Our investigation of logarithmic spirals is motivated by disparate experimental results: (i) the discovery of logarithmic spiral shaped precipitate formation in chemical garden experiments. Understanding precipitate formation in chemical gardens is important since analogous precipitates form in deep ocean hydrothermal vents, where conditions may be compatible with the emergence of life. (ii) The discovery that logarithmic spiral shaped waves of spreading depression can spontaneously form and cause macular degeneration in hypoglycemic chick retina. The role of reaction–diffusion mechanisms in spiral formation in these diverse experimental settings is poorly understood. To gain insight, we use the topological shooting to prove the existence of 0-bump stationary logarithmic spiral solutions, and rotating logarithmic spiral wave solutions, of the Kopell–Howard lambda–omega reaction–diffusion model.

Experimental evidence indicates that reaction–diffusion mechanisms play a meaningful role in (i) the formation of logarithmic spiral shaped precipitates in chemical gardens and (ii) the formation of rotating logarithmic spiral shaped waves in the chick retina. To model these phenomena, we focus on the Kopell–Howard lambda–omega reaction–diffusion equations with the goal of answering the following basic question: can we prove the existence of logarithmic spiral solutions that share qualitative properties with observed spirals? Our approach to resolving this issue is to make use of a topological shooting technique. This technique, which closely corresponds to the numerical construction of solutions, is well suited to proving the existence of solutions that are qualitatively similar to observed spirals.

Our investigation is motivated by disparate experimental results: (i) in 2014, Haudin et al.14 discovered logarithmic spiral shaped precipitate formations in chemical gardens (Fig. 1, upper left). Understanding the underlying mechanisms responsible for precipitate formation is fundamentally important since analogous precipitates form in deep ocean hydrothermal vents. Hydrothermal vents have been identified as locations where conditions are compatible with the emergence of life.2,8,14,19,22,23 (ii) In 2012, Yu et al.29 discovered that logarithmic spiral shaped waves of spreading depression (SD) can form and cause macular degeneration in hypoglycemic chicken retina (Fig. 1, middle). The role of reaction–diffusion mechanisms in a spiral formation in these experimental settings is poorly understood. Thus, in Theorem 1.2, we prove the existence of logarithmic spiral wave solutions of the Kopell–Howard reaction–diffusion model. The remainder of this section consists of the following subsections: (I) chemical gardens, (II) spreading depression, (III) the model and previous results, (IV) goals and statement of main results, and (V) conclusions.

FIG. 1.

Upper left: a logarithmic spiral shaped cobalt chloride precipitate in a chemical garden. Note that, as radial distance r from the spiral tip increases from r=0, silicate concentration increases from a relatively low level to a higher level. Image is from Haudin et al. [Fig. 2(d) of Ref. 14]. Upper middle: a rotating logarithmic spiral shaped wave of depolarization in hypoglycemic chicken retina. Image is from Yu et al. (Fig. 1 of Ref. 29). Upper right: a Belousov–Zhabotinsky reaction Archimedean spiral. The image is from Brandeis University Dynamics Group. Lower left: superimposition of part of a logarithmic spiral, obtained from empirical formula (1.1), on a spiral precipitate. The image is from Haudin et al. (Fig. 3 of Ref. 14). Lower right: approximation to a stationary logarithmic spiral solution given by reaction–diffusion based formula (1.29).

FIG. 1.

Upper left: a logarithmic spiral shaped cobalt chloride precipitate in a chemical garden. Note that, as radial distance r from the spiral tip increases from r=0, silicate concentration increases from a relatively low level to a higher level. Image is from Haudin et al. [Fig. 2(d) of Ref. 14]. Upper middle: a rotating logarithmic spiral shaped wave of depolarization in hypoglycemic chicken retina. Image is from Yu et al. (Fig. 1 of Ref. 29). Upper right: a Belousov–Zhabotinsky reaction Archimedean spiral. The image is from Brandeis University Dynamics Group. Lower left: superimposition of part of a logarithmic spiral, obtained from empirical formula (1.1), on a spiral precipitate. The image is from Haudin et al. (Fig. 3 of Ref. 14). Lower right: approximation to a stationary logarithmic spiral solution given by reaction–diffusion based formula (1.29).

Close modal

Discovered in 1646 by Glauber10 are experiments in which metal salts (e.g., cobalt chloride) come into contact with liquid water glass, a syrupy solution of sodium silicate. Within minutes, twisted precipitates sprout from the point of contact.8,10 These chimney-like tubes have porous membrane structures that rupture and re-form as pressures build up when water passes through the membranes.8 Understanding chemical garden precipitate formation is a fundamentally important issue because of (i) the potential of obtaining a new understanding of material synthesis under non-equilibrium conditions and (ii) close similarities with the precipitate formation in deep ocean hydrothermal vents. For three decades, hydrothermal vents have been investigated as locations where conditions may be compatible with the origin of life.2,8,14,19,22,23 Haudin et al.14 emphasize that, in 3D space, three fundamental mechanisms are involved in precipitate formation, (i) buoyancy, (ii) osmosis, and (iii) reaction–diffusion.3,5 To gain insight, Haudin et al.14 performed an innovative series of experiments in which a silicate solution was injected at a fixed concentration, and constant flow rate, into a cobalt chloride solution in a quasi-2D Hele–Shaw cell. They make clear that, in this setting, the effects of buoyancy and osmosis are reduced; hence, one can focus on understanding the role of reaction–diffusion mechanisms in precipitate formation. For a range of concentrations and flow rates, Haudin et al.14 discovered the formation of logarithmic shaped spiral shaped cobalt silicate precipitates. Figure 1 (upper left) shows a spiral precipitate, which separates the cobalt chloride solution from the silicate solution. A key feature of Haudin et al.’s (Fig. 3 of Ref. 14) investigation is the development of empirically based geometrical formula

(1.1)

which they use to approximate shapes of experimental spiral precipitates (Fig. 1, lower left). To obtain further advances, it is necessary to gain new understanding of precipitate properties, in particular, underlying reaction–diffusion dynamics influencing the development of spiral wall thickness. For this, we observe the following thickness property of the spiral precipitate in Fig. 1 (upper left):

(E1) As radial distance r from the spiral tip increases from r=0, the magnitude of silicate concentration continuously increases from a low level to its maximum.

