: 6 − 2|d|τT > 0, min{Q∓(w) : w ∈ [0, ˜w]} > L6, and
max{Q±(w) : w ∈ [0, ˜w]} < L6.
It can be seen that under condition (B1)+(respectively, (B1)−), there exists a unique
positive zero to Q+(w) (respectively, Q−(w)), say ω+∗ (respectively, ω∗−), and no
positive zeros to Q+(w) (resp Q−(w)). We thus obtain the following lemma.
Lemma 4.1. Under condition (B1)+ (respectively, (B1)−), there exist exactly
one pair of purely imaginary roots, say ±iω+∗ (respectively, ±iω∗−), for character-istic equation ∆(λ) = 0. In particular, ±iω+∗ (respectively, ±iω−∗) are the roots of ∆+(λ) = 0 (respectively, ∆−(λ) = 0).
Remark 4.1. (i) Note that Q+(0) = (1 − d)2 and Q−(0) = (1 + d)2. Therefore
is a necessary condition for (B1)+ (respectively, (B1)−) to hold. Moreover, d =
cL 6= 0 is necessary, as seen from the inequality (4.7); in particular, c = d/L > 0 (respectively, c = d/L < 0) is necessary for condition (B1)+ (respectively, (B1)−).
(ii) The following weaker condition
L6 > (1 − d)2 ( respectively, L6 > (1 + d)2) (4.8) can also provide the existence of zero to Q+(w) = 0 (respectively, Q−(w) = 0), but
the uniqueness of positive zero can not be guaranteed. However, the situation of multiple zeros can be ruled out with assistance of numerical computation. Basically, from (4.8), it can be observed that larger L is advantageous to the occurrence of Hopf bifurcation, hence oscillation.
To find the value of τI such that ±iω+∗ (respectively, ±iω∗−) are the purely
imaginary roots of ∆+(λ) = 0 (respectively, ∆−(λ) = 0), we divide the first equation
of (4.3) by the second one and obtain tan(3τIw) = S±(w)/C±(w),
S±(w) := (w3− 3w) ± (w2− 1)d sin(τTw) ± 2d cos(τTw)w,
C±(w) := (1 − 3w2) ∓ (1 − w2)d cos(τTw) ∓ 2d sin(τTw)w.
Let us define ηk+, k ∈ N, as follows:
ηk±:= 1 3ω±∗ 3π/2 + 2(k − 1)π, if C±(ω±∗) = 0, S±(ω∗±) < 0; π/2 + 2(k − 1)π, if C±(ω±∗) = 0, S±(ω∗±) > 0; tan−1(S±(ω∗±)/C±(ω∗±)) + 2kπ, if C±(ω±∗) > 0; tan−1(S±(ω∗±)/C±(ω∗±)) + (2k − 1)π, if C±(ω±∗) < 0. (4.9) Herein, ηk+ (respectively, η−k) is positive and the critical value of bifurcation param-eter with respect to τI, at which ∆(λ) = 0 has exactly one pair of purely imaginary
roots ±iω∗+ (respectively, ±iω−∗). To apply the Hopf bifurcation theory, it remains to verify the transversality condition:
Condition (B2)±: [R±(ω∗±, ηk±)] 2+ [I
±(ω±∗, η±k)]
2 6= 0, and Λ
where R±(ω, τI) := [−3 − 9τI± 3dτIcos(τTω) ∓ dτT cos(τTω)]ω2 ±(2 + 6τI + 2τT)d sin(τTω)ω + 3 + 3τI± (−2 − 3τI+ τT)d cos(τTω) I±(ω, τI) := −3τIω3± (τT − 3τI)d sin(τTω)ω2 +[6 + 9τI± (−2 − 6τI+ 2τT)d cos(τTω)]ω ± (2 + 3τI − τT)d sin(τTω) Λ±(ω) := [9 ± 3τTd cos(τTω)]ω4± (15 + 3τT)d sin(τTω)ω3 +[9 + 6d2± (−12 + 3τT)d cos(τTω)]ω2+ (3 + 3τT)d sin(τTω)ω.
