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.
I. INTRODUCTION
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.
Upper left: a logarithmic spiral shaped cobalt chloride precipitate in a chemical garden. Note that, as radial distance from the spiral tip increases from 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).
Upper left: a logarithmic spiral shaped cobalt chloride precipitate in a chemical garden. Note that, as radial distance from the spiral tip increases from 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).
A. Chemical gardens
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
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):
(E) As radial distance from the spiral tip increases from 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.
B. Spreading depression (SD)
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
(E) “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.”
(E) Figures 1(c) and 1(d) of Ref. 29 show that each time a spiral wave goes through one complete rotation, over a period a sensor records one spike in potential, of amplitude duration plus refractory time
C. The model and previous results
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 (E), (E), and (E) 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
where and correspond to components of concentrations and and are functions of which denotes the magnitude of concentration referred to in property (E). 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 and
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
For this representative, canonical choice of and 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 and 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 , Then, Eq. (1.2) becomes
Next, let denote polar coordinates of the plane and assume that functions and exist such that
where is a constant. The corresponding solution of Eq. (1.2) is
When (1.6) represents a spiral wave in concentration that rotates with frequency about .4 When (1.6) represents a stationary spiral in concentration. The function denotes the magnitude of concentration. Further details for specific choices, and are given in properties (P), (P), and (P) following the statement of Theorem 1.2. Substituting (1.5) into Eq. (1.4), we conclude that and satisfy gives the system,
Cohen et al.4 investigate the existence of solutions of (1.7) such that and are bounded and regular at so that concentrations are bounded with bounded gradients. Thus, they impose standard initial conditions and As , they assume that approaches a constant and hence They make simple assumptions on and ,
(H exists such that
(H and exist such that . Under these assumptions, they make use of the Schauder fixed point theorem (p. 158 of Ref. 13) to prove
(Cohen et al.4)
3. Computations
To gain insight, Cohen et al.4 set and investigate Eq. (1.7) for the representative, generic choice,
Thus, they solve initial value problem
Here, and either or These parameters, as well as a rescaling of the spatial variable in Eq. (1.13), can be used to model spiral properties obtained from experimental data. When Cohen et al.4 compute solutions using numerical shooting, where is the shooting parameter. The first step is to observe that, when and there exists such that and (Lemma 2.3). This reduces the range of to , where a solution can exist such that Next, when , a computation shows that before As is increased, this procedure is continued until a maximal value, is found where and Cohen et al.4 show that by making small changes in this value of 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, of this solution satisfies and This solution satisfies chemical garden property (E 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), For the second solution (labeled II), is not strictly increasing. There is one relative maximum, followed by a relative minimum, and then increases to as Thus, we refer to the first solution, which satisfies property (E 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 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 and First, when 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 , we find (Fig. 2, lower panels) that a 0-bump solution exists, and, upon comparing with we obtain
Thus, following Cohen et al.,4 we conclude that the numerical shooting procedure gives an approximation to a true solution, which satisfies
An integration gives
where is an arbitrary constant. In part (P), 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.
Upper left: components of 0-bump and 1-bump solutions of Eqs. (1.13) and (1.14) computed in Ref. 4 when Upper right: components of 0-bump, 2-bump, 3-bump solutions when Lower left: component of 0-bump solution when Lower right: Comparison of the component of the solution with shows that
D. Goals and statement of main results
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 and 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 0-bump solutions exist for each but 1-bump solutions, and possibly other multi-bump solutions, exist only over a limited range of Thus, our goal in Theorem 1.2 is to prove the existence of 0-bump solutions for the representative choice of and 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 (E)–(E)–(E).
1. Qualitative properties of solutions
Below, in (P)–(P), we describe qualitative properties of solutions of (1.13) and (1.14), which are predicted by our results.
(P) The primary difference between Theorems 1.1 and 1.2 is that Theorem 1.2 proves the existence of 0-bump solutions where ; however, it is not possible to conclude if the solution in Theorem 1.1 is a 0-bump, 1-bump, or multi-bump solution.
(P) 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 , 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 and ( means “is asymptotic to”) as where are constants. These properties, together with the fact that imply that, as radial distance from the spiral tip increases from the magnitude of concentration continuously increases [Fig. 2, sol. (I)] until at a value and (1.6) satisfies
over a finite range where depends on Thus, qualitative property (E) holds. Below, we complete our derivation of logarithmic spiral approximations by separately considering the cases and
2. (P3) Stationary logarithmic spirals
When (1.20) becomes
Thus, system (1.21) corresponds to a solution of constant concentration
along the stationary logarithmic spiral
where is a constant. Substituting and into Eq. (1.23) gives the logarithmic spiral formula
where if and if Formula (1.24), which holds when radial distance from the spiral tip exceeds 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 and in formula (1.1) satisfy
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 replaced by 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 and rescale the spatial variable 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 and in (1.21); hence,
Next, let and introduce the scaling Then, approximation (1.26) becomes
Property (1.27) implies that along the stationary logarithmic spiral
Solving Eq. (1.28) for gives the equivalent formula
where and lie in the range (1.25). The only effect of in Eq. (1.29) is to cause a rotation of axes in Fig. 1 (lower right), which gives the graph of (1.29) over the range 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 , we investigate the behavior of solutions of the underlying spatially independent kinetic system,
at a fixed spatial point, where 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 The resultant solution of Eq. (1.30) is not oscillatory. Instead, it remains constant and satisfies This is consistent with the fact that logarithmic spiral (1.23) is stationary.