Haudin et al.14 also make clear that “control of concentrations of the reagents and of the flow rate will allow the study of the switch from dominant reaction–diffusion processes to flow driven ones.” In a 2015 follow-up study, Haudin and De Wit15 investigated this important issue. They performed a series of experiments to determine the extent to which a Hele–Shaw viscous fingering mechanism causes pattern formation. They clearly demonstrated that “viscous fingering shapes flower patterns at low metallic salt concentrations, but it is not the driving mechanism in the development of filaments and spirals at larger cobalt chloride concentrations.” Thus, unless carefully determined evidence is presented to the contrary, it is reasonable to conclude that a range of parameters exists where a reaction–diffusion mechanism plays a meaningful role in spiral shaped precipitate formation.

Discovered by Leao21 in 1944 is a wave of depolarization of gray matter or retina that disrupts neuronal function.29 In 1983, Gorelova and Bures11 reported spiral waves of SD in chick retina. In 1997, Dahlem and Muller7 describe the oscillatory nature of SD spiral waves. In 2010, Dahlem et al.6 posited that SD waves resulted from a reaction–diffusion process. Next, before describing the results of the 2012 investigation by Yu et al.,29 we first need to summarize key findings of recent (2021) experiments by de Lima et al.,20 which categorize patterns of SD waves in chick retina. De Lima et al. conclude (Sec. 3.1 of Ref. 20) that the most probable SD wave patterns are either solitary circular waves or rotating logarithmic spirals. We now summarize important aspects of Yu et al.’s29 investigation of rotating SD spirals in chick retina. They conclude that

(E2) “Spontaneous spiral wave episodes of SD can occur, resulting in irreversible punctate lesions in the macula. These lesions can act as sites of origin for SD spiral waves (Fig. 1, middle) of SD that progressively enlarge the lesions.”

(E3) Figures 1(c) and 1(d) of Ref. 29 show that each time a spiral wave goes through one complete rotation, over a period 3.5min±0.2min, a sensor records one spike in potential, of amplitude 20Mv, duration 60s, plus refractory time 100s.

We use the topological shooting to prove the existence of logarithmic spiral solutions of the Kopell–Howard lambda–omega reaction–diffusion model,17 which satisfies qualitative properties (E1), (E2), and (E3) stated above. Before stating our main result (Theorem 1.2), we describe the model and previous results. The Kopell–Howard17 model consists of the equation

(1.2)

where U and V correspond to components of concentrations and λ and ω are functions of R=U2+V2, which denotes the magnitude of concentration referred to in property (E0). This system, which is also known as a lambda–omega system, arises as a model reaction–diffusion system, and as the normal form near Hopf bifurcations from spatially homogeneous rest states.4,17,18 Note that, for specific applications, the spatial variables can be scaled, resulting in diffusion coefficients multiplying 2U and 2V.

1. Archimedean spiral solutions

Equation (1.2) has played an important role in modeling Archimedean spiral waves in the Belousov–Zhabotinsii (BZ) reaction. The BZ reaction, discovered in 1959 by Belousov,1 exhibits bulk temporal oscillations when the reaction takes place in a stirred beaker. In 1964, Zhabotinsky28 published further details of the oscillations discovered by Belousov. In 1972, Winfree27 investigated the properties of plane waves and Archimedean spiral waves in a thin layer of BZ reagent in a Petri dish (Fig. 1, upper right). Recently, in 2003, Vanag and Epstein26 gave a comprehensive description of the BZ spirals. In 1973, Kopell and Howard17 made use of Eq. (1.2) to model plane waves. In a seminal 1974 paper, Field and Noyes9 derived a five step mechanism to explain the chemical basis of the BZ reaction. In 197,8 Cohen et al. (Sec. 5 of Ref. 4) showed how “Eq. (1.2) arises naturally as the dominant system in the asymptotic analysis of more general reaction–diffusion systems actually describing specific physical processes.” In 1980, Greenberg12 investigated Eq. (1.2) when

(1.3)

For this representative, canonical choice of λ(R) and ω(R) his asymptotic analysis indicates that Eq. (1.2) has an Archimedean spiral wave solution, i.e., a solution qualitatively similar to a BZ Archimedean spiral (Fig. 1, upper right). Subsequently, in 1982, Kopell and Howard18 considered more general λ and ω and proved the existence of target patterns and Archimedean spirals, which are also qualitatively similar to those observed in the BZ reaction. In 1986, Keener and Tyson16 modeled BZ spirals with a FitzHugh–Nagumo based system. Their asymptotic analysis showed that spiral solutions are Archimedean. In 1988 and 1997, Scheel24,25 extended these studies and proved the bifurcation of solutions to spiral waves and rotating waves with infinitely many arms.

