Phase transition of oscillators and travelling waves in a class of relaxation systems

The main purpose of this article is to investigate the phase transition of oscillation solutions and travelling wave solutions in a class of relaxation systems as follows 
\begin{eqnarray} 
\left\{ 
 \begin{array}{ll} 
 \frac{\partial u}{\partial t}=\pm u(u-a)(u-b)-v+D \frac{\partial ^{2}u}{\partial^{2} x},~~~a\neq b, 
 \\\frac{\partial v}{\partial t}=\varepsilon( mu + nv + p ), ~~~~0<\varepsilon\ll 1,\nonumber 
\end{array} 
\right. 
\end{eqnarray} 
where $a,b,m,n,p$ are parameters in this system. By using the orbit analysis method of planar dynamical system and the homoclinic bifurcation theory, the phase transitions of the solitary oscillators, kink oscillators, periodic oscillators and travelling waves in the relaxation system above are studied. Various critical parameters of the phase transition are obtained under different parametric conditions, while various sufficient conditions to guarantee the existence of the above oscillation solutions and travelling waves are given. As some applications, this paper studied the FitzHugh-Nagumo equation, the van der Pol-equation and the Winfree generic system.


1.
Introduction. The relaxation oscillator is a type of limit cycle which repeatedly alternates between two states and also alternates the speeds between fast and slow. It is applied to dynamical systems in diverse areas of science such as geothermal geysers, networks of firing nerve cells, thermostat controlled heating systems, chemical reactions and the beating human heart, see [2], [17], [13]. The general form of the oscillation system is widely studied in [14] and [8], which is shown as follows where u is called fast variable, and v is slow variable, λ = λ(λ 1 , λ 2 , λ 3 , . . .) is the parameter in the system (1). More detailed introductions and results of (1) can be found in [11] and [6]. In this paper we shall focus on the relaxation oscillators of the following class ∂u ∂t = ±u(u − a)(u − b) − v + D ∂ 2 u ∂ 2 x , a = b, ∂v ∂t = ε(mu + nv + p), 0 < ε 1. (2) The system (2) generates from many famous relaxation oscillation systems including the Fitzhugh-Nagumo model (see [11], [5]), the van der-Pol equation (see [14]) and the Winfree generic system (see [17]), moreover some physical phenomena tell us there are relaxation oscillators in these systems under some parametric conditions. As we know, the system with homoclinic orbit or heteroclinic orbit is structurally unstable, so we can deduced that the solitary oscillation solutions and the kink oscillation solutions may disappear for the perturbations of the system (2). Based on the orbit analysis theories including the homoclinic bifurcation theory we established a method to study the phase transition of the oscillation solutions and travelling waves in (2). Various critical parameters of the phase transition are obtained, and also various sufficient conditions to guarantee the existence of the different types of oscillation solutions are given. The main conclusions in this paper is shown as follows: • (1)In system (1) let the parameter D = 0 and the parameters b, m, n, p be constant. If a is an adjusted parameter for the system (2), then a = 0 is a critical point of the system (1), and there is a fast-slow relaxation oscillator bifurcates from the point a = 0.
• (2)In system (1) let the parameter D = 0 and the parameters a, b be constant, and m, n, p are the adjusted parameters of the system (2), with the application to Andronov-Leontovich Theorem, there are two critical points p 0 = mu A +nA n and p 1 = mu B +nA n in this case, and there is a fast-slow relaxation oscillator bifurcates from the point p 0 and p 1 .
• (3)Note D = 0, the parameters a, b be constant, if m, n are the adjusted parameters of the system (2), then some bifurcation theorems of the oscillators are also obtained in this case.
• (4)Let D = 0, let the parameters b be constant, and a is the an adjusted parameter of the system (2), then the bifurcation theorem of the travelling waves is given and we applied it to study the Fitzhuge-Nagumo equation. As the applications of the method we obtained, we study the following equations, they are all famous relaxation oscillation systems in nonlinear science. The theorems in ((1),(2),(3),(4) are applied to the model FHN model, Van der Pol equation, and the Winfree generic system.

