# Bifurcation analysis on a hybrid systems model of intermittent hormonal therapy for prostate cancer

**ABSTRACT** Hybrid systems are widely used to model dynamical phenomena that are characterized by interplay between continuous dynamics and discrete events. An example of biomedical application is modeling of disease progression of prostate cancer under intermittent hormonal therapy, where continuous tumor dynamics is switched by interruption and reinstitution of medication. In the present paper, we study a hybrid systems model representing intermittent androgen suppression (IAS) therapy for advanced prostate cancer. Intermittent medication with switching between on-treatment and off-treatment periods is intended to possibly prevent a prostatic tumor from developing into a hormone-refractory state and is anticipated as a possible strategy for delaying or hopefully averting a cancer relapse which most patients undergo as a result of long-term hormonal suppression. Clinical efficacy of IAS therapy for prostate cancer is still under investigation but at least worth considering in terms of reduction of side effects and economic costs during off-treatment periods. In the model of IAS therapy, it depends on some clinically controllable parameters whether a relapse of prostate cancer occurs or not. Therefore, we examine nonlinear dynamics and bifurcation structure of the model by exploiting a numerical method to clarify bifurcation sets in the hybrid system. Our results suggest that adjustment of the normal androgen level in combination with appropriate medication scheduling could enhance the possibility of relapse prevention. Moreover, a two-dimensional piecewise-linear system reduced from the original model highlights the origin of nonlinear phenomena specific to the hybrid system.

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**ABSTRACT:**In hybrid dynamical systems including both continuous and discrete components, an interplay between a continuous trajectory and a discontinuity boundary can trigger a sudden qualitative change in the system dynamics. Grazing phenomena, which occur when a continuous trajectory hits a boundary tangentially, are well known as a representative of such phenomena. We demonstrate that a grazing phenomenon of a chaotic attractor can result in its sudden disappearance and initiate chaotic transients. The mechanism of this grazing-induced crisis is revealed in an illustrative example. Furthermore, we derive a formula to obtain the critical exponent of the power law on the mean duration of chaotic transients.Physics Letters A 08/2009; 373(35):3134-3139. · 1.63 Impact Factor - SourceAvailable from: Rui Liu[Show abstract] [Hide abstract]

**ABSTRACT:**Non-smooth or even abrupt state changes exist during many biological processes, e.g., cell differentiation processes, proliferation processes, or even disease deterioration processes. Such dynamics generally signals the emergence of critical transition phenomena, which result in drastic changes of system states or eventually qualitative changes of phenotypes. Hence, it is of great importance to detect such transitions and further reveal their molecular mechanisms at network level. Here, we review the recent advances on dynamical network biomarkers (DNBs) as well as the related theoretical foundation, which can identify not only early signals of the critical transitions but also their leading networks, which drive the whole system to initiate such transitions. In order to demonstrate the effectiveness of this novel approach, examples of complex diseases are also provided to detect pre-disease stage, for which traditional methods or biomarkers failed.Quantitative Biology. 06/2013; 1(2). - [Show abstract] [Hide abstract]

**ABSTRACT:**The behaviors of system which alternate between Duffing oscillator and van der Pol oscillator are investigated to explore the influence of the switches on dynamical evolutions of system. Switches related to the state and time are introduced, upon which a typical switched model is established. Poincaré map of the whole switched system is defined by suitable local sections and local maps, and the formal expression of its Jacobian matrix is obtained. The location of the fixed point and associated Floquet multipliers are calculated, based on which two-parameter bifurcation sets of the switched system are obtained, dividing the parameter space into several regions corresponding to different types of attractors. It is found that cascading of period-doubling bifurcations may lead the system to chaos, while fold bifurcations determine the transition between period-3 solution and chaotic movement.Communications in Nonlinear Science and Numerical Simulation 03/2014; 19(3):750-757. · 2.57 Impact Factor

