Textbook Nonlinear Systems Vol 1 Mathematical Theory and Computational Methods Victoriano Carmona Ebook All Chapter PDF
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Understanding Complex Systems
Victoriano Carmona
Jesús Cuevas-Maraver
Fernando Fernández-Sánchez
Elisabeth García-Medina Editors
Nonlinear
Systems,
Vol. 1
Mathematical Theory and
Computational Methods
Springer Complexity
Springer Complexity is an interdisciplinary program publishing the best research and
academic-level teaching on both fundamental and applied aspects of complex systems—
cutting across all traditional disciplines of the natural and life sciences, engineering, economics,
medicine, neuroscience, social and computer science.
Complex Systems are systems that comprise many interacting parts with the ability to
generate a new quality of macroscopic collective behavior the manifestations of which are
the spontaneous formation of distinctive temporal, spatial or functional structures. Models
of such systems can be successfully mapped onto quite diverse “real-life” situations like
the climate, the coherent emission of light from lasers, chemical reaction-diffusion systems,
biological cellular networks, the dynamics of stock markets and of the internet, earthquake
statistics and prediction, freeway traffic, the human brain, or the formation of opinions in
social systems, to name just some of the popular applications.
Although their scope and methodologies overlap somewhat, one can distinguish the
following main concepts and tools: self-organization, nonlinear dynamics, synergetics,
turbulence, dynamical systems, catastrophes, instabilities, stochastic processes, chaos, graphs
and networks, cellular automata, adaptive systems, genetic algorithms and computational
intelligence.
The three major book publication platforms of the Springer Complexity program are the
monograph series “Understanding Complex Systems” focusing on the various applications
of complexity, the “Springer Series in Synergetics”, which is devoted to the quantitative
theoretical and methodological foundations, and the “Springer Briefs in Complexity” which
are concise and topical working reports, case studies, surveys, essays and lecture notes of
relevance to the field. In addition to the books in these two core series, the program also
incorporates individual titles ranging from textbooks to major reference works.
Fernando Fernández-Sánchez
Elisabeth García-Medina
Editors
123
Editors
Victoriano Carmona Fernando Fernández-Sánchez
Departamento de Matemática Aplicada II Departamento de Matemática Aplicada II
Universidad de Sevilla Universidad de Sevilla
Sevilla, Spain Sevilla, Spain
This Springer imprint is published by the registered company Springer Nature Switzerland AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
To Prof. Antonio Castellanos Mata, founder
and head of the Group of
Electrohydrodynamics and Cohesive
Granular Media of the University of Seville.
Antonio directed research projects for more
than 30 years, and this made it possible for
him to organize two laboratories at the
University. Dedicating a lot of efforts to pure
science, Antonio was also interested in
practical problems and collaborated with
industry (Xerox Corporation, Novartis, Dow
Corning, IFPRI).
Antonio belonged to a generation that
played an important role in the revival of
physics in Spain. In 2013, he was awarded
the Prize FAMA for the research career by
the University of Seville. In his last years,
Antonio worked on thermodynamics in
relativity and on triboelectricity in fine
powders As a researcher, Antonio combined a
strong theoretical mind, experimental
intuition, profound understanding of physics
of phenomena, and passionate love for
science.
Preface
vii
viii Preface
xi
xii Contents
xiii
xiv Contributors
1 Introduction
The famous Lorenz system was derived from a simplified model of convection in
the atmosphere: a two-dimensional fluid cell is warmed from below and cooled
from above and the resulting convective motion is modeled by a partial differential
equation. This system is obtained after a Galerkin approximation, that is, the variables
are expanded into an infinite number of modes and all except three of them are put
to zero [86, 97]: ⎧
⎨ ẋ = σ (y − x),
ẏ = ρx − y − x z, (1)
⎩
ż = −bz + x y.