2. Logarithmic spiral solutions

In 1978, Cohen et al.4 used a functional analysis approach to prove that Eq. (1.2) has logarithmic spiral wave solutions. For this, they let U=Rcos(Θ), V=Rsin(Θ). Then, Eq. (1.2) becomes

(1.4)

Next, let (r,θ) denote polar coordinates of the plane and assume that functions ρ(r) and ψ(r) exist such that

(1.5)

where Ω is a constant. The corresponding solution of Eq. (1.2) is

(1.6)

When Ω0,(1.6) represents a spiral wave in concentration that rotates with frequency Ω about r=0.4 When Ω=0,(1.6) represents a stationary spiral in concentration. The function ρ=U2+V2 denotes the magnitude of concentration. Further details for specific choices, λ and ω are given in properties (P2), (P3), and (P4) following the statement of Theorem 1.2. Substituting (1.5) into Eq. (1.4), we conclude that ρ(r) and ψ(r) satisfy gives the system,

(1.7)

Cohen et al.4 investigate the existence of solutions of (1.7) such that ρ and ψ are bounded and regular at r=0 so that concentrations U,V are bounded with bounded gradients. Thus, they impose standard initial conditions ρ(0)=0,ρ(0)>0, and ψ(0)=0. As r, they assume that ρ approaches a constant and ψ0, hence Ωω(ρ())=ψ()=0. They make simple assumptions on λ and ω,

(H1)a>0 exists such that λC1[0,a],λ(ρ)>0ρ[0,a),λ(a)=0,λ(a)<0.

(H2)ωC[0,a], and ε>0,μ>0 exist such that |ω(ρ)ω(a)|ε|aρ|1+μ. Under these assumptions, they make use of the Schauder fixed point theorem (p. 158 of Ref. 13) to prove

(Cohen et al.4)
Theorem 1.1
(Cohen et al.4)
If 0<ε1, then there exists Ω=ω(a) and a solution of Eq. (1.7) satisfying ρ(0)=0,ρ(0)>0,ψ(0)=0 such that
(1.8)
(1.9)
(1.10)
(1.11)

3. Computations

To gain insight, Cohen et al.4 set a=1 and investigate Eq. (1.7) for the representative, generic choice,

(1.12)

Thus, they solve initial value problem

(1.13)
(1.14)

Here, μ>0 and either ε>0 or ε<0. These parameters, as well as a rescaling of the spatial variable r in Eq. (1.13), can be used to model spiral properties obtained from experimental data. When (μ,ε)=(1,1) Cohen et al.4 compute solutions using numerical shooting, where ρ(0)=α>0 is the shooting parameter. The first step is to observe that, when α1312 and |ε|>0, there exists rα(0,1] such that ρ(r)>0r[0,rα] and ρ(rα)=1 (Lemma 2.3). This reduces the range of α to 0<α<1312, where a solution can exist such that 0<ρ(r)<1r>0. Next, when 0<α1, a computation shows that ρ=0 before ρ=1. As α is increased, this procedure is continued until a maximal value, αmax>0, is found where ρ(r)>0r(0,20] and ρ(20)1. Cohen et al.4 show that by making small changes in this value of α, they were “able to make the exponential instability in the numerical solution positive or negative, which gives confidence that the procedure actually approximates a true solution.” Thus, the magnitude of concentration, ρ(r) of this solution satisfies ρ(r)>0r>0 and ρ()=1. This solution satisfies chemical garden property (E0). They also use this procedure to compute a second solution. In Fig. 2 (upper left), we use this procedure to reproduce their two solutions. For one solution (labeled I), ρ(r)>0r>0. For the second solution (labeled II), ρ(r) is not strictly increasing. There is one relative maximum, followed by a relative minimum, and then ρ increases to ρ=1 as r. Thus, we refer to the first solution, which satisfies property (E0), as a “0-bump” solution, the second solution as a “1-bump” solution. Cohen et al. (Figs. 4 and 7 of Ref. 4) find that “islands” appear in contour plots of U(r,θ,0), corresponding to the 1-bump solution. This is due to the oscillatory nature of the solution. No island appears in contour plots for 0-bump solutions (Fig. 3 of Ref. 4). To gain further understanding, we use numerical shooting to compute solutions of Eqs. (1.13) and (1.14) when (μ,ε)=(1,2) and (μ,ε)=(.8,2). First, when (μ,ε)=(1,2) we find (Fig. 2, upper right) that 0-bump, 2-bump, and 3-bump solutions coexist. However, we did not find a 1-bump solution. Second, when (μ,ε)=(.8,2), we find (Fig. 2, lower panels) that a 0-bump solution exists, and, upon comparing ψ with 1.31r, we obtain

(1.15)

Thus, following Cohen et al.,4 we conclude that the numerical shooting procedure gives an approximation to a true solution, which satisfies

(1.16)

An integration gives

(1.17)

where ψ0 is an arbitrary constant. In part (P3), following the statement of Theorem 1.2, we use properties (1.16) and (1.17) to construct a logarithmic spiral solution, which approximates the shape of a chemical garden precipitate.

FIG. 2.

Upper left: ρ components of 0-bump and 1-bump solutions of Eqs. (1.13) and (1.14) computed in Ref. 4 when (μ,ε)=(1,1). Upper right: ρ components of 0-bump, 2-bump, 3-bump solutions when (μ,ε)=(2,1). Lower left: ρ component of 0-bump solution when (μ,ε)=(.8,2). Lower right: Comparison of the ψ component of the solution with 1.31r shows that ψ(r)1.31rr15.

FIG. 2.

Upper left: ρ components of 0-bump and 1-bump solutions of Eqs. (1.13) and (1.14) computed in Ref. 4 when (μ,ε)=(1,1). Upper right: ρ components of 0-bump, 2-bump, 3-bump solutions when (μ,ε)=(2,1). Lower left: ρ component of 0-bump solution when (μ,ε)=(.8,2). Lower right: Comparison of the ψ component of the solution with 1.31r shows that ψ(r)1.31rr15.