Page 1

Physica D 237 (2008) 2616–2627

www.elsevier.com/locate/physd

Bifurcation analysis on a hybrid systems model of intermittent hormonal

therapy for prostate cancer

Gouhei Tanakaa,∗, Kunichika Tsumotoa,b, Shigeki Tsujia,b, Kazuyuki Aiharaa,b

aInstitute of Industrial Science, University of Tokyo, Tokyo 153-8505, Japan

bERATO Aihara Complexity Modelling Project, JST, Tokyo 151-0065, Japan

Received 16 August 2007; received in revised form 30 January 2008; accepted 22 March 2008

Available online 4 April 2008

Communicated by A. Mikhailov

Abstract

Hybrid systems are widely used to model dynamical phenomena that are characterized by interplay between continuous dynamics and discrete

events. An example of biomedical application is modeling of disease progression of prostate cancer under intermittent hormonal therapy, where

continuous tumor dynamics is switched by interruption and reinstitution of medication. In the present paper, we study a hybrid systems model

representing intermittent androgen suppression (IAS) therapy for advanced prostate cancer. Intermittent medication with switching between on-

treatment and off-treatment periods is intended to possibly prevent a prostatic tumor from developing into a hormone-refractory state and is

anticipated as a possible strategy for delaying or hopefully averting a cancer relapse which most patients undergo as a result of long-term hormonal

suppression. Clinical efficacy of IAS therapy for prostate cancer is still under investigation but at least worth considering in terms of reduction of

side effects and economic costs during off-treatment periods. In the model of IAS therapy, it depends on some clinically controllable parameters

whether a relapse of prostate cancer occurs or not. Therefore, we examine nonlinear dynamics and bifurcation structure of the model by exploiting

anumericalmethodtoclarifybifurcationsetsinthehybridsystem.Ourresultssuggestthatadjustmentofthenormalandrogenlevelincombination

with appropriate medication scheduling could enhance the possibility of relapse prevention. Moreover, a two-dimensional piecewise-linear system

reduced from the original model highlights the origin of nonlinear phenomena specific to the hybrid system.

c ? 2008 Elsevier B.V. All rights reserved.

PACS: 05.45.-a; 87.19.Xx; 87.53.Tf

Keywords: Hybrid systems; Piecewise linear systems; Grazing bifurcations; Limit cycle; Chaos; Prostate cancer; Intermittent androgen suppression

1. Introduction

Following great progresses of life science and nonlinear

science in recent years, a mathematical approach is becoming

a more promising methodology for advanced studies in

biomedical science. In particular, much attention has been

paid to mathematical and computational modeling of cancer

dynamics involving nonlinear biological interactions [1,2]. In

the context of prostate cancer, for example, dynamical systems

have been extensively used to describe tumor growth [3,4] and

temporal variations of biomarkers [5–7] in efforts to predict

∗Corresponding author. Tel.: +81 3 5452 6693; fax: +81 3 5452 6694.

E-mail address: gouhei@sat.t.u-tokyo.ac.jp (G. Tanaka).

medical conditions and help appropriate diagnoses. The main

concern of these studies is to understand the mechanism of

treatment-resistant tumor growth, or a cancer relapse, after

a remission period induced by hormone deprivation therapy.

A possible strategy to delay or prevent the progression to

hormone-resistance caused by prolonged hormone suppression

is to incorporate treatment interruption, which is known as

intermittent hormonal therapy repeating cycles of on-treatment

and off-treatment periods [8–15]. In this paper, we deal with

nonlinear dynamics and bifurcations in a mathematical model

of intermittent hormonal therapy for prostate cancer.

Prostate cancer is a disease characterized by uncontrolled

growth of cancer cells within the prostate gland in males.