Proposition 4.2. Assume that conditions (B1)+ and (B2)+ (respectively, (B1)−
and (B2)−) hold for some fixed k ∈ N. The Hopf bifurcation occurs at τI = ηk+
(respectively, ηk−), and a periodic orbit is bifurcated from the zero solution of (1.2). Proof. We only prove the first case. The latter one can be verified similarly. First, we derive that ∂ ∂λ∆+(λ)|λ=iω∗+,τI=η+k = {2(1 + λ)2(1 + λ − de−λτT) + (1 + λ)2(1 + dτ Te−λτT) + 3τIL3e−3λτI}|λ=iω∗+,τI=η+k = {2(1 + λ)2(1 + λ − de−λτT) + (1 + λ)2(1 + dτ Te−λτT) +3τI(1 + λ)2(1 + λ − de−λτT)}|λ=iω∗ +,τI=ηk+ = R+(ω∗+, η + k) + iI+(ω∗+, η + k).
Thus, ∂λ∂∆+(λ)|λ=iω+∗,τI=η+k 6= 0, under condition (B2)+; hence there exists some
δ > 0 and a smooth function λ : (ηk+− δ, η+k + δ) → C such that ∆+(λ(η+k)) = 0 and
λ(ηk+) = iω+∗. Differentiating ∆+(λ(τI)) = 0 with respective to τI at τI = ηk+, we get
λ0(ηk+) = Q1+ iQ2 W1+ iW2 , where Q1 = (6 − 3d cos(τTω+∗))(ω ∗ +)2+ 3d sin(τTω+∗)ω ∗ +, Q2 = 3(ω∗+) 3+ 3d sin(τ Tω+∗)(ω ∗ +) 2+ [−3 + 3d cos(τ Tω∗+)]ω ∗ +, W1 = −3ηk+(ω+∗) 2 + (τT − 3ηk+)d sin(τTω+∗)ω ∗ ++ (−2 − 3η + k + τT)d cos(τTω+) + 3 + 3η + k, W2 = [3 + 6η+k + (τT − 3ηk+)d cos(τTω+∗)]ω ∗ ++ (2 + 3η + k − τT)d sin(τTω+∗).
Therefore, Reλ0(ηk+) = {[9 + 3τTd cos(τTω∗+)](ω ∗ +) 4+ (15 + 3τ T)d sin(τTω+∗)(ω ∗ +) 3 +[9 + 6d2+ (−12 + 3τT)d cos(τTω+∗)](ω ∗ +) 2 + (3 + 3τT)d sin(τTω∗+)ω ∗ +}/(W 2 1 + W 2 2) 6= 0, under condition (B2)+.
As mentioned in Remark 4.1, (1 − d)2 < L6 and c > 0 are necessary for condition (B1)+. In such a situation, condition (S1) may hold under the same
parameters; but it is impossible that condition (S2) or (S3) also holds. From this view point, positive coupling strength c is advantageous to the birth of synchronous oscillation of (1.2). On the other hand, similarly, we observe that negative coupling strength c is advantageous to the birth of anti-phase oscillation. Motivated by these observations, Theorems 3.3, 3.7, and Proposition 4.2, we summarize the following conclusion.
Theorem 4.3. Assume that conditions (S1) (respectively, (A1)), (B1)+, and (B2)+
(respectively, (B1)−, and (B2)−) hold for some fixed k ∈ N. Then there exists a
synchronous (respectively, anti-phase) periodic solution bifurcated from the trivial equilibrium, at internal delay τI = ηk+(respectively, τI = ηk−) for the coupled K-loop
(1.2).
5
Numerical illustrations
We provide two numerical examples to illustrate the present theory.
Example 1. Consider the coupled 3-loops (1.2) with g(ξ) = tanh(0.999ξ), τI =
11.2, τT = 0.001, and c = 400/999. Then (1.2) satisfies condition (S1), as c =
400/999 > 0; τTcL(2 + cL)(1 + c) = 0.001344384384 < c; 1 + c ˜Lc− τTcL(2 + cL +
c ˜Lc) = 1.085305978 > L3. By Theorem 3.3, (1.2) attains global synchronization.