Close modal

In the BZ modeling studies described above,12,17,18,24,25 the authors do not study specific details of the underlying chemistry. Instead, they show, under appropriate assumptions, how reaction–diffusion system (1.2) has Archimedean spiral wave solutions that are qualitatively similar to observed BZ waves (Fig. 1, upper right). Here, we follow the same approach. That is, we also do not study specific details of underlying chemistry. Instead, our goal is to show, under simple, representative assumptions on λ and ω, how system (1.2) has logarithmic spiral solutions that are qualitatively similar to observed spirals (Fig. 1, upper left and upper middle). For this, we first note that since the Schauder fixed point (p. 158 of Ref. 13) approach is used in Ref. 4 to prove Theorem 1.1, it is not possible to conclude whether the solution proved in Theorem 1.1 is a 0-bump, 1-bump, or multi-bump solution. Second, our numerical study suggests that, for fixed ΩR, 0-bump solutions exist for each μ>0, but 1-bump solutions, and possibly other multi-bump solutions, exist only over a limited range of μ. Thus, our goal in Theorem 1.2 is to prove the existence of 0-bump solutions for the representative choice of λ and ω given by (1.12). Following the statement of Theorem 1.2, we show how our results generate logarithmic spiral solutions that are qualitatively similar to observed spirals and satisfy qualitative properties (E1)(E2)(E3).

Theorem 1.2
Let ΩR and μ>0. If 0<|ε|1, there is a solution of initial value problems (1.13) and (1.14) such that
(1.18)
(1.19)

1. Qualitative properties of solutions

Below, in (P1)–(P4), we describe qualitative properties of solutions of (1.13) and (1.14), which are predicted by our results.

(P1) The primary difference between Theorems 1.1 and 1.2 is that Theorem 1.2 proves the existence of 0-bump solutions where ρ(r)>0r>0; however, it is not possible to conclude if the solution in Theorem 1.1 is a 0-bump, 1-bump, or multi-bump solution.

(P2)Figure 1, lower left panel, shows the superimposition of the Haudin et al.14 logarithmic spiral formula (1.1) onto a spiral precipitate. For appropriately chosen r0,θ0, Haudin et al.14 show how their formula gives a close approximation to a large part of the precipitate where concentration is maximal. Next, we show how Theorem 1.2 can be used to generate a solution of system (1.2) whose concentration is maximal along logarithmic spiral formula equation (1.24) below, which is equivalent to the Haudin et al. formula (1.1). The first step is to conclude from properties (1.18) and (1.19) in Theorem 1.2 that ρ1 and ψ(r)ψ0cln(r/r^) ( means “is asymptotic to”) as r, where r^,ψ0 are constants. These properties, together with the fact that ρ(r)>0r>0, imply that, as radial distance r from the spiral tip increases from r=0, the magnitude ρ(r) of concentration continuously increases [Fig. 2, sol. (I)] until ρ(r)1 at a value r^>0, and (1.6) satisfies

(1.20)

over a finite range r^rr^L, where L depends on |c|. Thus, qualitative property (E0) holds. Below, we complete our derivation of logarithmic spiral approximations by separately considering the cases Ω=0 and Ω0.

2. (P3) Stationary logarithmic spirals

When Ω=0,(1.20) becomes

(1.21)

Thus, system (1.21) corresponds to a solution of constant concentration

(1.22)

along the stationary logarithmic spiral

(1.23)

where A is a constant. Substituting c=θ0 and A=ψ0 into Eq. (1.23) gives the logarithmic spiral formula

(1.24)

where θ0<0 if c>0, and θ0>0 if c<0. Formula (1.24), which holds when radial distance r from the spiral tip exceeds r^, is equivalent to the Haudin et al.14 formula (1.1). Formula (1.1) was not designed to model boundary conditions, e.g., the interaction of a spiral precipitate with the boundary wall of the Hele–Shaw cell. Instead, it is used to approximate the shape of part of a logarithmic shaped precipitate (Fig. 1, lower left) within the cell. Their analysis of 193 precipitates shows that r0 and θ0 in formula (1.1) satisfy

(1.25)

Next, we show how our approach can be used to generate a stationary logarithmic spiral solution (Fig. 1, lower right) satisfying formula (1.24), with θ replaced by θπ2, for parameter values, which lie in the range (1.25). Following Haudin et al.,14 we also do not take into account boundary conditions, instead focusing on deriving a formula to approximate the shape of part of a precipitate within the cell. For this, we set (μ,ε)=(0.8,2) and rescale the spatial variable r. Recall (Fig. 2, lower panels) that numerical shooting gives a 0-bump solution satisfying system (1.15)(1.17). From properties (1.17), we conclude that c=1.31 and r^=18 in (1.21); hence,

(1.26)

Next, let ψ0=0 and introduce the scaling r~=0.5r18. Then, approximation (1.26) becomes

(1.27)

Property (1.27) implies that (U,V)(0,1) along the stationary logarithmic spiral

(1.28)

Solving Eq. (1.28) for r^ gives the equivalent formula

(1.29)

where r~(π2)=.5 and θ0=1.31 lie in the range (1.25). The only effect of π2 in Eq. (1.29) is to cause a 90° rotation of axes in Fig. 1 (lower right), which gives the graph of (1.29) over the range π2θ7π4. Thus, formula (1.29) gives an approximation which is qualitatively comparable to the approximation in Fig. 1 (lower left). Finally, to complete our understanding of the case Ω=0, we investigate the behavior of solutions of the underlying spatially independent kinetic system,

(1.30)

ρ=U2+V2, at a fixed spatial point, where rr^, along the logarithmic spiral (1.23). At such a point, because the concentration of the logarithmic spiral solution satisfies (1.22), we assume that the corresponding initial concentration for Eq. (1.30) is (U(0),V(0))=(cos(A),sin(A)). The resultant solution of Eq. (1.30) is not oscillatory. Instead, it remains constant and satisfies (U(t),V(t))=(cos(A),sin(A))t0. This is consistent with the fact that logarithmic spiral (1.23) is stationary.