The prostate gland is dependent on hormonal secretion by

0167-2789/$ - see front matter c ? 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.physd.2008.03.044

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G. Tanaka et al. / Physica D 237 (2008) 2616–2627

2617

the testes for growth and development. When production of

the male hormone (androgen) decreases, the prostate begins

to degenerate. Since this androgen-dependent (AD) nature of

the prostatic cells is shared by unusual malignant tumor cells,

namely AD cells of prostate cancer, androgen deprivation has

been the mainstay for treating advanced or metastatic prostate

cancer [16]. Most androgens synthesized and secreted from the

testes can be suppressed by surgical or chemical castration. The

influence of adrenal androgens remaining after castration can

be eliminated by the addition of nonsteroidal androgen receptor

antagonists, or antiandrogens. Combination of castration and

antiandrogens, referred to as combined or maximum androgen

blockade, can facilitate tumor regression [17]. The serum

prostate-specific antigen (PSA), which is a widely used

biomarkerforprostatecancer,enablesmonitoringofthedisease

progression through blood tests. Androgen deprivation initiated

in a state with a high PSA level promotes apoptotic death

of malignant cells and leads to a significant decrease of the

PSA value implying tumor regression. However, continuation

of androgen suppression not only causes side effects from

the treatment but also mostly results in a relapse by recurrent

tumor growth after a remission period [18]. A relapse of

prostate cancer is considered to be primarily due to the

progression to an androgen-independent (AI) state where the

tumor is no longer responsive to androgen deprivation [19].

Many efforts have been made to understand the properties of

androgen receptors and signal-transduction pathways leading

to a hormone-refractory prostate cancer [20–23]. Intermittent

androgen suppression (IAS) is an alternative approach to

maintain tumor sensitivity to androgen deprivation with the

possibility to prolong a relapse [8–15]. Although clinical

trials are still in progress for establishing the clinical efficacy

of IAS therapy [11,14,15], it has already been confirmed

that side effects and economic costs are at least reduced in

comparison with continued androgen suppression. Following

these advances in experimental and clinical studies of IAS, a

mathematical model of IAS therapy for prostate cancer was

presented [24–26].

IAS therapy aims at delaying prostate cancer relapse by

keeping androgen dependency of the tumor through repeating

on-treatment and off-treatment cycles. Correspondingly, the

IAS therapy model alternates between two different kinds of

dynamics. The androgen concentration tends to recover and

maintain the normal level a0 (nmol/l) during off-treatment

periods whereas it decays to almost zero during on-treatment

periods. These different androgen levels make much difference

in tumor dynamics regulated by the androgen concentration

and in variations of the serum PSA concentration. Interruption

and reinstitution of therapeutic administration for androgen

suppression are determined based on changes of the serum PSA

level in the model as well as in real treatments. It is assumed in

our model that administration is suspended when a decreasing

PSA level drops to a lower threshold value r0(ng/ml) whereas

it is resumed when an increasing PSA level reaches an upper

threshold value r1 (ng/ml) [24–26]. If this dosing strategy

is successful in relapse prevention, then the serum PSA

level remains around a region between r0and r1. Otherwise,

recurrent tumor growth eventually happens and thereby the

serum PSA level continues to increase explosively. In the model

study, occurrence and prevention of a relapse are characterized

by divergent and nondivergent solutions, respectively. Thus,

parameter conditions for relapse prevention can be revealed by

investigating the transition between two qualitatively different

regimes. In this paper, we examine bifurcation phenomena with

respect to changes of clinically controllable parameters such as

a0, r0, and r1. For this purpose, we utilize shooting methods

for locating bifurcation sets of a limit cycle in hybrid systems.

Moreover, the origin of nonlinear and chaotic dynamics of the

IAS therapy model is clarified through return plot analysis of

a piecewise linear system derived from the original model by

simplification.

Hybrid dynamical systems involving both continuous and

discrete variables have been pervasive especially in computer

science and control engineering [27–29]. Analysis and control

of nonlinear phenomena in hybrid dynamical systems are

practicallysignificantforindustrialapplicationssuchasprocess

control [30,31], mechanical systems [32], robotic control [33],

power systems [34], and power electronics [35,36]. A number

of methodologies have been developed for analysis and control

designofhybrid(piecewisesmooth)dynamicalsystems[37].In

nonlinear science, stability and bifurcations in hybrid systems

have been the main focus since early studies [38,39], because

interactions between a trajectory and borders for discrete events

can bring about complex phenomena. Recently much attention

has been paid to grazing phenomena [40] and border-collision

bifurcations [41]. Return map analysis is often effective for

understanding chaotic behavior in low-dimensional switched

dynamical systems [42]. Numerical methods for locating

bifurcation points of a periodic solution have been proposed by

appropriately formulating conditional equations [43]. Similar

methods have been developed to formulate grazing phenomena

of periodic and non-periodic trajectories [44].

