Bifurcation of traveling waves in extrinsic semiconductors

Physica D 144 (2000) 1–19

Bifurcation of traveling waves in extrinsic semiconductors B. Katzengruber, M. Krupa, P. Szmolyan∗ Institut für Angewandte und Numerische Mathematik, Technische Universität, Vienna, A-1040, Austria Received 25 August 1997; received in revised form 14 January 2000; accepted 31 January 2000 Communicated by C.K.R.T. Jones

Abstract We analyze the bifurcation of traveling waves in a standard model of electrical conduction in extrinsic semiconductors. In scaled variables the corresponding traveling wave problem is a singularly perturbed nonlinear three-dimensional o.d.e. system. The relevant bifurcation parameters are the wave speed s and the total current j . By means of geometric singular perturbation theory it suffices to analyze a two-dimensional reduced problem. Depending on the relative size of s and a dimensionless small parameter β different types of traveling waves exist. For 0 ≤ s  β the only waves are fronts corresponding to heteroclinic orbits. For β  s similar fronts — but with left and right states reversed — exist. The transition between these regimes occurs for s = O(β) in a complicated global bifurcation involving a Hopf bifurcation, bifurcation of multiple periodic orbits, and heteroclinic and homoclinic bifurcations. We present a consistent bifurcation diagram which is confirmed by numerical computations. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Traveling waves; Singular perturbations; Global bifurcations; Semiconductors

1. Introduction Semiconductor instabilities — based on nonlinear generation and recombination of charge carriers — provide a rich source of dynamical systems with complicated spatio–temporal behavior. We refer to [18] as a general reference for carrier transport in semiconductors and to [16] for background material and further references on generation and recombination instabilities. In many cases this dynamical behavior is caused by moving high-field domains in a bulk semiconductor. The best known effect of this kind is the Gunn-effect in semiconductor materials which exhibit bulk negative differential resistivity [14,17]. In a mathematical idealization the high-field domains correspond to traveling wave solutions of the underlying p.d.e. system on the infinite line (see [1,19,21]). These traveling wave solutions are crucial in an asymptotic analysis of the corresponding initial boundary-value problem on finite intervals [1,22]. In a series of papers [2,3,23] similar questions were considered for a standard dynamical model of electrical transport in an extrinsic p-type semiconductor which includes effects of electric field dependent impurity capture, impact ionization, and free carrier velocity saturation. In scaled dimensionless variables the governing ∗

Corresponding author.

0167-2789/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 0 0 ) 0 0 0 3 0 - 0

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equations are At = βκ(E)P (ρ(E) − A),

(1.1a)

Pt = (δPx − v(E)P )x + At ,

(1.1b)

Ex = P − A,

(1.1c)

with A the concentration of ionized acceptors minus a uniform constant donor concentration, P the free hole concentration, and E the electric field. Differentiation with respect to the spatial variable x ∈ R and time t > 0 is denoted by subscripts x and t, respectively. The known functions κ and ρ in Eq. (1.1a) can be expressed in empirical functions of E describing the rate of impact ionization of neutral acceptors and the rate of recombination onto ionized acceptors. Eq. (1.1b) is the charge continuity equation for the free hole concentration; the function v is the velocity of the free holes due to their acceleration by the electric field. Eq. (1.1c) is a Poisson equation which relates the electric field E and the space charge P − A. The small parameters β and δ are due to the scaling; typical orders of magnitude are β = 10−5 , δ = 10−2 . For details on the model and the scaling we refer to [2]. Our analysis relies on a few properties of the functions κ, ρ and v, which appear to hold for the experimental functions. These assumed properties will be specified below. Traveling waves are solutions of (1.1a)–(1.1c) which are functions of ξ = x − st, where s is the wave speed. By inserting E(x, t) = E(ξ ), A(x, t) = A(ξ ), P (x, t) = P (ξ ) into (1.1a)–(1.1c) and by integrating (1.1b) once, we obtain the three-dimensional autonomous system s A˙ = βκ(E)P (A − ρ(E)),

δ P˙ = v(E)P − j − s(P − A),

E˙ = P − A,

(1.2)

governing all traveling waves of (1.1a)–(1.1c). The integration constant j is the total current. We restrict ourselves to waves traveling to the right and standing waves, i.e. s > 0 and s = 0, respectively. In this case the interesting dynamic behavior of (1.2) occurs on an appropriate open set U ⊂ R+ × R+ × R+ . Three basic types of traveling waves which occur for (1.1a)–(1.1c) are periodic wave trains, fronts, and pulses. These waves correspond to periodic, heteroclinic, and homoclinic orbits of (1.2), respectively. Recall that heteroclinic orbits are trajectories which have two distinct equilibria as their α- and ω-limit sets and homoclinic orbits are trajectories whose α- and ω-limit sets consist of the same equilibrium. Additionally we find traveling waves corresponding to connections between an equilibrium and a periodic orbit and to a connection between periodic orbits. The equilibria of (1.2) correspond to the solutions of the equations P = A,

A = ρ(E),

v(E)P = j,

(1.3)

which reduce to the single equation ρ(E)v(E) = j.

(1.4)

In the following we discuss qualitative properties of the smooth functions κ, v, ρ. The function κ is positive. The velocity v is a monotonically increasing function which satisfies v(0) = 0 and saturates at a value vs which equals 1 due to the scaling. Teitsworth [23] obtains formulas for the experimental functions ρ, v and κ. Teitsworth [23] argues that (ρv) has the following property: (P) The function (ρv)(E) = ρ(E)v(E) has a unique local maximum at E12 and a unique local minimum at E23 with E12 < E23 . Both E12 and E23 are non-degenerate, i.e. (ρv)00 (E∗ ) 6= 0,

E∗ ∈ {E12 , E23 }.