2.
The phase transition of oscillators and travelling waves.
2.1. Preliminary knowledge. Before presenting our results, we give some background ideas and notations, which can also be found in [14], [1], [15]. Relaxation oscillators are characterized by two alternating processes on different time scales: a long relaxation period during which the system approaches an equilibrium point, alternating with a short impulsive period in which the equilibrium point shifts, the typical relaxation oscillators approach u(x, t) is a type of limit cycle of (1) which repeatedly alternates between two states and also alternates the speeds between fast and slow. Suppose that u(x, t) is a continuous solution of (1) for t ∈ (−∞, +∞), and lim t→∞ u(x, t) = c, lim t→−∞ u(x, t) = d, where c and d are both equilibrium states of (1). It is well known that where λ is a parameter, f and g are smooth functions, and (u, v) ∈ R 2 . Denote by ϕ t λ the flow of the (3), let (u 0 , v 0 ) be an equilibrium of the system (3) and it is a saddle point, let the eigenvalues be µ 1 (0), µ 2 (0) at λ = 0, and µ 2 (0) < 0 < µ 2 (0). The saddle quantity σ 0 (σ 0 = µ 1 (0) + µ 2 (0)) of a hyperbolic equilibrium is the sum of the real parts of its eigenvalues, where µ 2 is an unstable eigenvalue and µ 1 is a stable eigenvalue.
An orbit Γ 0 starting at a point (u, v) in R 2 is called homoclinic orbit to the equilibrium point (u 0 , v 0 ) of (3) at λ = 0 if lim t→±∞ ϕ t λ (u, v) = (u 0 , v 0 ). Generically, we assume σ 0 = 0 in the system (3). A homoclinic orbit to a hyperbolic equilibrium of (3) is structurally unstable, which means that the phase portrait in the neighborhood of Γ 0 ∪ (u 0 , v 0 ) becomes topological nonequivalent to the original one, as we shall see, in [7], it is shown that the homoclinic orbit simply disappears for perturbations of the system.
The presence of a homoclinic orbit implies a global codimension-one bifurcation of (3)at λ = 0, since the homoclinic orbit disappears for all sufficiently small perturbation. Moreover, the disappearance of a homoclinic orbit leads to the creation or destruction of one (or more) limit cycle nearby. When such a cycle approaches the homoclinic orbit Γ 0 as λ → 0, its period tends to infinity, which is shown in Figure 2, a homoclinic orbit forms at λ = 0, and a periodic orbit exists for λ > 0 but not for λ < 0. As λ → 0 + , the period T of the periodic orbit diverges to infinity. The Andronov-Leontovich Theorem (see [2], [13], [16]) is a basic tool to study the bifurcation of the oscillators, which is shown as follows.
Theorem 2.1. (Andronov-Leontovich, 1971 ) For any generic one-parameter system (3) having a saddle equilibrium point (u 0 , v 0 ) with a homoclinic orbit Γ 0 at λ = 0, the separated function is β(x)( which is defined in [14]), if the following assumptions hold true (A1)The saddle quantity σ 0 = µ 1 (0) + µ 2 (0) = 0; (A2)β (0) = 0; then there exists a neighbourhood of Γ 0 ∪ (u 0 , v 0 ) in which a unique limit cycle L λ bifurcates from Γ 0 as λ passes through zero. 2.1.2. The stability of the orbits. In the system (3), at the case λ = 0, if the the saddle quantity σ 0 of the homoclinic orbit Γ 0 is negative, then Γ 0 is a stable orbit, in the other word, it can attract the orbits around it, and also the limit cycle L λ bifurcates from Γ 0 is stable. However, if the saddle quantity σ 0 is positive, then Γ 0 is an unstable orbit, and the limit cycle L λ bifurcates from Γ 0 is unstable. The stability of the orbits leads to the stability of the oscillators, which means, the stable oscillator attracts the oscillators around it as t → +∞, the unstable oscillator repels the oscillators around it as t → +∞.