Tumor growth under intermittent therapy can also be viewed

as a hybrid system composed of continuous tumor dynamics

and discrete events with interruption and reinstitution of

medication. In the model study, the clinical issue of how to

schedule on–off cycles of administration for preventing an

explosion of malignant cell populations can be reduced to

a mathematical problem of how to adequately set adjustable

parameter values for confining a solution in a finite region in

continuous state space. Therefore, it is significant to understand

nonlinear phenomena peculiar to hybrid systems, such as a

grazing bifurcation which occurs when a trajectory tangentially

hits a boundary set related to a discrete event [40]. Present

research on the IAS therapy model not only supports a

possible clinical advantage of IAS therapy but also highlights

the complexity generated as a fundamental feature of hybrid

systems.

In Section 2, we will describe shooting methods to specify

bifurcation sets of a limit cycle in a hybrid system by defining

an appropriate Poincar´ e map. Results of bifurcation analysis

of the IAS therapy model will be shown in Section 3. The

essential mechanism of nonlinear and complex behavior in the

IAS therapy model will also be elucidated by analysis of a

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G. Tanaka et al. / Physica D 237 (2008) 2616–2627

simplified model. Finally, the last section will be devoted to

conclusions.

2. Method

2.1. Hybrid systems

In this section, we introduce a method to analyze

bifurcations of a hybrid system consisting of two ordinary

differential equations described as follows:

Su:dx

where t ∈ R, x ∈ Rn, and u ∈ {0,1} denote time, continuous

state vector, and the binary discrete variable, respectively. For

simplicity, we assume that the vector field fu (u = 0,1)

defined on Rnis a C∞-class map for any state variables and

parameters. We consider a hybrid automaton [37] exhibiting

transitionsbetweensubsystems S0and S1,whichdescribestime

evolution of the continuous state x and the discrete state u.

We assume that switching from S1−uto Su(u = 0,1) occurs

when a trajectory in continuous state space hits the following

codimension-one surfaces:

dt

= fu(x),

u = 0,1,

(1)

Πu= {x ∈ Rn| qu(x) = 0},

where qu is a scalar-valued function of the continuous state

vector x. The solution of subsystem Su is represented as

follows:

u = 0,1,

(2)

x(t) ≡ ϕu(t,x0),

where x0= x(0) is an initial condition of the continuous state.

2.2. Limit cycle and Poincar´ e map

u = 0,1,

(3)

Let us begin with constructing a Poincar´ e map in order to

formulate the conditions of a limit cycle and its bifurcations

in a hybrid system. We consider a situation where a trajectory

of the hybrid system (1) exhibits a nearly periodic motion

with repeating transitions between S0and S1alternately. Fig. 1

shows a schematic illustration of a part of a trajectory crossing

the sections Π0and Π1alternately in continuous state space. A

trajectory leaving from the initial point x0on Π0intersects with

Π1atx1withtravelingtimeτ0underthedynamicsinsubsystem

S0. At the moment of intersection, the subsystem changes from

S0to S1, and then the trajectory returns to Π0with traveling

time τ1. We define maps from one section to the other one as

follows:

T0: Π0→ Π1;x0?→ x1≡ ϕ0(τ0,x0),

T1: Π1→ Π0;x1?→ ϕ1(τ1,x1).