The function (ρv) has no other critical points and neither j12 = (ρv)(E12 ) nor j23 = (ρv)(E23 ) corresponds to a global extremum.

B. Katzengruber et al. / Physica D 144 (2000) 1–19

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Fig. 1. Qualitative behavior of (ρv)(E).

Property (P) implies the following result. Lemma 1. For any j ∈ (j23 , j12 ) there exist three fixed points Fi = (Ei , Ai , Pi ), i = 1, 2, 3 (Fi are ordered by the increasing values of Ei ). For j < j23 (j > j12 ) only the fixed point F1 (F3 ) exists. At j = j23 (j = j12 ) the fixed point F1 (F3 ) and a nonhyperbolic fixed point F23 (F12 ) exist from which a pair F2 , F3 (F1 , F2 ) of fixed points bifurcates in a saddle–node bifurcation as j varies. The unfoldings of both saddle–node bifurcations are nondegenerate. Proof. The bifurcation sequence is illustrated in Fig. 1. Nondegeneracy of the saddle–node bifurcations follows from nonvanishing of the second derivative, see property (P).  Remark 2. Negative differential resistance (NDR), i.e. the phenomenon that the current j = (ρv)(E) is a decreasing function in a certain interval of the electric field E, is one of the main sources of semiconductor instabilities [16]. While the question of the existence of fixed points is answered easily, a bifurcation analysis of periodic, homoclinic and heteroclinic orbits of (1.2) is difficult due to the global nature of the problem, knowledge of no more than qualitative features of the empirical functions κ, v, and ρ, and the occurrence of four parameters and β, δ, s, and j . However, the occurrence of the small parameters β and δ is actually advantageous and basic to our approach, because it allows us to use (regular and singular) perturbation theory to reduce the three-dimensional problem (1.2) to a two-dimensional problem. The limiting problem β = 0 and δ = 0 has been discussed in [2] where some parts of the bifurcation diagram presented in this paper have been obtained. Furthermore, it has been conjectured there that these results can be extended to the case of small, positive β and δ. We show that this is indeed the case, however, the roles of β and δ are quite different. More precisely, we consider wave speeds of three different orders of magnitude with respect to β, i.e. the cases (i) s  β, (ii) s = O(β), and (iii) s  β; in all cases δ is an additional small parameter. Our basic approach is the use of geometrical singular perturbation theory to reduce the analysis of problem (1.2) to the analysis of two- or even one-dimensional problems which are more accessible to a bifurcation analysis. The necessary results from [11,20] are briefly explained in Section 2. Cases (i) and (iii) are analyzed in Section 3. In case (i) we use s/β as singular perturbation parameter to prove the existence of a two-dimensional invariant manifold in which (1.2) has exactly two traveling waves which correspond

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to heteroclinic orbits connecting the unstable node F2 to the saddles F1 (F3 ). In case (iii) we use β/s as singular perturbation parameter to prove the existence of a one-dimensional manifold in which (1.2) has exactly two traveling waves which are heteroclinic orbits connecting F1 (F3 ) to F2 . The intermediate case (ii) is analyzed in Section 4. We use δ as singular perturbation parameter to prove the existence of a two-dimensional invariant manifold of (1.2) in which the transition from case (i) to case (iii) occurs in a complicated global bifurcation. Depending on the value of j bifurcations of saddle–saddle connections, homoclinic loops, a heteroclinic loop, a Hopf bifurcation, and a saddle–node bifurcation of multiple periodic orbits occur as s varies. The parameter β is just a regular perturbation parameter which does not affect the dynamics. We wish to point out that for s = O(1) and certain choices of β and δ the simple approach taken in this work will not work, see Section 4 for more details. The analysis of this case will not be attempted in this article. Due to the global nature of the problem, rigorous analysis of all bifurcations in case (ii) seems impossible, however, we identify certain well known bifurcations of codimensions one and two and collect them in a consistent global bifurcation diagram. In Section 5 we report on numerical computations which give strong evidence for the correctness of the conjectured bifurcation diagram. Using the continuation package AUTO [5] we computed the bifurcation diagram for a set of functions ρ, κ, and v having the qualitative features of the experimental functions and obtained qualitative agreement with the partially proved and partially conjectured theoretical results. 2. Geometric singular perturbation theory The dynamical systems approach to singular perturbation problems goes back to [11]. A recent survey of geometric singular perturbation theory is [15]. In [20] a method — based on this invariant manifold approach — is formulated to study the existence and bifurcation of heteroclinic and homoclinic orbits of singularly perturbed differential equations. In this section we briefly summarize the necessary results from [11] which allow us to reduce the analysis of (1.2) to two- or even one-dimensional problems. We consider singularly perturbed systems of differential equations in the standard form x˙ = f (x, y, ε),

εy˙ = g(x, y, ε),

(2.1)

with ε ∈ [0, ε0 ), ε0 > 0 small, and (x, y) ∈ U ⊂ Rn × Rl open. We assume that f : U × [0, ε0 ) → Rn and g : U × [0, ε0 ) → Rl are smooth functions. By setting ε = 0 we obtain the reduced problem x˙ = f (x, y, 0),

0 = g(x, y, 0).