3. (P4) Rotating logarithmic spiral waves

When Ω0 and r1,(1.20) corresponds to a spiral wave of constant concentration (U,V)(cos(A),sin(A)) along the logarithmic spiral Ωt+θ+ψ0+cln(rr^)=A, where A is a constant, which rotates with frequency Ω about r=0.4 In particular, when ψ0=0,A=π2, and r1,(1.20) corresponds to a spiral wave of constant concentration

(1.31)

along the rotating logarithmic spiral

(1.32)

Thus, within the context of (E2)(E3) described above for chicken retina, (1.31) and (1.32) correspond to a logarithmic spiral wave of depolarization, which rotates with frequency Ω and period T=2πΩ. Next, we show how our rotating spiral wave generates behavior corresponding to action potential properties described in (E3). For this, we focus on a fixed spatial point corresponding to the location of a sensor in the tissue. At this point, we again analyze the behavior of the underlying nonlinear kinetic system,

(1.33)

where ρ=U2+V2. When the rotating spiral wave reaches the spatial point, we assume, because of (1.31), that it generates initial stimulus (U(0),V(0))=(0,1) for (1.33), corresponding to a stimulus of depolarization given at the location of the sensor in the tissue by the SD rotating spiral wave. The resultant solution of (1.33) is (U,V)=(sin(Ωt),cos(Ωt)). It has period T=2πΩ, which is exactly the period of one rotation of the spiral wave solution. Thus, corresponding to the action potential behavior described in (E3), we conclude that, over the time interval [0,T], the rotating wave goes through one rotation, and the corresponding solution of the underlying spatially independent kinetic system also goes through exactly one oscillation.

Numerical computations predict the existence of an abundance of solutions of initial value problem (1.13) and (1.14), which correspond to logarithmic spiral solutions of the Kopell–Howard lambda–omega model. In this work, we have developed a topological shooting technique to prove the existence of a specific solution, which has the qualitative property ρ(r)>0r>0. Our approach has also allowed us to prove other qualitative properties, e.g., (P1)(P4), of solutions of (1.13) and (1.14), which agree with observed phenomena. In the near future, we hope to combine our topological shooting techniques with the functional analysis approach of Cohen et al.4 to prove the existence of more complex families of solutions such as multi-bump solutions shown in Fig. 2. A second important direction of future theoretical research is the development of techniques to prove stability properties of logarithmic spiral solutions of the Kopell–Howard system.

In Sec. II, we give the proof of Theorem 1.2.

The first step of the proof is to set

(2.1)

Then, when |ε|>0, initial value problem (1.13) and (1.14) becomes

(2.2)
(2.3)
(2.4)

When |ε|>0, it is easily verified that solving Eqs. (2.2)(2.4) is equivalent to solving Eqs. (1.13) and (1.14). It follows from standard theory that the solution of Eqs. (2.2)(2.4) exists for r>0 as long as ρ(r)0 and 0<ρ(r)1. Thus, to prove Theorem 1.2, it suffices to prove

Theorem 2.1
Let μ>0. If 0<|ε|1, there is a solution of Eqs. (2.2)(2.4) such that
(2.5)
(2.6)

Outline of proof. We first give an outline of the basic steps of the proof.

(I) We use a topological shooting method to prove Theorem 2.1. Toward this end, the first step is to show that two ranges of μ,ε,α exist where (i)ρ(r)=0 before ρ(r)=1, and (ii)ρ(r)=1 before ρ(r)=0. This is done in Lemmas 2.2 and 2.3.

(II) Because we are assuming 0<|ε|1, it is essential to first completely determine the behavior of solutions when ε=0. This is done in Lemma 2.4.

(III) In Lemma 2.5, we use Lemma 2.4 to further restrict ranges of μ,ε,α where solutions of (2.2)(2.4) satisfy (i)ρ=0 before ρ=1, and (ii)ρ=1 before ρ=0.

(IV) In (2.68), we define the shooting set A to consist of α>0 values such that ρ=0 before ρ=1, and prove that Aϕ and bounded above.

(V) We prove that A is open. This is done with the help of three technical results (Lemmas 2.6–2.8).

(VI) Complete the proof of Theorem 2.1.

Step I. First, in Lemma 2.2, we determine a range of μ,ε,α where ρ=0 before ρ=1.

Lemma 2.2
Let
(2.7)
Then, rα=rα(ε)(0,8] exists such that the ρ component of the solution of Eqs. (2.2)(2.4) satisfies
(2.8)
Proof.
We focus on the interval 0<r8. It follows from Eqs. (2.3) and (2.4) that
(2.9)
First, we conclude from Eqs. (2.3), (2.4), and (2.9) and an integration that
(2.10)
for r(0,8] as long as ρ(r)>0 and 0<ρ(r)1. Since 0|ε|0.18, it follows from Eq. (2.10) that 0ε2h20.01 for r(0,8] as long as ρ(r)>0 and 0<ρ(r)0.99 Second, we conclude from Eq. (2.2) that, if 0|ε|ε0, then
(2.11)
for r(0,8] as long as ρ(r)>0 and 0<ρ<.99 Thus, (ρ+ρr)<0 and two integrations give
(2.12)
when 0|ε|ε0 and 0<αα0, as long as ρ>0. Next, we assume for contradiction that μ,ε,α exist in the range (2.7) such that
(2.13)
It follows from (2.2) and (2.13) that (ρ+ρr)αr, and two integrations give
(2.14)
Because 0<ρ(r)0.992, we can substitute (2.14) into (2.11), integrate, and get
(2.15)
Combining (2.14) and (2.15) gives
(2.16)
Since 0<αα0, it is easily verified that the right side of Eq. (2.16) is negative when r=6. This contradicts assumption (2.13). Thus, when 0|ε|ε0 and 0<αα0, an rα(0,8] exists such that ρ(r)>0r(0,rα),ρ(rα)=0,ρ(rα)<0.992 and ρ(rα)0. It remains to prove that ρ(rα)<0. Suppose that ρ(rα)=0. It follows from Eqs. (2.2), (2.3), and (2.10) that
(2.17)
since 0|ε|<0.18,rα8,0<h(rα)rα, and h(rα)(1ρ(rα))1+μ1. Thus, ρ(rα)<0, hence ρ(r)>0 and ρ(r)<0 on a positive interval (rαη,rα), contradicting ρ(r)>0r(0,rα). Thus, ρ(rα)<0. This completes the proof.