The aim of this survey is to comment some results on the Lorenz system obtained
by means of the Local Bifurcation Theory (see, for instance, [72, 82, 102]), in the way
we briefly summarize below. When an autonomous system is analyzed an usual target
is the knowledge of its dynamics in certain zones of the parameter space. In practice,
the parameter space is divided in regions, bounded by bifurcation loci, and the goal
is to determine the qualitative behaviour in each of such regions. This can be done in
several steps. First, the detection of the equilibrium points and the analysis of the lin-
earization around such equilibria, allow to show the presence of linear degeneracies
(nonhyperbolicities) for some values of the control parameters. Second, the computa-
tion of approximations of the center manifold (and also of the reduced system on the
center manifold) enables to reduce the dimension of the problem, transforming the
reduced system into the corresponding normal form by means of changes of variables
(sometimes a reparametrization of the time is also needed). Symbolic computation
algorithms greatly facilitate this task. Third, the analysis of the unfolding of the nor-
mal form in the nondegenerate cases, provides local information on the bifurcation
sets. Furthermore, possible nonlinear degeneracies giving rise to a higher codimen-
sion bifurcation problem can be detected at this step. Finally, from the information
achieved in the study of local bifurcations, good starting points for the application
of adequate numerical techniques can be obtained. This will provide a global picture
of the dynamics of the system in the parameter space (see, for instance, [64]).
To illustrate how the method described above allows to obtain a deep knowledge of
the dynamical system under consideration, we mention now two three-dimensional
systems with a very rich dynamics. On the one hand, for a modified van der Pol-
Duffing electronic oscillator, interesting information can be found in the following
references on some local and global bifurcations: Hopf and Takens–Bogdanov [17],
Hopf-pitchfork [18, 19, 21], triple-zero [67], periodic orbits bifurcations [4, 32,
59], homoclinic connections and some degeneracies [45, 65], T-points and some
degeneracies [5, 8, 58, 60, 62]. Secondly, in the case of the widely studied Chua’s
equation, the following references clarify how to deal with the corresponding local
and global bifurcations: Hopf [27], Takens–Bogdanov [20], Hopf-pitchfork [28],
triple-zero [29], homoclinic connections and some degeneracies [33–35], T-points
and some degeneracies [6, 25, 26, 61].
This work is organized as follows. In Sect. 2 we enumerate the linear degeneracies
that the equilibrium at the origin of the Lorenz system can exhibit. The analysis of the
pitchfork bifurcation is considered in Sect. 3. In Sect. 4 we present results on Hopf
bifurcations. Section 5 is devoted to Takens–Bogdanov bifurcations of equilibria as
well as of periodic orbits. Section 6 is dedicated to the study of resonances, whose
presence is motivated by the existence of torus bifurcations of periodic orbits. Finally,
some conclusions are reported in Sect. 7.
6 A. Algaba et al.
2 Linear Degeneracies
Along this work we consider Lorenz system (1) where σ , ρ and b are real parameters.
We exclude two degenerate situations: the system is linear if σ = 0 and non-isolated
equilibria on the z-axis exist for b = 0.
The Lorenz system (1) is invariant to the change (x, y, z) → (−x, −y, z). The
origin E 0 = (0, 0, 0) is always one equilibrium
√ √ for b(ρ − 1) > 0, two
point and,
symmetric nontrivial equilibria, E ± = (± b(ρ − 1), ± b(ρ − 1), ρ − 1), exist.
The linearization matrix of system (1) at the origin is
⎛ ⎞
−σ σ 0
⎝ ρ −1 0 ⎠ , (2)
0 0 −b
p = λ3 + p 1 λ 2 + p 2 λ + p 3 ,
where
p1 = b + 1 + σ, p2 = σ (1 + b − ρ) + b, p3 = −bσ (ρ − 1).