(2.2)

The basic idea is to obtain orbits of the singularly perturbed problem (2.1), for small values of ε, as smooth perturbations of orbits of the reduced problem (2.2). The following results are contained in [11, Theorem 9.1]. Theorem 3. In addition to the assumptions made above in this section, assume that 1. the equation g(x, y, 0) = 0 defines a manifold C0 which is a graph of a smooth function h : V ⊂ Rn → Rl ; 2. there exist integers ls and lu with ls + lu = l, such that the partial Jacobian gy has ls eigenvalues with negative real part and lu eigenvalues with positive real part, for all points of C0 . Then the reduced problem (2.2) defines a flow on C0 , and the following assertions hold in an appropriate neighborhood of C0 ∩ K, where K ⊂ U is any compact set satisfying K ∩ C0 6= ∅. There exists ε1 > 0 such that C0 can be extended to a smooth family of manifolds Cε , ε ∈ [0, ε1 ). The manifolds Cε are locally invariant under the flow of the singularly perturbed problem (2.1), and the restriction of this flow to Cε is a smooth perturbation of the reduced flow on C0 . The above theorem is basic for the following proposition.

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Proposition 4. Under the assumptions of Theorem 3 all structurally stable properties of the reduced problem (2.2) persist for the restriction of the singularly perturbed problem (2.1) to the invariant manifold Cε for small ε. In particular, (normally) hyperbolic (manifolds of) fixed points and periodic orbits of the reduced problem, their associated stable and unstable manifolds, and transversal intersections of these perturb smoothly. Hyperbolicity and transversality extend to the singularly perturbed problem (2.1) (without restriction to Cε ) with dimensions of stable and unstable manifolds increased by ls and lu , respectively. Bifurcations in singularly perturbed systems which depend smoothly on additional parameters µ ∈ Rk — as e.g. (1.2) does — can be studied by adding the trivial equation µ˙ = 0 to the system. For details we refer to [11,20], and for background material to [13].

3. The extreme cases s  β and s  β First we consider case (i), i.e. extremely slow waves with 0 ≤ s  β such that s/β is small. The case s = 0 corresponds to a standing wave. Thus (1.2) can be treated as a singularly perturbed system (2.1) with x = (E, P ), y = A, and ε = s/β, ε A˙ = κ(E)P (A − ρ(E)),

δ P˙ = v(E)P − j − εβ(P − A),

E˙ = P − A.

(3.1)

The assumptions of Theorem 3 are satisfied for P > 0 with the manifold C0 given as the graph of the function h(E, P ) = ρ(E), and lu = 1. The flow of the corresponding reduced problem on C0 is governed by δ P˙ = v(E)P − j,

E˙ = P − ρ(E),

(3.2)

which is dynamically rather simple. Lemma 5. For the reduced problem (3.2) the unstable node F2 is connected to the saddles F1 and F3 by transversal heteroclinic orbits for j ∈ (j23 , j12 ). For j = j23 (j = j12 ) the nonhyperbolic fixed point F23 (F12 ) is connected to F1 (F3 ) by a heteroclinic orbit. For j < j23 (j > j12 ) only the saddle F1 (F3 ) and no periodic and homoclinic orbits exist. Proof. Clearly, the fixed points of the reduced problem are the same as for (1.2). For j ∈ (j23 , j12 ) the nullclines of (3.2) are as shown in Fig. 2. F1 and F3 are saddles, F2 is an unstable node for j ∈ (j23 , j12 ). Except at the fixed points the flow is in the outward direction at the boundaries of the two lens-shaped regions bounded by the nullclines. The stable manifold of F1 (F3 ) points into these lens-shaped regions, thus, transversal heteroclinic orbits from F2 to F1 (F3 ) exist. For j = j23 (j = j12 ) the same argument implies that the nonhyperbolic fixed point F23 (F12 ) is connected to F1 (F3 ) by a heteroclinic orbit. For j < j23 (j > j12 ) only the saddle F1 (F3 ) exists. Since there are no other fixed points no periodic or homoclinic orbits of (3.2) exist.  Theorem 6. For β > 0 set ε := s/β. Then there exists an ε1 > 0 such that for 0 ≤ s < βε1 all traveling waves of (1.1a)–(1.1c) with speed s are given by the following orbits of (3.1) in a two-dimensional invariant manifold Cε : heteroclinic orbits connecting the unstable node F2 to the saddles F1 and F3 for j ∈ (j23 , j12 ) or a heteroclinic orbit connecting the nonhyperbolic fixed point F23 (F12 ) to the saddle F1 (F3 ) for j = j23 (j = j12 ).

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Fig. 2. Phase portrait with the nullclines.

Proof. We conclude from Theorem 3 the existence of ε1 (β) > 0 such that (1.2) has a two-dimensional repelling invariant manifold Cε for 0 ≤ ε < ε1 (β). For fixed j ∈ (j23 , j12 ) the assertion of the theorem follows immediately from Lemma 5 by means of Proposition 4. For j = j23 (j = j12 ) the nonhyperbolic fixed point F23 (F12 ) which does not depend on s persists, and hence the heteroclinic orbit connecting this fixed point to the saddle F1 (F3 ) exists in the two-dimensional invariant manifold for small ε by the same argument as in the proof of Lemma 5. Thus, ε1 can be chosen uniformly for j ∈ [j23 , j12 ]. Since Cε is repelling, no other traveling waves exist for small ε.  In case (iii) we set β = εs with small ε > 0. By transforming (1.2) to the new independent variable η = εξ and letting overdot denote d/dη we obtain A˙ = κ(E)P (A − ρ(E)),

εδ P˙ = v(E)P − j − s(P − A),

ε E˙ = P − A.