In Lemma 2.3, we determine a range of μ,ε,α, where ρ=1 before ρ=0.

Lemma 2.3
Let μ>0,|ε|0 and α1312. There exists rα(0,1] such that the solution of Eqs. (2.2)(2.4) satisfies
(2.18)
Proof.
First, Eq. (2.2) can be written as
(2.19)
We conclude from Eqs. (2.4) and (2.19) and the derivation of Eq. (2.10) that
(2.20)
Next, it follows from Eq. (2.2) that
(2.21)
for r(0,1] as long as ρ(r)>0 and 0<ρ(r)1; hence, two integrations give
(2.22)
The right side of Eq. (2.22) is greater than or equal 1 when r=1 and α1312. From this property [Eqs. (2.20) and (2.22)], we conclude that Eq. (2.18) holds at some r=rα(0,1]. This completes the proof.

Step II. Next, we determine the behavior of solutions when ε=0. In this case, initial value problem (2.2)(2.4) reduces to the decoupled system,

(2.23)
(2.24)
(2.25)
Lemma 2.4

Let μ>0 and ε=0. There exists α0>0 such that the solution of initial value problem (2.23)(2.25) satisfies

(i) if 0<α<α0, then rα>0 exists such that
(2.26)
(ii) If α=α0 then L0>0 and L1>0 exist such that
(2.27)
(2.28)
(2.29)
(2.30)
(iii) If α>α0 then rα>0 exists such that
(2.31)
Proof.
First, for each α>0, it follows from Eq. (2.24) that
(2.32)
for r0 as long as ρ(r)>0 and 0ρ(r)1. This accounts for the upper and lower bounds on h(r) in Eqs. (2.26), (2.30), and (2.31). Second, Greenberg12 proved the existence of a unique α0>0 such that the solution, denoted by ρ0, of initial value problem,
(2.33)
satisfies
(2.34)
We use these properties below in proving (i)–(iii). We now prove (i). Let α1(0,α0) and let ρ1 satisfy
(2.35)
Let [0,rmax) denote the maximal interval of existence of ρ1. We assume, for contradiction, that property (2.26) does not hold, and that
(2.36)
Then, it follows from Eqs. (2.33)(2.35) that
(2.37)
and an integration gives
(2.38)
We observe that ρ0(r)ρ1(r)>0 on a small interval (0,δ) since α1<α0. Suppose that there is a first r^(0,rmax) such that ρ0(r^)ρ1(r^)=0. Then, the right side of (2.38) is positive at r=r^,ρ0(r^)ρ1(r^)0, and we conclude from (2.38) that
(2.39)
However, the left side of Eq. (2.39) is non-positive, contradicting property (2.39). We conclude that
(2.40)
Next, if rmax=, then it follows from Eq. (2.35) that ρ1()=1, contradicting the uniqueness of the Greenberg solution, which satisfies (2.33) and (2.34). If rmax<, then it must be the case that ρ1 is unbounded on (0,rmax), contradicting (2.40). Therefore, property (2.36) cannot hold, and a first rα1(0,rmax) exists such that ρ1(rα1)=0. Then, ρ1(rα1)0 which implies, by (2.35), that 0<ρ1(rα1)11rα12<1. If ρ1(rα1)=0, then (2.35) implies that ρ1(rα1)=2ρ1(rα1)rα12<0, hence a positive interval (rα1η,rα1) exists such that ρ1(r)>0 and ρ1(r)<0r(rα1η,rα1), contradicting ρ1(r)>0r(0,rα1). Thus, ρ1(rα1)<0, which completes the proof of (i).

Next, we prove (iii). For this, let α1>α0, let ρ1 denote the corresponding solution of (2.35), and again let [0,rmax) denote the maximal interval of existence of ρ1. Then, we note that Eq. (2.38) holds for all r[0,rmax). Next, since α1>α0, then ρ1(r)ρ0(r)>0 and ρ1(r)ρ0(r)>0 on a small interval (0,γ). Suppose that a first r^(0,rmax) exists such that ρ1(r^)ρ0(r^)=0. Then, ρ1(r)ρ0(r)>0 for all r(0,r^) and it follows from Eq. (2.38) that

(2.41)

However, the left side of Eq. (2.41) is positive, a contradiction. We conclude that

(2.42)

Next, we assume, for contradiction, that property (2.31) does not hold, and that

(2.43)

If rmax=, then it follows from Eqs. (2.35), (2.42), and (2.43) that ρ1()=1, contradicting the uniqueness of the Greenberg solution ρ0. If rmax<, then ρ1(r) must be unbounded on (0,rmax), contradicting (2.43). Thus, we conclude that there is a first rα1(0,rmax) such that ρ1(rα1)=1. This completes the proof of (iii).