3 Pitchfork Bifurcation
For ρ = 1, the linearization matrix (2) has the eigenvalues 0, −(σ + 1), −b. There-
fore, as a consequence of its symmetry, the Lorenz system (1) exhibits a pitchfork
bifurcation. To study this bifurcation, we examine the Lorenz system at the critical
values of the parameters and use the linear change of variables given by
⎛ ⎞ ⎛ ⎞⎛ ⎞
x 1 σ 0 u
⎝ y ⎠ = ⎝ 1 −1 0 ⎠ ⎝ v ⎠ (3)
z 0 0 1 w
1 2
v = 0 + O(u3 ), w = u + O(u3 ),
b
and the third-order reduced system on the center manifold is
σ
u̇ = − u3 .
b(σ + 1)
Theorem 1 The locus in the (σ, ρ, b)-parameter space where the origin of the
Lorenz system undergoes a nondegenerate pitchfork bifurcation is defined by
ρ = 1, σ = 0, −1, b = 0.
4 Hopf Bifurcations
In this section we precis the principal results obtained in Ref. [2], devoted to the
analysis of Hopf bifurcations and their degeneracies in the Lorenz system (to do
that, the computation of some Lyapunov coefficients of the corresponding normal
form is needed [70, 72, 82, 102]). First, we consider the Hopf bifurcation of the
origin and later, the Hopf bifurcation exhibited by the nontrivial equilibria E ± .
As was mentioned in Sect. 2, the origin E 0 undergoes a Hopf bifurcation if σ =
−1, ρ > 1, b = 0. The corresponding normal form to third order, obtained with
the recursive algorithm developed in Ref. [66] is (see the details in Ref. [2, Sect. 2])
ṙ = a1 r 3 + · · · ,
(5)
θ̇ = 1 + b1r 2 + · · · ,
−b − 2
a1 = √ .
8 ρ − 1 4(ρ − 1) + b2
−1 −1
z z
−2
−3
−2 4
2
5 0
2
0 y −2 1
0
y −5 1 2 −4 −1
−2 −1 0
−2 x
x
Fig. 1 Two different perspectives of the phase space of the Lorenz system (1) for b = −2, σ = −1,
ρ = 2 where the origin undergoes a degenerate Hopf bifurcation of infinite codimension. Some
periodic orbits on the center manifold are drawn. Reproduced with permission from [2]. Copyright
(2015) by Springer
A Review on Some Bifurcations in the Lorenz System 9
The following statement sum up all the results on the Hopf bifurcation of the
origin.
Theorem 2 ([2, Theorem 1]) The locus in the (σ, ρ, b)-parameter space where the
origin of the Lorenz system undergoes a Hopf bifurcation is defined by
σ = −1, ρ > 1, b = 0.
This bifurcation is supercritical when b > −2 and subcritical if b < −2. A degen-
erate Hopf bifurcation of infinite codimension occurs if b = −2.
The rest of this section is devoted to the Hopf bifurcation of the nontrivial equilibria
(all the details can be found in Ref. [2, Sect. 3]). The standard techniques used in
the study of a Hopf bifurcation allow to determine, in a first step, the locus where it
occurs and to compute, in a second step, the Lyapunov coefficients that lead to the
detection of all the degeneracies this bifurcation can have. The results obtained are
summarized below.
Proposition 1 ([2, Proposition 2]) The nontrivial equilibria of the Lorenz system
experiment a Hopf bifurcation in the surface parameterized in explicit form by
−σ 2 − (3 − ρ)σ − ρ
Shnt = σ, ρ, b = : (σ, ρ) ∈ Ω ,
σ +ρ
with Ω = Ω1 ∪ Ω2 ∪ Ω3 ∪ Ω4 ∪ Ω5 , where
σ 2 + 3σ
Ω1 = (σ, ρ) ∈ R : σ < −1, ρ < ,
σ −1
Ω2 = {(σ, ρ) ∈ R : σ = −1, ρ < 1} ,
Ω3 = {(σ, ρ) ∈ R : −1 < σ < 0, ρ < −σ } ,
σ 2 + 3σ
Ω4 = (σ, ρ) ∈ R : 0 < σ < 1, ρ < ,
σ −1
σ 2 + 3σ
Ω5 = (σ, ρ) ∈ R : σ > 1, ρ > .