(3.3)

System (3.3) is singularly perturbed to the form (2.1) with x = A and y = (P , E). The assumptions of Theorem 3 are satisfied with C0 given as the graph of the function h(A) = (v −1 (j/A), A) and lu = 1, ls = 1. The flow of the corresponding one-dimensional reduced problem on C0 is governed by A˙ = κ(E(A))A(A − ρ(E(A))),

E(A) = v −1 (j/A).

(3.4)

Lemma 7. For the reduced problem (3.4) the repelling fixed points F1 and F3 are connected to the attracting fixed point F2 by (trivially transversal) heteroclinic orbits for j ∈ (j23 , j12 ). For j = j23 (j = j12 ) the repelling fixed point F1 (F3 ) is connected to the nonhyperbolic fixed point F23 (F12 ) by a heteroclinic orbit. For j < j23 (j > j12 ) only the saddle F1 (F3 ) exists. Proof. The lemma follows directly from the properties of the functions ρ and v.



Theorem 8. For any s > 0 there exists ε1 > 0 such that for β < ε1 s (1.1a)–(1.1c) has traveling waves with speed s which are given by the following orbits of (1.2) in the one-dimensional invariant manifold Cε : heteroclinic orbits connecting the saddles F1 and F3 to the saddle F2 for j ∈ (j23 , j12 ), a heteroclinic orbit connecting the saddle F1 (F3 ) to the nonhyperbolic fixed point F23 (F12 ) for j = j23 (j = j12 ). There are no other traveling waves corresponding to orbits which are uniformly bounded as ε1 → 0.

B. Katzengruber et al. / Physica D 144 (2000) 1–19

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Proof. The existence assertions are proved similar to the proof of Theorem 6. The nonexistence of other uniformly bounded orbits follows because all orbits of (3.3) which are not in Cε leave any compact neighborhood of Cε in finite (forward or backward) time.  The traveling waves given by Theorem 6 are almost standing waves, the waves given by Theorem 8 travel with speeds of the order of the carrier speed. For the stability of traveling wave solutions as solutions of the underlying p.d.e. system the monotonicity properties of these waves are often crucial. Lemma 9. The traveling waves given by Theorem 6 are fronts along which the variables P and E are monotone. The traveling waves given by Theorem 8 are fronts along which the variables A, P and E are monotone. Proof. The monotonicity is obvious for the orbits of the corresponding reduced problems and persists for sufficiently small ε. 

4. The case s = O(β) We rescale s in (1.2) by s = βc, where c > 0 and c = O(1), obtaining cA˙ = κ(E)P (A − ρ(E)),

(4.1a)

δ P˙ = v(E)P − j − cβ(P − A),

(4.1b)

E˙ = P − A.

(4.1c)

When the problem is posed for unbounded c there arises the possibility that the derivative with respect to P of the RHS of (4.1b) vanishes, thus making the reduction to a slow manifold no longer applicable. In this work we wish to avoid this potentially interesting, yet difficult situation. To this end we restrict the parameter region under investigation in the following way. Note that for some (possibly very large) c¯ the flow is as described in Theorem 8. Henceforth we restrict our investigation to c ∈ [0, c], ¯ making β so small that the above mentioned problem does not occur. We consider (4.1a)–(4.1c) as a singularly perturbed system of the form (2.1) with x = (A, E), y = P , and ε = δ. The assumptions of Theorem 3 are satisfied with C0 given as the graph of the function h(E) = j/v(E) and lu = 1. The flow of the corresponding two-dimensional reduced problem on C0 is governed by the equations cA˙ = j

κ(E) (A − ρ(E)) + O(β), v(E)

E˙ =

γ˙ − A + O(β). v(E)

(4.2)

The planar problem (4.2) governs the essential dynamics of the system (4.1a)–(4.1c). In the subsequent analysis we set κ ≡ 1. This simplifies the computations making the arguments more concise and easier to follow. For a general κ, conditions on κ, v and ρ for which our analysis still holds can easily be derived. We also neglect the O(β) terms. It will become clear that these terms do not alter the qualitative picture of the dynamics as long as β remains sufficiently small. Rescaling by the positive factor v(E) transforms (4.2) to the equivalent ODE j E˙ = j − Av(E). (4.3) A˙ = (A − ρ(E)), c

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4.1. Codimension one bifurcations of equilibria 4.1.1. Saddle–node bifurcations Proposition 10. There exist numbers j12 > j23 such that for an arbitrary c and j ∈ (j23 , j12 ) (4.3) has three equilibria F1 , F2 and F3 . The lines SN12 = {(c, j ) : j = j12 }

and SN23 = {(c, j ) : j = j23 }

correspond to saddle–node bifurcations involving F1 and F2 , respectively F2 and F3 . For j > j12 or j < j23 there is a unique equilibrium of saddle type. 

Proof. In the introduction. 4.1.2. Hopf bifurcation

/ Proposition 11. The equilibria F1 and F3 remain hyperbolic within the region of their existence as long as j ∈ {j12 , j23 }. The equilibrium F2 undergoes a Hopf bifurcation along a smooth curve H. Moreover, there exists a smooth and strictly decreasing function cH (j ) so that H = {(cH (j ), j ) : j ∈ (j12 , j23 )}. Proof. We parametrize the equilibria by E, i.e. A = ρ(E),

J = (ρv)(E).

(4.4)

The linearization of (4.3) at an equilibrium defined by (4.4) is given by w˙ = Lw, where L=



γ˙ /c −v(E)

−(γ˙ /c)ρ 0 (E) −Av 0 (E)

 .