Finally, we prove (ii). The first step is to observe that the ρ component of the solution of Eqs. (2.23)(2.25) is the unique solution, found by Greenberg,12 of Eq. (2.33) satisfying property (2.34). Next, we show that limrrρ(r)=0. First, if δ>0 exists such that lim infrrρ(r)>δ>0, then ρ(r)>δ2r when r1, and an integration gives limrρ(r)=, contradicting limrρ(r)=1. Thus, lim infrrρ(r)=0. If lim suprrρ(r)=L>0 for some L>0, then a positive, increasing, unbounded sequence (rN) exists such that

(2.44)

However, it follows from Eqs. (2.23) and (2.44) that

(2.45)

contradicting (rρ(r))|r=rN0. Thus, lim suprrρ(r)=0, and we conclude that limrrρ(r)=0. Also, since limr0rρ(r)=0 and rρ(r) is continuous on (0,), then L0>0 exists such that rρ(r)L0r0. This proves property (2.28). Next, we prove that limrr2(1ρ(r))=1. The first case to eliminate is limrr2(1ρ(r))>1. Then, since limrρ(r)=1, there exists k>1 such that

(2.46)

An integration gives limrrρ=, contradicting limrrρ=0. Thus, it is not possible that limrr2(1ρ(r))>1. The next case to eliminate is 0limrr2(1ρ(r))<1. Then, since limrρ(r)=1, there exists k(0,1) such that

(2.47)

An integration gives limrrρ=, contradicting limrrρ=0. Thus, 0limrr2(1ρ(r))<1 is not possible. Next, suppose that

(2.48)

If lim suprr2(1ρ(r))>1, then there is a positive, increasing, unbounded sequence (rN), and an L>1 such that for all N1,

(2.49)

The first step in obtaining a contradiction to assumption (2.49) is to differentiate r2(1ρ(r)), and get

(2.50)

Combining the assumption (r2(1ρ(r)))|r=rN=0 with (2.50), we conclude that

(2.51)

Next, a differentiation of (2.50) gives

(2.52)

Combining Eq. (2.23) for ρ and (2.52), we obtain

(2.53)

Finally, we set r=rN in (2.53), combine the resulting equation with (2.51), and get

(2.54)

It follows from the first inequality in (2.49), and the fact that limrρ(r)=1, that the right side of (2.54) is positive when N1, contradicting (r2(1ρ(r)))|r=rN0. Thus, lim sup(r2(1ρ(r)))1N1. Next, suppose that

(2.55)

Then, a positive, increasing, unbounded sequence (rN), and L(0,1) exist such that

(2.56)

It follows from (2.56) and the properties limrrρ(r)=0 and limrρ(r)=1 that the right side of (2.54) is negative when N1, contradicting (r2(1ρ(r)))|r=rN0. Thus, property (2.48) cannot hold; hence, limrr2(1ρ(r)) exists, and we conclude that limrr2(1ρ(r))=1. Next, since limr0+r2(1ρ(r))=0 and r2(1ρ(r)) is continuous on (0,), then L1>0 exists such that r2(1ρ(r))L1r0. This proves property (2.29). Finally, property (2.30) follows from formula (2.32) combined with properties (2.27) and (2.29). This proves (ii).

Step III. In Lemma 2.5, we further restrict ranges of μ,ε,α where solutions of Eqs. (2.2)(2.4) satisfy (i)ρ=0 before ρ=1, and (ii)ρ=1 before ρ=0.

Lemma 2.5

Let μ>0. There exist ε>0,α1(0,α0) and α2(α0,1312) such that solutions of Eqs. (2.2)(2.4) satisfy

(i) if 0<|ε|ε and 0<αα1, then rα>0 exists such that
(2.57)
(ii) if 0<|ε|ε and αα2, then rα>0 exists such that
(2.58)
(iii) there exists r100 such that, if 0|ε|ε and α1αα2, then
(2.59)
(2.60)
Remark

The values r,L0, and L1 are independent of specific choice of ε and α in the range 0|ε|ε and α1αα2. This property will play a key role in completing our topological shooting proof of Theorem 2.1.

Proof.
First, we prove (iii). Recall from Lemma 2.2 that α0=1308. It follows from Lemma 2.2, Lemma 2.4 [part (ii)], and continuity of solutions with respect to parameters, which δ>0 and εδ>0 exist, with
(2.61)
such that if 0<|ε|εδ and α1αα2, then r100 exists such that (2.59) and (2.60) hold. This proves (iii). Next, we prove (i). In Lemma 2.2, we proved that if 0<|ε|ε0 and 0<αα0 then rα>0 exists such that
(2.62)
Hence, property (2.57) holds. Next, since [α0,α1] is compact, it follows from part (i) of Lemma 2.4, and continuity of solutions with respect to parameters, that an ε1>0 exists such that if 0<|ε|ε1 and α0αα1, then there is an rα>0 such that property (2.57) holds. This proves (i). (ii) remains to be proved. First, it follows from Lemma 2.3 that if |ε|>0 and α>1312, then rα>0 exists such that
(2.63)
Since [α2,1312] is compact, it follows from part (ii) of Lemma 2.4, and continuity of solutions with respect to parameters, that an ε2>0 exists such that if 0<|ε|ε2 and α2α1312, then there is an rα>0 such that property (2.58) holds. Finally, we define ε=min{εδ,ε0,ε1,ε2}. It follows from the properties given above that the proofs of (i)–(ii)–(iii) are all valid when 0<|ε|ε¯.

Step IV. Next, we define the topological shooting set, A, and show that Aϕ and bounded above. First, we define constants

(2.64)
(2.65)
(2.66)
(2.67)

Next, let μ>0 and 0<|ε|ε¯ be fixed and define the topological shooting set A={α>α1,|thenthereexistsrα>0suchthatthesolutionof(2.2)(2.4) satisfies

(2.68)

The definition of rα implies that ρ(rα)>0 and ρ(rα)0. From these properties and Eq. (2.2), it follows that ρ(rα)11rα2<1. Thus, 0<ρ(rα)<1αA. We need to prove that Aϕ, open and bounded. Lemma 2.5 [part (i)] and continuity of solutions with respect to parameters imply that αA if 0<αα11, hence Aϕ. Lemma 2.5 [part (ii)] implies that αAαα2; hence, A is bounded above.