2.2. The phase transition of oscillation solutions. In the case D = 0, we study the phase transition of oscillation solutions in the following two dynamical systems : And In the system (4), remark that the curve , and the line = (u, v) : mu + nv + p = 0.
In the system (4), remark that f (u) = −u 3 + (a + b)u 2 − abu, u 1 and u 2 are the roots of the equation f (u) = 0, where Note f (u 1 ) = A = −u 3 1 + (a + b)u 2 1 − abu 1 and f (u 2 ) = B = −u 3 2 + (a + b)u 2 2 − abu 2 , u A is a root of the equation f (u) = A such that u A = u 1 , and also u B is a root of the equation f (u) = B such that u B = u 2 . Then we have the following theorems: Theorem 2.2. In the system (4), let m, n, a, b be constant, and m > 0, n > 0, a 2 + b 2 = 0, if p is an adjusted parameter in the system (4), then we have the following conclusions: (1)If the system (4) satisfies the following conditions: is the critical point of the system (4), namely if p = p 0 , there is a solitary oscillator solution in (4), if p < p 0 , there is a fast-slow relaxation oscillator bifurcates from the solitary oscillator solution in (4); (2)If the system (4)satisfies the following conditions: is the critical point of the system (4).If p = p 1 , there is a solitary oscillator solution in (4),if p > p 1 , there is a fast-slow relaxation oscillator solution bifurcates from the solitary oscillator solution in (4) Proof. We applied the Theorem 2.1 to proof (1), the proof of (2) in this theorem is similar to (1). Figure 3 is the phase plane of the system (2.2), we see the curve Γ divided the uv plane into two parts, it is easy to check that on the upper part of the curve Γ, du dt < 0 holds true, and on the lower half part of the curve Γ, du dt > 0 holds true. On the upper part of the line , dv dt > 0 holds true, and on the lower half part of the curve , dv dt < 0 holds true, then it is easy to find that there is a homoclinic orbit in (4)   It is easy to check that the point H is a saddle point in (4), the saddle quantity The separated function is β(p−p 0 ) = h 2 −h 1 , it is obvious that β (0) > 0. With the application of Andronov-Leontovich Theorem, the presence of a homoclinic orbit implies a global codimension-one bifurcation of (4) at p = p 0 , since the homoclinic orbit disappears for all sufficiently small |p − p 0 |. Moreover, the disappearance of a homoclinic orbit leads to the creation or destruction of one limit cycle nearby such that the cycle approaches the homoclinic orbit as p → p 0 , its period tends to infinity. It implies that there is a fast-slow relaxation oscillator in (4) if p < p 0 .  Which leads to there is a solidary oscillator solution in system (4), and also in case a < 0, O is still an equilibrium point, the flow from the point A arrived at the point B fast on the line AB, and then come to the point C slowly on the curve Γ, later it leaves the point C to arrived at the point D fast on the line CD, and then come back to the point A slowly on the curve Γ, that is, there is a limit cycle ABCD in (4) in figure 5, which leads to there is a periodic oscillator solution in system (4).  Proof. If 0 < − m n < B u2 , then figure 6 is the phase plane of the system (2.2) in this case, it is easy to check that on the upper part of the curve Γ, du dt < 0 holds true, and on the lower half part of the curve Γ, du dt > 0 holds true. On the upper part of the line , dv dt < 0 holds true, and on the lower half part of the curve , dv dt > 0 holds true, M 1 and M 2 are two equilibrium points, the flow from the point M 1 arrived at the point on curve Γ fast, which is shown in figure 6, then it arrived at the point M 2 fast on the curve Γ, that is, there is a heteroclinic orbit in (4), which leads to the existence of a kink oscillator solution in (4). Proof. Case 1. − p m < u 1 . Figure 7 is the phase plane of the system (2.2) in this case, it is easy to check that on the upper part of the curve Γ, du dt < 0 holds true, and on the lower half part of the curve Γ, du dt > 0 holds true. On the left part of the line , dv dt < 0 holds true, and on the right half part of the curve , dv dt > 0 holds true. It is easy to find that there is no homoclinic orbit or periodic orbit in Figure  7. Figure 8 is the phase plane of the system (2.2) in this case, it is easy to check that on the upper part of the curve Γ, du dt < 0 holds true, and on the lower half part of the curve Γ, du dt > 0 holds true. On the left part of the line , dv dt < 0 holds true, and on the right half part of the curve , dv dt > 0 holds true. It is easy to find that there is a homoclinic orbit in Figure 8. Figure 9 is the phase plane of the system (2.2) in this case, it is easy to check that on the upper part of the curve Γ, du dt < 0 holds true, and on the lower half part of the curve Γ, du dt > 0 holds true. On the left part of the line , dv dt < 0 holds true, and on the right half part of the curve , dv dt > 0 holds true. It is easy to find that there is a periodic orbit in Figure 9.  3. The applications. Based on the theory we established in this paper, we studied the phase transition of the oscillators and travelling waves in the following systems, all of them are famous Relaxation Oscillation systems.