The eigenvalues of L are given by the formula   s 2  1 j j j − ρ(E)v 0 (E) + 4 (ρv)0 (E) . λ± =  − ρ(E)v 0 (E) ± 2 c c c

(4.5)

Recall that F1 and F3 correspond to values of E satisfying E < E12 and E > E23 , respectively. This fact and the expression (4.5) imply that F1 and F3 are saddles. The equilibrium F2 corresponds to E ∈ (E12 , E23 ), where (ρv)0 is negative. The expression (4.5) implies that when j/c − ρ(E)v 0 (E) < 0 then F2 is stable and when j/c − ρ(E)v 0 (E) > 0 then F2 is unstable. The condition j − ρ(E)v 0 (E) = 0 (4.6) c defines the locus of a Hopf bifurcation. From (4.4) and (4.6) we obtain the corresponding value of c, which is given by cH (E) =

v(E) . v 0 (E)

(4.7)

B. Katzengruber et al. / Physica D 144 (2000) 1–19

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The Hopf curve H is now defined by (c, j ) = (cH (E), (ρv)(E)),

E ∈ (E12 , E23 ).

(4.8)

Assume that the function v is increasing and that v 0 is decreasing (this seems to hold for the experimental functions). It follows that as j decreases along H the corresponding value of c must increase.  4.2. Codimension two bifurcations 4.2.1. Takens–Bogdanov bifurcations Proposition 12. Assume that, in addition to (P ) the following inequalities hold: ρ 00 (E∗ ) 6= 0,

ρ 0 (E∗ )v(E∗ ) 6= 0,

E∗ ∈ {E23 , E12 }.

(4.9)

Then there exist precisely two nondegenerate Takens–Bogdanov points TB1 ∈ SN12 and TB3 ∈ SN23 . Proof. Fix E∗ ∈ {E23 , E12 } and let A∗ = ρ(E∗ ), j∗ = (ρv)(E∗ ) and c∗ = j∗ /(ρ(E∗ )v 0 (E∗ )). Let A˜ = A − A∗ and E˜ = E − E∗ . Replacing the RHS of (4.3) by its Taylor expansion at (A∗ , E∗ ) up to second order we obtain ˙˜ = j (A˜ − ρ 0 (E )E˜ − 1 ρ 00 (E )E˜ 2 ), A ∗ ∗ 2 c   j∗ 1 ˙ ˜ ˜ ˜ A − E − v 0 (E∗ )A˜ E˜ − 21 A∗ v 00 (E∗ )E˜ 2 . E = j − j∗ + c∗ ρ 0 (E∗ ) Defining j∗ t˜ = t, c∗

µ1 =

(j − j∗ )c∗ , j∗

c1 = − 21 ρ 00 (E∗ )µ2 ,

c2 =

µ2 =

jc∗ , cj∗

v 0 (E∗ ) , ρ 0 (E∗ )v(E∗ )

a∗ = ρ 0 (E∗ ), c3 =

1 ρ(E∗ )v 00 (E∗ ) , 2 ρ 0 (E∗ )v(E∗ )

letting overdot denote d/dt and dropping the tilde we obtain:          A c1 E 2 µ2 −µ2 a∗ A˙ 0 + + . = 1/a∗ −1 E c2 AE + c3 E 2 E˙ µ1

(4.10)

(4.11)

Considering the terms of order 0 and 1 we see that the Takens–Bogdanov point corresponds to (µ1 , µ2 ) = (0, 1) and the condition µ1 = 0 defines the locus of the saddle–node bifurcation. The fact that the trace of the linear part changes sign as µ2 passes through 1 implies that the unfolding of the terms up to order 1 is non-degenerate. Set (µ1 , µ2 ) = (0, 1). We now perform the linear transformation bringing the linear part of (4.11) to Jordan form. This transformation is given by ˜ A = a∗ A,

˜ E = A˜ − E.

Dropping the tilde we obtain c1 A˙ = E + (A − E)2 , a∗

c1 E˙ = (A − E)2 − c2 a∗ A(A − E) − c3 (A − E)2 . a∗

(4.12)

Applying a transformation of the form A˜ = A + k1 A2 + k2 AE,

E˜ = E + k3 A2 + k4 AE,

(4.13)

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with suitably chosen k1 , . . . , k4 we bring (4.11) to Takens normal form up to order two. Note that the transformation (4.13) does not change the coefficient of the E 2 term in either of the equations. Hence the Takens normal form up to order two is A˙ = E + b1 A2 ,

E˙ = b2 A2 ,

(4.14)

with b1 =

c1 , a∗

b2 =

c1 1 (ρv)00 (E∗ ) . − c2 a∗ − c3 = − 0 a∗ 2 ρ (E∗ )v(E∗ ) 

The result follows. Remark 13. A reasonable assumption on ρ is ρ 00 (E) > 0

for all

E ∈ (E12 , E23 ).

(4.15)

Under the assumption (4.15), b1 has the same sign at TB1 and TB3 , whereas, due to (P), b2 has different signs at both points. 4.3. Bifurcation diagram obtained by local analysis Based on the results of the previous section and the generic unfolding theory for the Takens–Bogdanov bifurcation [13] we get a complete picture of the local dynamics near the two bifurcation points. The description of this dynamics is given in the following proposition, see Fig. 3 for the relevant bifurcation diagram. Proposition 14. Assume that (4.15) holds. For (j, c) sufficiently close to TB1 there exist precisely three codimension one bifurcation curves, namely SN12 , the Hopf bifurcation curve H, and a curve hom1 , corresponding to the locus of a homoclinic orbit to F1 . Close to TB1 the Hopf bifurcation is supercritical. Both H and hom1 are tangent to the line SN12 and exist for j < j12 . There exists a function chom1 (j ) so that hom1 = {(chom1 (j ), j ) : j < j12 , j12 − j  1}. The functions chom1 and cH are monotonically decreasing and chom1 (j ) < cH (j ). In the region bounded by H and hom1 there exists a unique stable periodic orbit, which limits on F2 as (c, j ) approaches H and on the homoclinic orbit as (c, j ) approach hom1 .