By treating Π0as a Poincar´ e section, a Poincar´ e map T :

Π0→ Π0is defined by a composite of two submaps (4) and (5)

as follows [43]:

(4)

(5)

T = T1◦ T0,

where the total return time is given by τ = τ0+ τ1. It should

(6)

Fig. 1. Schematic illustration for construction of the Poincar´ e map P on the

local coordinate for a trajectory of the hybrid system (1), where the trajectory

under consideration exhibits a nearly periodic motion crossing two sections of

discrete events alternately.

be noted here that the global coordinate x0 ∈ Π0 ⊂ Rn

is redundant to identify any point on the codimension-one

Poincar´ e section. Thus, we define a local coordinate w ∈ Σ ⊂

Rn−1and introduce the projection map h and its inverse h−1:

h : Π0→ Σ;x0?→ w,

h−1: Σ → Π0;w ?→ x0,

such that the Poincar´ e map on the local coordinate is

represented as follows:

(7)

(8)

P : Σ → Σ;

Similarly, a Poincar´ e map for an m-folded limit cycle crossing

the Poincar´ e section for m times is given by Pm= h◦Tm◦h−1.

w ?→ h ◦ T ◦ h−1(w).

(9)

2.3. Stability and local bifurcation

A limit cycle in continuous state space corresponds to a

fixed point of the Poincar´ e map P, which satisfies the following

equation:

Flc(w) ≡ P(w) − w = 0.

In general this equation is not analytically solvable as usual for

nonlinear systems; it can be numerically solved with respect to

(n −1) unknown variables w by using numerical methods such

as Newton’s method. The Jacobian matrix of Flcrequired for

the numerical computation is given by:

(10)

DFlc(w) =∂P

where Indenotes the n×n identity matrix. See Appendix A for

details on the calculation procedure of first-order derivatives of

the Poincar´ e map P.

Stability of a fixed point of P, which is equivalent to that of

a corresponding limit cycle, can be evaluated by characteristic

(Floquet) multipliers of the Jacobian matrix of P at the fixed

point, which are the solutions of the following characteristic

equation:

?∂P

If all the exponents are located inside the unit circle on the

complex plane, then the fixed point is stable. Loss of stability

of the fixed point occurs when one of the exponents crosses

∂w− In−1,

(11)

χ(µ) ≡ det

∂w− µIn−1

?

= 0.

(12)

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G. Tanaka et al. / Physica D 237 (2008) 2616–2627

2619

Fig. 2. A grazing bifurcation point λ can be characterized by a trajectory

tangent to the event-triggering section Π0at x = x0. A trajectory hits the

section at λhit> λ whereas it misses to touch the section at λmiss< λ.

Fig. 3. Construction of maps¯Tkand cumulative traveling time ¯ τkfor an m-

folded limit cycle.

the unit circle outward from the inside by changing values of

system parameters, which is accompanied with a codimension-

one local bifurcation.

For specifying a local bifurcation of a fixed point, the fixed

point condition (10) is combined with the eigenvalue condition

as follows:

?P(w) − w

These equations are simultaneously solved with respect to n

unknown variables (w,λ) where λ is a bifurcation parameter.

We fix the value of µ for a specific type of local bifurcation:

µ = 1 for saddle-node bifurcation, µ = −1 for period-

doubling bifurcation, and µ

bifurcation. The Jacobian matrix of Flbneeded to solve Eq. (13)

by Newton’s method is written as follows:

The first- and second-order derivatives of P required to

calculate components of the matrix (14) are described in

Appendix A.

Flb(w,λ) ≡

χ(µ)

?

= 0.

(13)

=

eiθfor Neimark–Sacker

DFlb=

∂P

∂w− In−1

∂χ(µ)

∂w

∂P

∂λ

∂χ(µ)

∂λ

.

(14)

2.4. Grazing bifurcation

A grazing bifurcation exhibited by hybrid systems is

concerned with a sudden change in the interaction between a

trajectory and an event-triggering boundary for switching of

subsystems [44]. When a trajectory is transversal but almost

tangent to the Poincar´ e section Π0as illustrated in Fig. 2, a

slight change of a system parameter value can lead to loss

of the intersection and induce a change of dynamics. This

critical change in system behavior, called grazing bifurcation,

can be characterized by a trajectory tangent to the event-

triggering section. If such a qualitative change occurs for a

periodic solution, then it is called periodic grazing [44]. At

the right grazing bifurcation with respect to the section Π0,

the directional vector of the trajectory is orthogonal to the

normal vector of the section at the point of tangency. Since the

transversality condition (A.14) at Π0(see Appendix A) does

not hold for a tangency point of a grazing trajectory, we need

to sort out the conditions of periodic grazing by regarding the

return time of a limit cycle as an unknown variable.