3.1. The application to Fitzhugh-Nagumo equation. The general Fitzhugh-Nagumo model (see [5]) is shown as follows, where u is a function of x and t. The model is a family of two-variable reaction diffusion equations that capture the essential properties of spatially distributed excitable media, which also described the propagation of electrical signals in nerve axons and other biological tissues (see [11], [6], [4]). Here u(x, t) is the voltage inside the axon at position x ∈ R and time t. The first equation in (15) is Kirchhoffs law, expressing that the change ∂ 2 u ∂ 2 x of the current ∂u ∂x along the axon is compensated by the currents passing through the cell membrane: a capacitance based current ∂v ∂t and a resistance based current −u(u − g)(u − 1) − v. v describes a part of the transmembrane current that passes through slowly adapting ion channels. The oscillators and travelling waves of Fitzhugh-Nagumo equation is widely studied in many papers, including [3], [15], [9], [4], [10].
In [17], the original Fitzhugh-Nagumo model was formulated as a reduction of the Hodgkin-Huxley model without the diffusion effect of the u, so firstly we studied the following system Let Note that u A is a root of the equation f (u) = −u 3 1 + (g + 1)u 2 1 − gu 1 such that u A = u 1 , u B is a root of the equation f (u) = −u 3 2 + (g + 1)u 2 2 − gu 2 such that u B = u 2 . Then the Corollary 3 follows from the Theorem 2.3.
Corollary 3. In the system (16), let e, r, k be constant and e > 0, k = 0, r > 0. If e r > B u2 then g is the an adjusted parameter in the system (16), we have the following conclusions: (1)If g > 0, there is no oscillator solution in (16); (2)If g = 0, there is a solidary oscillator solution in (16); (3)If g < 0, there is a fast-slow relaxation oscillator solution in (16); Corollary 4. In the system (16), let r, g, e be constant, and e > 0, r < 0. If k is an adjusted parameter of the system (16), then we have the following conclusions: (1)If the system (16)satisfies the following conditions: Then k = k 0 = −eu A +rA r is the critical point of the system (16). If k = k 0 , there is a solitary oscillator solution in (16), if k < k 0 , there is a fast-slow relaxation oscillator solution bifurcates from the solitary oscillator solution in (16); (2)If the system (16)satisfies the following conditions: then k = k 1 = −eu B +rB r is the critical point of the system (16). If k = k 1 , there is a solitary oscillator solution in (16), if k > k 1 , there is a fast-slow relaxation oscillator that bifurcates from the solitary oscillator solution in (16) The Corollary 4 follows from the Theorem 2.2.
The Corollary 5 follows from the Theorem 2.4. The Corollary 6 follows from the Theorem 2.5. In the next section, we will study the bifurcation of travelling waves in system (15) with D = 0 (which is studied in [6], [15], [18]), the method can be found in [17], some other papers including [19] also studied the bifurcation of travelling waves.
Then we look for A and c to make (37) satisfying (35), by directing calculation, we obtained where A and c (the wave speed) are uniquely determined: ).