Fig. 3. Bifurcation diagram obtained by local analysis (horizontal lines indicate stability of F2 ).

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Fig. 4. The first Liapunov coefficient as a function of E (the corresponding values of c and j are given by (4.8)).

Analogously, for (j, c) sufficiently close to TB3 , there exist precisely three bifurcation curves, namely SN23 , H , and hom3 with hom3 = {(chom3 (j ), j ) : j > j23 , j − j23  1}, for some smooth, strictly decreasing function chom3 (j ). Close to TB3 the Hopf bifurcation is subcritical, chom3 (j ) > cH (j ), and in the region bounded by H and hom3 there exists a unique unstable periodic orbit. Proof. The result follows from Propositions 10–12, the unfolding theory of Takens–Bogdanov bifurcations and  from the properties of the coefficients of the normal forms at the points TB1 and TB3 , see Remark 13. Proposition 14 has the following interesting implication. Corollary 15. There exists a point DH along H where the first Liapunov exponent vanishes. Proof. The result follows from the fact that the Hopf bifurcation changes criticality along H .



Remark 16. Choosing the nonlinearity as in (5.1) we have obtained an implicit formula for the first Liapunov coefficient using Maple. It does not appear that a proof of the existence of a unique zero can be obtained analytically. A Maple plot of the coefficient does, however, indicate the existence of a unique zero. This plot is shown in Fig. 4. 4.4. Saddle–saddle connections We prove the following result. Proposition 17. There exist points (c, j ) with j close to j12 (j23 ) on the right (left) hand side of the curve hom1 (hom3 ) corresponding to heteroclinic connections from F3 to F1 . Similarly, there exist points (c, j ) with j close to j12 (j23 ) on the left (right) hand side of the curve hom1 (hom3 ) corresponding to heteroclinic connections from F1 to F3 .

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Proof. We prove the existence of connections between F3 and F1 to the right of hom1 . The other cases are similar. Let j = j12 . Recall that for c¯ there exists a connection from F3 to F12 along W c (F12 ). Let V be a neighborhood of F12 where the local analysis of either the Takens–Bogdanov unfolding or the saddle–node unfolding holds. By ¯ We compactness there exists an ε > 0 such that this neighborhood can be chosen uniformly in c ∈ [cTB1 − ε, c]. can choose V so that for c > cTB1 and j = j12 each trajectory entering V must be either attracted to F12 or leave V along W c (F12 ) and for j < j12 with j12 − j small, every trajectory entering V must either be attracted to F2 , intersect W s (F1 ) or leave along W u (F1 ). Suppose that the connection from F3 to F12 along W c (F12 ) persists till TB1 . Then there exists a connection from F3 to F1 in the unfolding of TB1 . To see this first consider j < j12 and c > cTB1 . If j − j12 is small enough then a branch of W u (F3 ) is attracted to F2 . For c = chom1 it is clear that W u (F3 ) must be outside of the homoclinic orbit and thus follows the unbounded branch of W u (F1 ). Hence along a path between the two points W u (F3 ) and W s (F1 ) intersect. We now consider the case when the connection does not persist till TB1 . Let C∗ be the set of points c > cTB1 for which there exists a sequence {ck }k=1,2,... accumulating on c from the left, such that for each (c, j ) = (ck , j12 ) the manifold W u (F3 ) passes through V and exits V along W c (F12 ). Suppose C∗ is not empty and let c∗ = sup(C∗ ). Then for every c ≥ c∗ there exists a connection from F3 to F12 (possibly along W c (F12 )). Hence there exists an ε > 0 (by compactness independent of c) such that for every j < j12 with and j12 − j < ε and c ∈ [c∗ − ε, c] ¯ the manifold W u (F3 ) enters V . Then there exists j = jhet31 and some c ∈ [c∗ − ε, c∗ ] so that for (c, jhet31 ) the manifold W u (F3 ) exits V along W u (F1 ). Since for (c, ¯ jhet31 ) there exists a connection from F3 to F2 a heteroclinic ¯ must exist for some chet31 ∈ [c∗ − ε, c]. If C∗ is empty then W u (F3 ) enters V for every c ∈ [cTB1 , c]. ¯ Fix c0 > cTB1 close enough to cTB1 so that the Takens–Bogdanov unfolding holds. Then, for j near j12 and c = c0 , either there exists a heteroclinic orbit in the unfolding of TB1 or there is a parameter point for which W u (F3 ) exits V along W u (F1 ) and a heteroclinic orbit ¯  must exist for some c ∈ (c0 , c). Remark 18. Typically saddle–saddle-connections exist along smooth curves in parameter space. Thus one would expect that the heteroclinic orbits given by Proposition 17 exist along smooth curves het13 and het31 with endpoints on SN12 and SN23 . Due to the position of the endpoints these curves would have to intersect in a point HC corresponding to a heteroclinic cycle. Unfortunately, we are not able to prove these assertions rigorously by establishing transversal intersection of W u (F3 ) and W s (F1 ) with respect to variation of c and j. Transversality would follow if we could show that the corresponding Melnikov integrals do not vanish, which appears to be difficult. Without transversality we must rely on topological arguments which weakens the result and makes the proof complicated. 4.5. The structure of the global dynamics We have not been able to show that the bifurcations corresponding to the saddle–saddle-connections and to the degenerate Hopf bifurcation are unique and non-degenerate. However, based on a thorough numerical investigation, which we describe in the next section, we make the following conjecture. Conjecture 19. 1. The curves het 13 and het31 are unique and monotonic in c. Their endpoints lie on SN12 and SN23 and they intersect in a unique point HC. 2. The curves hom1 and hom3 are unique and terminate at the point HC. 3. There exists a unique curve SNP corresponding to the locus of saddle–node bifurcations of periodic orbits with endpoints at DH and HC. 4. All bifurcations unfold generically and no other bifurcations take place.