We consider a grazing bifurcation of an m-folded limit

cycle corresponding to an m-periodic point of the Poincar´ e

map for a natural number m in the following. Let us suppose

that a grazing trajectory starting from x0on Π0transversally

intersects with Π0and Π1for (2m − 1)-times alternately and

then finally becomes tangent to Π0. For convenience, we define

the following maps:

¯T2k: Π0→ Π0;x0?→ x2k≡ (T1◦ T0)k(x0),

k = 0,1,...,m − 1,

¯T2k+1: Π0→ Π1;x0?→ x2k+1≡ T0◦ (T1◦ T0)k+1(x0),

k = 0,1,...,m − 1,

as illustrated in Fig. 3. Correspondingly, the traveling time from

xk to xk+1 is denoted by τk(xk) for k = 0,1,...,2m − 1.

Then, the cumulative traveling time from x0to xk, denoted by

¯ τk, satisfies the following formula:

¯ τk+1(x0) = ¯ τk(x0) + τk(xk),

where ¯ τ0(x0) ≡ 0. The above setting provides a map dependent

on the total return time τ and the initial condition x0as follows:

(15)

(16)

k = 0,...,2m − 1,

(17)

¯T(τ,x0) = ϕ1(τ − ¯ τ2m−1(x0),x2m−1),

where τ > ¯ τ2m−1(x0).

By using the map¯T, the conditions of a grazing bifurcation

of an m-folded periodic solution can be formulated as follows:

∂x

The first equation means that a trajectory leaving from x0on Π0

is periodic with return time τ. The second equation indicates

that the trajectory is located on Π0at time τ. The third equation

represents the condition that the trajectory is tangent to section

Π0at time τ. The set of Eq. (19) are simultaneously solved

with respect to (n +2) unknown variables (x0,τ,λ) where λ is

a bifurcation parameter. The Jacobian matrix of Fgbrequired to

numerically solve Eq. (19) is given by:

(18)

Fgb(x0,τ,λ) ≡

¯T(τ,x0) − x0

q0(¯T(τ,x0))

∂q0

· f1(¯T(τ,x0))

= 0.

(19)

DFgb

=

∂¯T

∂x0

∂q0

∂x

?∂2ϕ1

− In

∂¯T

∂x0

∂¯T

∂τ

∂¯T

∂λ

∂q0

∂x

?∂2ϕ1

∂¯T

∂τ

∂q

∂x

?∂2ϕ1

∂¯T

∂λ+∂q0

∂¯T

∂λ

∂q0

∂x

·

∂t∂x

∂¯T

∂x0

?

∂q0

∂x

·

∂t∂x

∂¯T

∂τ

?

∂q0

∂x

·

∂t∂x

∂λ+∂2ϕ1

∂t∂λ

?

.

(20)

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G. Tanaka et al. / Physica D 237 (2008) 2616–2627

Refer to Appendix B for details of the calculation procedure of

the derivatives of¯T.

3. Results

3.1. Mathematical model of IAS therapy

The numerical method for bifurcation analysis introduced

in the previous section is used to investigate the mathematical

model of IAS therapy for prostate cancer [24,25]. The

purpose of this section is to elucidate how dynamical behavior

is influenced by some control parameters with respect to

clinical importance in prostate cancer therapy. As introduced

in Section 1, at least two different kinds of prostatic cells

are involved in androgen suppression therapy. AD cells are

sensitivetoandrogensuppressionandarelikelytoinducetumor

regression by their apoptosis, while AI cells are insensitive to

androgen deprivation and considered to be responsible for a

relapse. The IAS therapy model represents growth of a tumor

consisting of a mixed dynamical assembly of AD and AI cancer

cells under intermittent administration. Population growths of

both AD and AI cells are dependent on androgen concentration

a (nmol/l) whose dynamics relies on the binary variable u

indicating whether medication of maximum androgen blockade

is administered (u = 1) or not (u = 0). Administration is

reinstated when an increasing PSA concentration y (ng/ml)

rises up to the upper threshold valuer1, whereas it is interrupted

when a decreasing PSA level falls to the lower threshold value

r0.