σ −1
−5
Ω − Ω5
−10
−10 −5 0 5
σ
where
Δ = σ 2 + (3 − ρ)σ + ρ,
N 1(σ, ρ) = 6σ 4 + 15σ 3 + 9σ 3 ρ + 35σ 2 ρ + σ 2 ρ 2
+21σρ 2 + 2σρ + σρ 3 + 5ρ 3 + ρ 4 ,
D1(σ, ρ) = 8(ρ − 1) σ 2 (ρ − 1)2 − (σ + ρ)2 Δ
× 4σ 2 (ρ − 1)2 − (σ + ρ)2 Δ ,
and of the second Lyapunov coefficient a2 of the fifth-order normal form for the
reduced system (whose expression appears in Ref. [2, Appendix A]) allow to find all
the degeneracies this Hopf bifurcation may experiment. This information is summa-
rized below.
Theorem 3 The nontrivial equilibria of the Lorenz system undergo a degenerate
Hopf bifurcation in the following cases:
1. The first Lyapunov coefficient a1 vanishes for all the values (σ, ρ) ∈ Ω where
the polynomial N 1(σ, ρ) is zero. A codimension-two bifurcation occurs in this
case when the second Lyapunov coefficient a2 is nonzero.
2. On the two points (σ, ρ, b) ∈ Shnt given by
and
P2 ≈ (−0.0100012, −0.0396965, −1.408456)
A Review on Some Bifurcations in the Lorenz System 11
15 0.1
ρ=σ(σ+3)/(σ−1) Ω ρ=σ(σ+3)/(σ−1)
5
σ+ρ=0 (A) σ+ρ=0
Ω−Ω5 Ω−Ω5
10
Ω 0.05
5 σ=0
σ=0 a =01
5 a1=0
ρ ρ=1 ρ 0
σ=1 (F)
0
(D)
−0.05
−5 (C) (D)
(B)
Ω − Ω5 (E)
−10 −0.1
−10 −5 0 5 −0.06 −0.04 −0.02 0 0.02 0.04 0.06
σ σ
Fig. 3 (Left) Projection onto the (σ, ρ)-plane of the locus where the first Lyapunov coefficient a1
is zero. When this curve is inside the region Ω, it corresponds to a degenerate Hopf bifurcation.
(Right) Zoom in a neighborhood of the origin. Reproduced with permission from [2]. Copyright
(2015) by Springer
a codimension-three Hopf bifurcation occurs because the first and the second
Lyapunov coefficient vanish simultaneously and the third one a3 is nonzero.
These are the unique codimension-three Hopf bifurcation points.
3. On the half-line given by σ = −1, b = −2, ρ < 1 a Hopf bifurcation of codi-
mension infinite occurs because the reduced system on the center manifold is
Hamiltonian (centers on center manifolds).
We would like to do several remarks on the above result. First, the region Ω is
split in six zones (see Fig. 3). A subcritical Hopf bifurcation occurs in the zones (A),
(C), (E) and (F), while on the contrary it is supercritical in the zones (B) and (D).
Remark that it is well-known that the Hopf bifurcation in the region where the three
parameters are positive (our region (A) that corresponds to Ω5 ) is always subcritical
[89, 95, 99].
Second, to guarantee the existence of the two codimension-three points P1 and P2
the Poincaré-Miranda theorem was used [81]. A detailed analysis of the roots of a
polynomial of degree 104 (it appears in the computation of the resultant of N 1(σ, ρ)
and the numerator of a2 ) is also needed.
Finally, as it occurs for the origin, the Hopf bifurcation of the nontrivial equilibria
has infinite codimension because the center manifold is an algebraic invariant surface,
namely x 2 + 2z = 0 (see Fig. 4). Moreover, the Hopf bifurcation of the nontrivial
equilibria only has this polynomial center manifold in the Lorenz system.