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Fig. 5. The conjectured global bifurcation diagram.

We summarize our results and conjectures in the following theorem. Theorem 20. Suppose that Conjecture 19 holds. Then the bifurcations of fixed points, periodic orbits, homoclinic and heteroclinic orbits of system (4.3) upon varying the parameters c, j occur as shown in the bifurcation diagram in Fig. 5. More specifically, there exist the following curves along which codimension one bifurcations occur: 1. there exist numbers j12 > j23 such that the fixed points F1 , F2 (F2 , F3 ) are generated in a saddle–node bifurcation along straight lines SN12 (SN23 ) given by j = j12 (j = j23 ); 2. a smooth curve H along which a periodic orbit is created in a Hopf bifurcation at F2 as (c, j ) crosses H; 3. two smooth curves het 13 (het31 ) along which heteroclinic orbits connecting the saddles F1 and F3 (F3 and F1 ) exist; 4. two smooth curves hom1 (hom3 ) along which orbits homoclinic to the saddle F1 (F3 ) exist; 5. a smooth curve SNP (saddle–node of periodic orbits) along which two periodic orbits are generated in a saddle–node bifurcation; 6. for points on H above the point DH (see (10) below) the Hopf bifurcation occurs supercritically, i.e. an attracting limit cycle is generated at H as c decreases; for points on H above HC the stable limit cycle exists until it disappears in the homoclinic orbit at hom1 ; for points on H between the points HC and DH the stable limit cycle generated in the Hopf bifurcation and a second unstable limit cycle generated at hom3 exist for decreasing values of c until both coalesce and disappear at the curve SNP; for points on H below the point DH the Hopf bifurcation occurs subcritically, i.e. an unstable limit cycle is generated as c increases and exists until it disappears in the homoclinic orbit at hom3 . Endpoints and intersection points of the bifurcation curves listed in (1)–(5) correspond to the following codimension two bifurcations: 7. four saddle–node connections (SNC) at the endpoints of het13 and het31 on SN12 and SN23 ; 8. one heteroclinic cycle (HC) at the point of transversal intersection of the curves het 13 and het31 . The point HC is also an endpoint for hom1 , hom3 and SNP;

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9. two Takens–Bogdanov bifurcations (TB); one at the point on SN12 where hom1 and H come together and the other on SN23 where hom3 and H come together; 10. one degenerate Hopf point (DH) corresponding to the point where SNP emanates from H. Theorem 21. There exist β0 > 0 and δ0 > 0 such that the assertions of Theorem 20 hold for the singularly perturbed problem (4.1a)–(4.1c) for β ∈ [0, β0 ] and δ ∈ [0, δ0 ]. Proof. The result follows for the restriction of system (4.1a)–(4.1c) to the slow manifold for δ sufficiently small from the stability of the generically unfolded bifurcations under the regular perturbation for sufficiently small β.  Remark 22. Our partially proved and partially conjectured bifurcation diagram corresponds to the “saddle”-case in the unfolding of a codimension three point analyzed by Dumortier et al. [9]. In the local setting considered there the validity of the bifurcation diagram is proved. We also refer to that paper for illustrations of corresponding phase portraits (see [9, p. 6]). 4.6. Existence of traveling waves It is clear that in case (ii) there exists a large variety of traveling waves. To obtain them one needs to list all possible bounded solutions of (4.1a)–(4.1c), which can be easily done using Theorem 20, and identify the corresponding traveling waves. Here we indicate the possible types of waves and leave the task of finding them on the bifurcation diagram to the interested reader. Fronts. For j ∈ (j23 , j12 ) there exist fronts corresponding to connecting orbits from F2 to F1 or F3 and from F3 or F1 to F2 . These correspond to structurally stable saddle–sink or source–saddle connections and thus represent families of traveling waves present for a given triple (β, δ, j ). Additionally, there exist fronts corresponding to connections from F3 to F1 and from F1 to F3 . These correspond to saddle–saddle connections and thus are unique for a given parameter triple. For j ∈ {j23 , j12 } there exist fronts corresponding to connections from F3 to F12 , F12 to F3 , F1 to F23 and F23 to F1 . Pulses. For j ∈ (j23 , j12 ) there exist pulses corresponding to the homoclinic orbits to F1 and to F3 . Periodic trains. For j ∈ (j23 , j12 ) there exist periodic trains corresponding to periodic orbits born from the Hopf bifurcation. For a given (β, δ, j ) there exists a family of periodic trains and, for some (β, δ, j ) there exist two waves with the same wave speed. This phenomenon corresponds to coexistence of periodic orbits. Other types of traveling waves. As indicated before we also encounter traveling waves corresponding to connections between equilibria and periodic orbits as well as to connections between periodic orbits. In particular there may be connections from F1 or F3 to a periodic orbit (or vice versa) as well as connections from F2 to a periodic orbit (or vice versa). Connections from one periodic orbit to another occur for parameters where periodic orbits coexist. 5. Numerical bifurcation diagram Using the path following package AUTO [5] we located the bifurcation curves and the bifurcation points whose existence was proved and conjectured in the previous section. To simplify the implementation of the numerics we used the functions E(E − 1)(E − 2) + v(E) 2 ρ(E) = , κ(E) ≡ 1 (5.1) v(E) = arctan(E), π v(E)