Step V. Prove that A is open. For this, we need to derive upper bounds for |ε|rh(r),rρ(r), and r2(1ρ(r)). This is done in Lemmas 2.6–2.8, with the help of constants B0,B1,M0,M,D,Q,ε¯ defined above. For each αA, define

(2.69)

Thus, [0,rM) is the largest subinterval of [0,rα) where 0r2(1ρ(r))<M.

Lemma 2.6
Let μ>0,0|ε|ε¯,α1<αα2 and αA. Then, the solution of Eqs. (2.2)(2.4) satisfies
(2.70)
Proof.
We consider two cases. First, suppose that 0<rMr. Then, 0<ρ(r)<1 and ρ(r)>0r(0,rM], and we conclude from Eq. (2.10) that
(2.71)
Next, suppose that rM>r. Since r2(1ρ(r))Mr(0,rM), then
(2.72)
We conclude from (2.66), (2.67), (2.71), and (2.72) that 0|ε|rh(r)0.1r[0,rM].
Lemma 2.7
Let μ>0,0|ε|ε¯,α1<αα2, and αA. Then, the solution of Eqs. (2.2)(2.4) satisfies
(2.73)
Proof.
Suppose, for contradiction, that r~ρ(r~)=2L0+3 for some r~(0,rM]. It follows from (2.59) and continuity that r^ exists, with r<r^<r~, such that
(2.74)
Since |ε|rh(r)0.1 (by Lemma 2.6), and 0<ρ(r)<1r[r^,r~], then Eq. (2.2) gives (rρ(r))1.01rr[r^,r~]. An integration from r^ to r shows that
(2.75)
Hence, exp(21.01)r^r~rM. Also, since 2L0+1rρ(r)r[r^,r~], and r^>r, then ρ(r^)0.95,ρ(r)1r and an integration from r^ to r gives
(2.76)
The right side of (2.76) is greater than one when r=r~ since exp(21.01)r^r~rM, contradicting 0<ρ(r)<1r(0,rM].
Lemma 2.8
Let μ>0,0|ε|ε¯,α1<αα2 and αA. Then, rM=rα and the solution of Eqs. (2.2)(2.4) satisfies
(2.77)
Proof.
Suppose, for contradiction, that rM<rα. Then, rM2(1ρ(rM))=M=B1M0>M0, and it follows from Eqs. (2.60), (2.64), and (2.65) and continuity that r<r^<rM exists such that
(2.78)
A differentiation gives (r2(1ρ(r)))=2r(r2(1ρ(r)))r2ρ2r(r2(1ρ(r))) for all r[r^,rM]. It follows from Eqs. (2.65) and (2.78) and an integration over [r^,rM] that
(2.79)
Thus,
(2.80)
Next, ρ(r)0.95r[r^,rM] since r^>r. Also, r2(1ρ(r))M0=2L1+3.01 for all r[r^,rM]. From these properties, Lemma 2.6 and Eq. (2.2), we obtain
(2.81)
Since rρ(r)2L0+3r[r^,rM] (by Lemma 2.7), an integration gives
(2.82)
The right side of (2.82) is zero when r=exp(2L0+31.9)r^. From this property and Eq. (2.80), it follows that there is an r¯ such that r^<r¯exp(2L0+31.9)r^rM and r¯ρ(r¯)=0, a contradiction since r¯<rα. We conclude that rM=rα, and the definition of rM implies that property (2.77) holds. This completes the proof of Lemma 2.8.

Finally, we prove that A is open. Let αA. Then, rα>0 exists such that

(2.83)

If ρ(rα)<0, then continuity of solutions with respect to initial conditions implies that

(2.84)

Thus, if ρ(rα)<0αA, then property (2.84) holds for each αA; hence A is open. Suppose, however, that an αA exists such that property (2.83) holds, and ρ(rα)=0. Then, we conclude from Eqs. (2.2)(2.3) that

(2.85)

It follows from Lemma 2.6 that 0|ε|rαh(rα)0.1. It follows from Lemma 2.8 and Eq. (2.67) that 0|ε|rα2(1ρ(rα))1+μ0.1. Combining these properties with (2.85) gives ρ(rα)1.98ρ(rα)rα3<0. Thus, a positive interval (rαη,rα) exists such that ρ(r)>0 and ρ(r)<0 for all r(rαη,rα), contradicting property (2.83) since ρ(r)>0 when r(rαη,rα). We conclude that ρ(rα)<0αA; hence, A is open.

Step VI. Complete the proof of Theorem 2.1. Define

(2.86)

The first property we need to prove is that the solution of (2.2) and (2.3), with initial conditions ρ(0)=h(0)=0andρ(0)=α, satisfies

(2.87)

Suppose that (2.87) does not hold. If r^>0 exists such that ρ(r)>0r(0,r^] and ρ(r^)=1, then continuity of solutions with respect to initial conditions implies the same property holds if αA and 0<αα1, contradicting the definition of α. Next, suppose that rα>0 exists with ρ(r)>0r(0,rα) and ρ(rα)=0. If ρ(rα)<1, then αA, a contradiction since A is open. If ρ(rα)=1, then ρ(rα)0. However it follows from Eq. (2.2) that ρ(rα)>0, a contradiction. Thus, rα does not exist and (2.87) holds. Next, we prove that r2(1ρ(r))Mr>0. If not, there is an r¯>0 such that r¯2(1ρ(r¯))>M. This property and continuity of solutions with respect to initial conditions imply that, if αA and 0<αα1, then rα>r¯ and r¯2(1ρ(r¯))>M, contradicting Lemma 2.8. Thus, 0<r2(1ρ(r))Mr>0, hence limrρ(r)=1. Finally, we conclude from (2.10) and the fact that 0<r2(1ρ(r))Mr>0, that

(2.88)

Thus, property (2.5) holds and the proof of Theorem 2.1 is complete.

This paper is dedicated to Professor Richard Field on the occasion of his 80th birthday.

The author thanks the referees for valuable suggestions.

The authors have no conflicts to disclose.

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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