In the following we provide the results of some numerical continuations, obtained
with AUTO [52], in order to illustrate the dynamical consequences of the presence
of a Hopf bifurcation of codimension-three (that occurs at P1 and at P2 ). From this
degeneracy, a curve of cusp bifurcations of periodic orbits appears [70]. Thus, two
curves of saddle-node bifurcation contact tangentially at the cusp point, giving rise to
a semicubic parabola. Three periodic orbits exists in the system for proximate param-
12 A. Algaba et al.
1 5
0
0
−1 z
−2 −5
−3
−10
z −4 8
−5 6
4
−6
2
−7
y 0
−8 −2
−4
10
−6
0 5
0 −8 2 3 4 5
−10 −5 −2 −1 0 1
y −5 −4 −3
x x
Fig. 4 Two different perspectives of the phase space of the Lorenz system (1) for b = −2, σ =
−1, ρ = −2, where the nontrivial equilibria undergo a degenerate Hopf bifurcation of infinite
codimension. Some periodic orbits on the center manifold appear. Reproduced with permission
from [2]. Copyright (2015) by Springer
-0,52 -0,65
sn2
σ σ sn 2 h
h
Dh1
h sn 1
-0,53
cu 1 h Dh1
-0,66
-6,6 -6,4 -6,2 -6,6 -6,4 -6,2
ρ ρ
Fig. 5 Two partial bifurcation sets in the (ρ, σ )-plane in a neighborhood of the point P1 : (Left)
for b = −1.6; (Right) for b = −1.72. Reproduced with permission from [2]. Copyright (2015) by
Springer
degenerate point Dh1 on the curve h, the Hopf bifurcation varies from subcritical
(on the left) to supercritical (on the right). Now the saddle-node curve sn2 emerges
from Dh1 and remains above h.
5 Takens–Bogdanov Bifurcations
In this section we summarize the results obtained in Ref. [3], devoted to the analysis of
Takens–Bogdanov bifurcations in the Lorenz system. In the first part we mention the
analytical results in the case of the Takens–Bogdanov bifurcation of the equilibrium
at the origin. Secondly, we precis some numerical results on the existence of Takens–
Bogdanov bifurcations exhibited by periodic orbits.
As was stated in Sect. 2, the origin E 0 exhibits a Takens–Bogdanov bifurcation
when
σ = −1, ρ = 1, b = 0. (7)
The corresponding normal form to third order for the reduced system on the center
manifold, obtained with the recursive algorithm developed in Ref. [66], is (see the
details in Ref. [3, Sect. 2])
u̇ = v,
(8)
v̇ = a3 u3 + b3 u2 v,
with
1 −2 − b
a3 = , b3 = .
b b2
Whereas the coefficient a3 cannot vanish, a degenerate Takens–Bogdanov bifur-
cation occurs when b3 = 0, i.e. when b = −2.
As it is well known (see, for instance, [47, 72]), when b3 = 0, the nondegenerate
Takens–Bogdanov bifurcation is of heteroclinic type if a3 > 0 and of homoclinic
type for a3 < 0. Therefore, a nondegenerate Takens–Bogdanov of heteroclinic type
exists in the Lorenz system if b > 0 and of homoclinic type for b < 0 (b = −2).
In symmetric systems, the Takens–Bogdanov point TB appears when a curve of
pitchfork bifurcations of the origin Pi collapses with a curve of Hopf bifurcations of
the same equilibrium H. In the heteroclinic case (see Fig. 7 (Left)), a curve of hetero-
clinic connections of the nontrivial equilibria He emerges from TB. In the homoclinic
case (see Fig. 10 (Left)), three curves emerge from TB: h (of Hopf bifurcations of
the nontrivial equilibria), Ho (of homoclinic connections to the origin) and SN (of
saddle-node bifurcations of symmetric periodic orbits).
When the coefficient b3 of the normal form (8) vanishes (if b = −2) a nonlinear
degeneracy appears. Specifically, as the center manifold is an algebraic invariant
surface, the Takens–Bogdanov bifurcation has infinite codimension: the origin is a
center in the Lorenz system when b = −2, σ = −1 and ρ = 1. This fact is illustrated
in Fig. 6.