It can be verified that condition (B1)+ holds by direct computation. Moreover,
ω∗+ = 0.5128010915; η1+ = 7.092622146, η2+ = 11.17685162. It can also be justified that condition (B2)+, k = 1, 2, hold by direct computation. By Theorem 4.3, there
Figure 3 illustrates that the solution of (1.2) tends to a synchronous periodic orbit as t → ∞. Figure 4 provides the the dynamics for each component of the solution in figure 3; in the panels, six different colors represent the evolutions of six different components.
Example 2. Consider the coupled 3-loops (1.2) with L = 1.02. As c = 0.5 τI = 2
and τT = 0.01, it can be checked that (1.2) satisfies condition (S1) as ˜Lc= g
0
(1+c) = 0.1745496000; 1/[L(2+cL)(1+c)] = 0.2603963232; 1+c ˜Lc−τTcL(2+cL+c ˜Lc)−L3 =
0.0128206985 > 0. In such a situation, figure 5 illustrates that the solution of (1.2), evolved from (1.6 + 0.1 sin t, −1.6 + 0.1t, 1.6 + 0.1 sin t · cos t, −1.6 + 0.1 sin t, −1.6 + 0.1t, −1.6+0.1 sin t·cos t) is synchronized. If we consider (1.2) with smaller c = 0.001 instead, which does not satisfy condition (S1), figure 6 illustrates that the solution of (1.2) converges to some asynchronous equilibrium point. If we consider (1.2) with the same parameters but with larger τT = 100 instead, which does not satisfy
condition (S1), figure 7 illustrates that the solution of (1.2) appears to be anti-phase, hence not synchronized. If we increase the coupling strength to c = 20 and still hold τT = 100, the system still exhibits anti-phase, hence not synchronized, as shown in
figure 8. −1.5 −1 −0.5 0 0.5 1 1.5 −1 0 1 −1.5 −1 −0.5 0 0.5 1 1.5 x1(t) x2(t) x3 (t) −1.5 −1 −0.5 0 0.5 1 1.5 −1 0 1 −1.5 −1 −0.5 0 0.5 1 1.5 y1(t) y2(t) y3 (t)
Figure 3: An orbit of (1.2) with g(ξ) = tanh(0.999ξ), τI = 11.2, τT = 0.001, and
c = 400/999, which is evolved from φ(t) = (sin t, cos t, t, t sin t, −t, sint · cos t). There exists a synchronous limit cycle.
6
Discussions and further results
As mentioned earlier, coupled network system can exhibit a variety of interesting behaviors. We plan to depict the dynamical scenario for the coupled K-loops (1.2) under the influence of coupling strength, the gain of the activation function, the internal delay, and the transmission delay. Let us first mention some properties of the single K-loop (1.1). Notably, the dynamics of (1.1) without internal delay τI
has been studied extensively in [3]. By similar approach as Lemma 2 in [12] and Lemma 4.1 in [29], it can be shown that the trivial equilibrium of (1.1) is stable for all τI ≥ 0 if L ≤ 1. On the other hand, there exist periodic solutions bifurcated
from the zero solution of (1.1) if L > 1. Therefore, the dynamical scenarios for cases L ∈ (0, 1) and L ∈ (1, ∞] are rather different. Below we shall discuss the two cases: L ∈ (0, 1) and L ∈ (1, ∞], for the coupled loops separately. Notice that the trivial equilibrium for the coupled loops (1.2) can become unstable as there exist periodic solutions bifurcated from the equilibrium, cf Theorem 4.3. This already shows an effect from the coupling between these two loops.
Notably, what we have derived for global synchronization, global convergence to the origin are theories with sufficient conditions. When we say that (1.2) does not admit certain dynamics (such as synchronization), we may need computer sim-ulations to support the arguments. In addition, caution must be used if saying that a system can not be synchronized when running a numerical simulation. It is not assured how long a simulation should last to exclude the possibility of synchroniza-tion. On the other hand, anti-phase is an evidence of desynchrony that one can assure from analysis or numerical simulation.