3. (P4) Rotating logarithmic spiral waves
When and (1.20) corresponds to a spiral wave of constant concentration along the logarithmic spiral where is a constant, which rotates with frequency about .4 In particular, when , and (1.20) corresponds to a spiral wave of constant concentration
along the rotating logarithmic spiral
Thus, within the context of (E)–(E) described above for chicken retina, (1.31) and (1.32) correspond to a logarithmic spiral wave of depolarization, which rotates with frequency and period Next, we show how our rotating spiral wave generates behavior corresponding to action potential properties described in (E). 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,
where When the rotating spiral wave reaches the spatial point, we assume, because of (1.31), that it generates initial stimulus 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 It has period which is exactly the period of one rotation of the spiral wave solution. Thus, corresponding to the action potential behavior described in (E), we conclude that, over the time interval the rotating wave goes through one rotation, and the corresponding solution of the underlying spatially independent kinetic system also goes through exactly one oscillation.
E. Conclusions
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 Our approach has also allowed us to prove other qualitative properties, e.g., (P)–(P), 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.
II. PROOF OF THEOREM 1.2
The first step of the proof is to set
When 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 as long as and Thus, to prove Theorem 1.2, it suffices to prove
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 exist where (i) before and (ii) before This is done in Lemmas 2.2 and 2.3.
(II) Because we are assuming it is essential to first completely determine the behavior of solutions when This is done in Lemma 2.4.
(III) In Lemma 2.5, we use Lemma 2.4 to further restrict ranges of where solutions of (2.2)–(2.4) satisfy (i) before and (ii) before
(IV) In (2.68), we define the shooting set to consist of values such that before and prove that and bounded above.
(V) We prove that 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 where before
In Lemma 2.3, we determine a range of , where before .
Step II. Next, we determine the behavior of solutions when In this case, initial value problem (2.2)–(2.4) reduces to the decoupled system,
Next, we prove (iii). For this, let let denote the corresponding solution of (2.35), and again let denote the maximal interval of existence of Then, we note that Eq. (2.38) holds for all Next, since then and on a small interval Suppose that a first exists such that Then, for all and it follows from Eq. (2.38) that
However, the left side of Eq. (2.41) is positive, a contradiction. We conclude that
Next, we assume, for contradiction, that property (2.31) does not hold, and that
If , then it follows from Eqs. (2.35), (2.42), and (2.43) that contradicting the uniqueness of the Greenberg solution If , then must be unbounded on contradicting (2.43). Thus, we conclude that there is a first such that This completes the proof of (iii).
Finally, we prove (ii). The first step is to observe that the 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 First, if exists such that then when and an integration gives contradicting Thus, If for some then a positive, increasing, unbounded sequence exists such that
contradicting Thus, and we conclude that Also, since and is continuous on then exists such that This proves property (2.28). Next, we prove that The first case to eliminate is Then, since there exists such that
An integration gives contradicting Thus, it is not possible that The next case to eliminate is Then, since there exists such that
An integration gives contradicting Thus, is not possible. Next, suppose that
If , then there is a positive, increasing, unbounded sequence and an such that for all
The first step in obtaining a contradiction to assumption (2.49) is to differentiate and get
Combining the assumption with (2.50), we conclude that
Next, a differentiation of (2.50) gives
It follows from the first inequality in (2.49), and the fact that that the right side of (2.54) is positive when contradicting Thus, Next, suppose that
Then, a positive, increasing, unbounded sequence and exist such that
It follows from (2.56) and the properties and that the right side of (2.54) is negative when contradicting Thus, property (2.48) cannot hold; hence, exists, and we conclude that Next, since and is continuous on then exists such that 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 where solutions of Eqs. (2.2)–(2.4) satisfy (i) before and (ii) before
The values , and are independent of specific choice of and in the range and This property will play a key role in completing our topological shooting proof of Theorem 2.1.
Step IV. Next, we define the topological shooting set, and show that and bounded above. First, we define constants
The definition of implies that and From these properties and Eq. (2.2), it follows that Thus, We need to prove that open and bounded. Lemma 2.5 [part (i)] and continuity of solutions with respect to parameters imply that if hence Lemma 2.5 [part (ii)] implies that ; hence, is bounded above.
Step V. Prove that is open. For this, we need to derive upper bounds for , and This is done in Lemmas 2.6–2.8, with the help of constants defined above. For each , define
Thus, is the largest subinterval of where
Finally, we prove that is open. Let Then, exists such that
If , then continuity of solutions with respect to initial conditions implies that
Thus, if then property (2.84) holds for each ; hence is open. Suppose, however, that an exists such that property (2.83) holds, and Then, we conclude from Eqs. (2.2)–(2.3) that
It follows from Lemma 2.6 that . It follows from Lemma 2.8 and Eq. (2.67) that . Combining these properties with (2.85) gives Thus, a positive interval exists such that and for all contradicting property (2.83) since when We conclude that ; hence, is open.
Step VI. Complete the proof of Theorem 2.1. Define
The first property we need to prove is that the solution of (2.2) and (2.3), with initial conditions satisfies
Suppose that (2.87) does not hold. If exists such that and then continuity of solutions with respect to initial conditions implies the same property holds if and contradicting the definition of Next, suppose that exists with and If , then a contradiction since is open. If , then However it follows from Eq. (2.2) that a contradiction. Thus, does not exist and (2.87) holds. Next, we prove that If not, there is an such that This property and continuity of solutions with respect to initial conditions imply that, if and , then and contradicting Lemma 2.8. Thus, hence Finally, we conclude from (2.10) and the fact that that
Thus, property (2.5) holds and the proof of Theorem 2.1 is complete.
DEDICATION
This paper is dedicated to Professor Richard Field on the occasion of his 80th birthday.
ACKNOWLEDGMENTS
The author thanks the referees for valuable suggestions.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
DATA AVAILABILITY
Data sharing is not applicable to this article as no new data were created or analyzed in this study.