14 A. Algaba et al.
0
z −1
−0.5
−2
z −1 4
−1.5 2
−2 y 0
5
2 −2
1
0 0
y −1 −4 1 2
−5 −2 x −2 −1 0
x
Fig. 6 Two different perspectives of the phase space of the Lorenz system (1) for b = −2, σ = −1,
ρ = 1, where the origin undergoes a degenerate Takens–Bogdanov bifurcation of infinite codimen-
sion. Some periodic orbits on the center manifold x 2 + 2z = 0 appear. Reproduced with permission
from [3]. Copyright (2016) by Elsevier
18
Pi
H 16
-1 TB
DHe1
DHe 2 14
z
σ 12
Pi He
DHe3 10
-2
0 1 2 3 4 5 -6 -4 -2 0 2 4 6
ρ x
Fig. 7 (Left) For b = 1, partial bifurcation set on the (ρ, σ )-plane in a neighborhood of the Takens–
Bogdanov point TB (heteroclinic case). Three degeneracies DHe1 , DHe2 and DHe3 are present
on the curve of heteroclinic connections He. (Right) For b = 1 and ρ = 15, projection onto the
(x, z)-plane of the Shilnikov heteroclinic loop He existing for σ ≈ −3.874338. Reproduced with
permission from [3]. Copyright (2016) by Elsevier
σ = −1, ρ = 1, b = 0.
a Hopf of the origin if ρ > 1, a Hopf of the nontrivial equilibria when ρ < 1 and a
Takens–Bogdanov of the origin for ρ = 1.
On the other hand, it is interesting to comment that, in these three situations,
it is possible to find analytical expressions for the period of the orbits existing in
the center manifold x 2 + 2z = 0 (see Figs. 1, 4 and 6). Thus, by taking limit in the
corresponding expressions, for finite values of the parameter ρ, the existence of
superluminal periodic orbits (periodic orbits with unbounded amplitude and whose
period tends to zero) is demonstrated. All the details can be found in Ref. [36]. In
this work, it is also numerically shown that superluminal periodic orbits also exists
in other situations of physical interest when the parameter ρ tends to infinity.
In the rest of this section, we highlight the most important numerical results on the
Takens–Bogdanov bifurcation of the origin presented in Ref. [3, Sect. 3], which have
been obtained with AUTO [52]. Specifically, we present two partial bifurcation sets
in the (ρ, σ )-parameter plane. The first one, for b = 1, illustrates the heteroclinic
case (b = 1) whereas the second one, for b = −1.6, corresponds to the homoclinic
case. Note that, as for b = 0 a triple-zero degeneracy occurs, we obtain information
on both sides of such rich bifurcation.
In Fig. 7 (Left), for b = 1, a partial bifurcation set is drawn in a neighborhood
of the Takens–Bogdanov point on the (ρ, σ )-plane. According to the well-known
results in the heteroclinic case [47, 72], the Takens–Bogdanov point TB is placed on
the curve where the origin exhibits a pitchfork bifurcation Pi. From that point a curve
of Hopf bifurcation of the origin H emerges. As it is a supercritical Hopf bifurcation, a
stable symmetric periodic orbit arises at H. This periodic orbit disappears in the curve
He, where a heteroclinic orbit to the nontrivial equilibria occurs. In Fig. 7 (Right) a
heteroclinic loop is drawn for ρ = 15.
Three degeneracies are numerically detected on the curve He. For their descrip-
tion, the eigenvalues of the nontrivial equilibria are denoted by α ± βi, λ, and the
saddle-quantity δ = |α/λ| is considered. The first degeneracy He1 appears when the
nontrivial equilibria change from real saddle to saddle-focus. As δ > 1, this global
connection remains tame [44] and, in this way, a symmetric stable periodic orbit is
born from the curve He. A second degeneracy, He2 , is present when δ = 1, namely
the eigenvalues are resonant. At this point the heteroclinic orbit changes from tame
to chaotic Shilnikov [82, 102].