Effect of coupling strength. Let us first consider the case that L ∈ (0, 1), c ≥ 0. It can be seen from Theorem 3.6 that if c, the coupling strength between two loops, is sufficiently small so that
c ≤ 1/L − 1, (6.1)
the coupled loops (1.2) attain global convergence to the trivial equilibrium (hence global synchronization) in spite of delay magnitude of τI and τT. In addition, if c is
larger so that (6.1) fails to hold, the coupled loops (1.2) can attain global synchro-nization if τT is small enough. Such an observation follows from that for arbitrarily
large c > 0, condition (S1) for Theorem 3.3 holds if τT is small enough. On the other
under certain criteria. It can be seen that the dominant condition (B1)+ can not
hold if c = 0 or is too small, cf Remark 4.1. On the other hand, for some suitable magnitude of c (not too large) such that (B1)+ holds, the nontrivial synchronous
periodic solutions of coupled loops can be induced by internal delays. Now, it is natural to ask what will occur if magnitude of c is quite large. In [26], it is found that system (1.2) with sufficiently large c achieves global convergence to nontrivial synchronous equilibria (hence globally synchronized) in spite of internal lag (τI) if
transmission lag (τT) is small enough. This result has justified the observation from
the numerical simulation in [3].
Accordingly, we summarize a dynamical scenario for (1.2) as c varies from zero to ∞:
global convergence to zero
→ global synchronization (τT small)
→ global synchronization with synchronous oscillation (induced by internal delays)
→ global convergence to nontrivial synchronous equilibria (τT small).
Notice that the effect of positive coupling strength on synchronization and negative coupling strength on anti-phase motion are counterparts to each other for the coupled loops. Indeed, if L ∈ (0, 1), as c varies from 0 to −∞, (1.2)B goes
through global convergence to zero → birth of anti-phase oscillation (induced by internal delays) → global convergence to nontrivial antisynchronous equilibria.
Roughly speaking, it is easier for the coupled loops (1.2) to attain global syn-chronization for the case L ∈ (0, 1) than L ∈ [1, ∞).
In the case that L ∈ [1, ∞), basically, wether if the coupled loops (1.2) can attain synchronization strongly relies on the interaction between two coupled loops. Observe condition (S1) in Theorem 3.3, it can be seen that coupling strength c is necessary (can not be zero), and τT must be small enough. It has been illustrated
in Example 2 that (1.2) fails to be synchronized if coupling strength is too small, and can be synchronized if coupling strength is large. However, it can be seen that large magnitude of c does not always favor condition (S1), due to the nonlinearity of activation function g. In the literatures, e.g., [22] concludes that oscillator with linear coupling achieves synchronization if coupling strength is large enough. There-fore, it is appealing to investigate whether if large magnitude of coupling strength
can be advantageous to synchronization for nonlinearly coupled oscillators. In our numerical computation in Example 2, for some situation as transmission is too large, large coupling strength c still can not synchronize the oscillators.
Effect of delays. As previous arguments, in some parameter regions of L and c, it is necessary that transmission delay is small enough for (1.2) to be synchronized. Example 2 has illustrated that the coupled loops can be synchronized as transmission lag τT is small enough, and the coupled loops can not be synchronized as τT is too
large. Notice that the coupled loops can not regain synchronization even if we increase the coupling strength c to be arbitrarily large, see Example 2. Restated, by our numerical evidence, strong coupling strength c can not resynchronize the system after the asynchrony has been induced by transmission delay. Under our formulation, whether if the coupled loops (1.2) can be synchronized does not depend on internal delays, cf Theorems 3.3, 3.4. But transmission delay plays a role in synchronization. It is then natural to ask how the internal delay affect the dynamics in (1.2). Following our result in Theorem 4.3, it can be seen that large magnitude of internal delay can create oscillation.
We have presented an approach to study global synchronization and global convergence to the equilibria for two delayed neural network loops coupled with transmission delay. We also demonstrated that the asymptotically synchronous phases could be a single equilibrium, multiple equilibria, and a periodic orbit. The conditions for the theory involve the gain and bound of activation function, the coupling strength, transmission and internal delays. Due to limitation of article size, the result on global convergence to multiple synchronous equilibria is summarized in another manuscript [26]. By similar formulation, we can also establish anti-phase periodic solutions bifurcated from the trivial equilibrium induced by transmission delay.