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rather than the experimental functions. We performed two sets of computations. One for the planar reduced problem (4.2) obtained by setting δ = 0 and one for the three-dimensional singularly perturbed system (4.1a)–(4.1c) with δ = 0.1. In the following we refer to these cases as the δ = 0 and the δ = 0.1 case, respectively. In both cases we obtained bifurcation diagrams which were qualitatively the same as the diagram given in Fig. 5. In all the computations we have set β = 0. We will now briefly describe each of the performed computations, see [6,7]. 5.1. Hopf curve H AUTO has the capacity to follow an equilibrium, detect a Hopf bifurcation, and follow the curve of Hopf bifurcations in two parameters. We carried out this procedure starting with F2 in the stability region. 5.2. Saddle–saddle connections het 31 and het13 Since AUTO contains a state-of-the-art collocation boundary value solver it is very well suited for computation of heteroclinic orbits. The numerical approximation of a saddle–saddle connection p to q is given by an orbit which intersects the unstable subspace E u (p) and the stable subspace E s (q) and needs a long passage time from one subspace to the other. If dimE u (p) = 1 then a good strategy is to start at p + εv u (p) with a small ε > 0 and integrate forward until an intersection with E s (q) is reached. Such an intersection can be found for a suitable choice of the parameters (not too far from a point where a connection exists). To find good parameter values one can use a phase plane analyzer, e.g. PhasePlane [10] or DSTOOL [12]. Further continuation is carried out under the constraint that the right end point of the solution is in E s (q). Using integration time as the continuation parameter and letting it increase one zooms in closely on the connection. The curve of connections is then obtained by fixing the integration time at a high value and continuing the solution in two parameters [8]. The above described approach was made automatic in the recent extension of AUTO called HomCont, see also [4]. In our computations we made use of HomCont. We located and continued the heteroclinic orbits het31 and het 13 . For δ = 0.1 we had to reverse time to guarantee that the sought heteroclinic connection has fixed points with one-dimensional unstable manifold at ξ = −∞. The point HC was located as the intersection point of the two curves. 5.3. Saddle–node of periodic orbits SNP At a point of a nondegenerate Hopf bifurcation AUTO can switch from the equilibrium solution to the bifurcating periodic orbit. AUTO monitors Floquet multipliers and can detect a multiplier 1 as well as follow the curve of multipliers 1 in two parameters. We followed a periodic orbit from a Hopf bifurcation for j ∈ (jDH , jHC ) and thus located the saddle–node bifurcation. Next we traced out the curve SNP from a point on H (which we identified as DH) to the point HC. 5.4. Homoclinic orbits hom1 and hom3 Homoclinic orbits can be found analogously as heteroclinic connections. Another way is to continue a periodic orbit until an orbit with very high period is reached. The homoclinic curve is then approximated by continuing the periodic orbit with fixed high period. We used the former method in the computation for δ = 0 and the latter in the computation for δ = 0.1. The numerical bifurcation diagrams are given in Figs. 6 and 7. A blow-up of the bifurcation diagrams showing the regions of interest is given in Figs. 8 and 9. In Figs. 10 and 11 we show sequences of homoclinic orbits limiting on the heteroclinic cycle.

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Fig. 6. Numerically computed global bifurcation diagram for δ = 0, β = 0.

Fig. 7. Numerically computed global bifurcation diagram for δ = 0.1, β = 0.

Fig. 8. Blow-up of the bifurcation diagram for δ = 0, β = 0.

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Fig. 9. Blow-up of the bifurcation diagram for δ = 0.1, β = 0.

Fig. 10. Sequence of homoclinic orbits from the Takens–Bogdanov points to the heteroclinic cycle for δ = 0, β = 0; (a) homoclinic orbits between TB1 and HC, (b) homoclinic orbits between TB3 and HC.

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Fig. 11. Sequence of homoclinic orbits from the Takens–Bogdanov points to the heteroclinic cycle for δ = 0.1, β = 0; (a) homoclinic orbits between TB1 and HC, (b) homoclinic orbits between TB3 and HC.

Acknowledgements The authors would like to thank Eusebius Doedel and Bernd Krauskopf for help with numerical computation of bifurcation diagrams. This research was funded by the Austrian Science Foundation through the START-project Y42-MAT. References [1] L.L. Bonilla, Solitary waves in semiconductors with finite geometry and the Gunn effect, SIAM J. Appl. Math. 51 (1991) 727–747. [2] L.L. Bonilla, S.W. Teitsworth, Theory of periodic and solitary space charge waves in extrinsic semiconductors, Physica D 50 (1991) 545–559. [3] L.L. Bonilla, Small signal analysis of spontaneous current instabilities in extrinsic semiconductors with trapping: application to p-type ultrapure germanium, Phys. Rev. B 45 (1992) 1142–1154. [4] A.R. Champneys, Yu.A. Kuznetsov, B. Sandstede, A numerical toolbox for homoclinic bifurcation analysis, Int. J. Bifurc. Chaos 6 (1996) 867–887. [5] E.J. Doedel, A.R. Champneys, T.F. Fairgrieve, Y.A. Kuznetsov, B. Sandstede, X.-J. Wang, in: AUTO 97, Continuation and Bifurcation Software for Ordinary Differential Equations, Concordia University, Montreal, 1997.

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