The third degeneracy He3 occurs because δ = 1/2 (null divergence). Since the
expression of the divergence in the Lorenz system (1) is divF = −(b + σ + 1) and
we have fixed b = 1, then divF = 0 along the straight line σ = −2. Observe that
divF has no dependence on the spatial variables but only on the system parameters.
This fact has important consequences on the bifurcations of the periodic orbits as we
briefly explain in the following. If γ (t) denotes a periodic orbit in the autonomous
system ẋ = F(x), then the variational equation is defined by the linear system
Perforating ulcer of the foot begins on the sole, beneath any of the
metatarso-phalangeal articulations, preferably the first or the fifth, or
under the heel, as a small pustule under the epidermis. This
ruptures, and the ulcer which results begins to extend in a direction
vertical to the surface, involving the deeper tissues or even opening
into the joint and destroying the bone. It appears rather like a sinus
than an ulcer, and is remarkable from the fact that it is not painful
and is insensitive to touch, although it may prevent the patient from
walking on account of extreme tenderness. The epidermis around
the sinus is thickened and insensitive, and there may be anæsthesia
of the entire sole of the foot, and even of the leg, although this is due
to the neuritis present, which is also the cause of the ulcer, rather
than to the ulceration. The circulation is sluggish in the affected
extremity; it becomes cyanotic on exposure to cold, and seems
peculiarly liable to become œdematous. The œdema may go on to
suppuration, and involve the articulations, and ankylosis of the
smaller joints may follow. This, too, is to be traced to the neuritis.
The skin of the foot becomes pigmented, and may be dry or covered
with offensive sweat. As the patient cannot walk while the ulcer
remains, the condition demands treatment. Rest, moist warm
applications, antiseptic lotions, scraping out the sinus, and other
surgical means appropriate to the treatment of ulcers and sinuses do
not often prove of benefit, and in obstinate cases recourse has been
had to amputation of the foot. Electrical treatment has been tried in
vain.
INDEX TO VOLUME V.
A.
684
in tubercular meningitis,
726
727
Abscess of the brain,
791
792
808
Achromatopsia, hysterical,
247
1224
1227
hypodermically in exacerbations of cerebral syphilis,
1015
40
41
Acrodynia,
1254
387
alcoholism,
586
cerebral anæmia,
776
encephalitis,
791
myelitis, spinal,
810
596
simple meningitis,
716
spinal meningitis,
749
pachymeningitis,
747
Æsthesodic system of encephalon, localization of lesions in,
81
69
153
750
of catalepsy,
315
of cerebral anæmia,
777
of chorea,
441
of chronic lead-poisoning,
680
686
688
of disseminated sclerosis,
883
951
of epilepsy,
470
of family form of tabes dorsalis,
871
77
707
of hysteria,
216
of hystero-epilepsy,
293
of insanity,
116
117
of intracranial hemorrhage and apoplexy,
927
of labio-glosso-laryngeal paralysis,
1173
of migraine,
406
1230
of myxœdema,
1271
of neuralgia,
1217
of paralysis agitans,
433
of progressive unilateral facial atrophy,
694
1000
of tabes dorsalis,
856
650
983
of tubercular meningitis,
725
spinal cord,
1090
of writers' cramp,
512
1147
Agraphia in hemiplegia,
957
42-44
Alalia (see
Speech, Disorders of
).
31
682
689
716
of cerebral hyperæmia,
765
of intracranial hemorrhage and apoplexy,
929
933
of tabes dorsalis,
854
of writers' cramp,
512
642
644
in cerebral anæmia,
789
in heat-exhaustion,
388
in insomnia,
380
381
677
673
675
in thermic fever,
397
Alcoholic abuse as a cause of epilepsy,
472
insanity,
175
202
630-633
LCOHOLISM
573
Classification,
573
Definition,
573
Diagnosis,
637
of acute alcoholism,
637
of chronic alcoholism,
638
of dipsomania,
639
of hereditary alcoholism,
639
Etiology,
575