Acknowledgments. This work is partially supported by The National Sci-ence Council and The National Center of Theoretical SciSci-ences, of R.O.C. on Taiwan.
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0 100 200 300 400 500 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 time t solution x1 ,y1 x1(t) y1(t) 0 100 200 300 400 500 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 time t solution x2 ,y2 x2(t) y2(t) 0 100 200 300 400 500 −1 −0.5 0 0.5 1 1.5 time t solution x3 ,y3 x3(t) y3(t)
0 20 40 60 80 100 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 time t solution x3 ,y3 x3(t) y3(t) 0 20 40 60 80 100 −2 −1.5 −1 −0.5 0 0.5 1 time t solution x2 ,y2 x2(t) y2(t) 0 20 40 60 80 100 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 time t solution x1 ,y1 x1(t) y1(t)
Figure 5: Evolutions of components (xi(t), yi(t)) for the solution of (1.2) with
g(ξ) = tanh(1.02ξ), τI = 2, τT = 0.01, and c = 0.5, starting from φ(t) =
(1.6 + 0.1 sin t, −1.6 + 0.1t, 1.6 + 0.1 sin t · cos t, −1.6 + 0.1 sin t, −1.6 + 0.1t, −1.6 + 0.1 sin t · cos t). It appears to be synchronized.
0 50 100 150 200 250 300 −2 −1.5 −1 −0.5 0 0.5 1 time t solution x2 ,y2 x2(t) y2(t) 0 50 100 150 200 250 300 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 time t solution x1 ,y1 x1(t) y1(t) 0 50 100 150 200 250 300 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 time t solution x3 ,y3 x3(t) y3(t)
Figure 6: Evolutions of components (xi(t), yi(t)) for the solution of (1.2) with
g(ξ) = tanh(1.02ξ), τI = 2, τT = 0.01, and c = 0.001, starting from φ(t) =
(1.6 + 0.1 sin t, −1.6 + 0.1t, 1.6 + 0.1 sin t · cos t, −1.6 + 0.1 sin t, −1.6 + 0.1t, −1.6 + 0.1 sin t · cos t). The solution converges to asynchronous steady state.
0 500 1000 1500 2000 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 time t solution x3 ,y3 x3(t) y3(t) 0 500 1000 1500 2000 −2 −1.5 −1 −0.5 0 0.5 1 time t solution x2 ,y2 x2(t) y2(t) 0 500 1000 1500 2000 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 time t solution x1 ,y1 x1(t) y1(t)
Figure 7: Evolutions of components (xi(t), yi(t)) for the solution of (1.2) with
g(ξ) = tanh(1.02ξ), τI = 2, τT = 100, and c = 0.5, starting from φ(t) =
(1.6 + 0.1 sin t, −1.6 + 0.1t, 1.6 + 0.1 sin t · cos t, −1.6 + 0.1 sin t, −1.6 + 0.1t, −1.6 + 0.1 sin t · cos t). The system tends to an anti-phase motion.
0 500 1000 1500 2000 −25 −20 −15 −10 −5 0 5 10 15 20 25 time t solution x3 ,y3 x3(t) y3(t) 0 500 1000 1500 2000 −2 −1.5 −1 −0.5 0 0.5 1 time t solution x2 ,y2 x2(t) y2(t) 0 500 1000 1500 2000 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 time t solution x1 ,y1 x1(t) y1(t)
Figure 8: Evolutions of components (xi(t), yi(t)) for the solution of (1.2) with g(ξ) =
tanh(1.02ξ), τI = 2, τT = 100, and c = 20, starting from φ(t) = (1.6+0.1 sin t, −1.6+
0.1t, 1.6+0.1 sin t·cos t, −1.6+0.1 sin t, −1.6+0.1t, −1.6+0.1 sin t·cos t). The system retains an anti-phase motion, even with larger coupling strength c.