Due to these discrete events concerning administration, the

IAS therapy model is formulated as a hybrid system [24,25]

consisting of subsystems described as follows:

dxD

dt

dxI

dt

da

dt

where continuous state variables xD

populations of AD and AI cancer cells, respectively. The

discrete variable u indicates on-administration periods for u =

1 or off-administration ones for u = 0. As in clinical practice,

suspension and reinstitution of administration are conducted in

the following way:

?1 → 0

Serum PSA concentration y serves as a biomarker for growth

of a prostatic tumor as follows: y = f (xD,xI) = cDxD+cIxI

where cD= cI= 1 for the sake of simplicity. Net growth rates

of AD and AI cells are denoted by gDand gI, respectively, and

the rate of mutation from AD cells to AI ones is indicated by

m. These factors deeply affecting tumor dynamics are given by

functions of androgen concentration as follows [24,25]:

= (gD(a) − m(a))xD,

= m(a)xD+ gI(a)xI,

= −γ(a − a0(1 − u)),

(21)

and xI

represent

u :

when y = r0and dy/dt < 0

when y = r1and dy/dt > 0.

0 → 1

(22)

gD(a) = αD

?

k1+(1 − k1)a

a + k2

?

− βD

?

k3+(1 − k3)a

a + k4

?

,

(23)

(24)

gI(a) = αI(1 − ea) − βI,

m(a) = m1

?

1 −a

a0

?

.

(25)

Here gD(a) is the difference between proliferation and

apoptosis rates of AD cells, which is approximately equal

to αD− βD in an androgen-rich environment. Values of αD

and βD can be estimated from the experimental data [45]

on cell proliferation and apoptosis rates of AD cells for

hormonally untreated patients, respectively. In an androgen-

depleted environment, AD cells are not able to proliferate

and their apoptosis rate can be estimated from real data [12–

15] of the serum PSA during androgen suppression. Plausible

functions with these conditions can be realized by adjusting

parameters k1, k2, k3, and k4. Growth rate of AI cells is also

dependent on the androgen level [46] because their growth

is still dependent on the androgen receptor, although details

of their androgen dependence remain unclear. Therefore we

assume that gI(a) is a linear function of the androgen level [24,

25] for the sake of simplicity, which is equal to αI− βIin an

androgen-depleted environment. Values of αI and βI can be

estimated from experimental data [45] for hormonally failing

patients. The androgen dependence of the proliferation rate of

AI cells is controlled by the parameter e. We also assume that

the mutation rate m(a) is linearly increasing with a decrease of

androgen level, because continuation of androgen suppression

is considered to enhance the mutation from AD cells to AI ones.

The IAS therapy model is regarded as a hybrid automaton

with continuous state vector x = (xD,xI,a) and discrete state

variable u. Event-triggering sections related to discrete events

(22) are defined as follows:

Π0= {x | q0(x) = f (xD,xI) − r0= 0},

Π1= {x | q1(x) = f (xD,xI) − r1= 0}.

A Poincar´ e map T with respect to the Poincar´ e section Π0

can be derived as demonstrated in Section 2. By choosing the

following transformation between local and global coordinates

on Π0:

(26)

(27)

h : Π0→ Σ;(xD,xI,a) ?→ (w1,w2) = (xD,a),

h−1: Σ → Π0;(w1,w2) ?→ (xD,xI,a)

= (w1,r0− w1,w2),

we obtain the Poincar´ e map P = h ◦ T ◦ h−1on the local

coordinate. Consequently, bifurcation analysis of a limit cycle

in the IAS therapy model is reduced to that of a fixed or a

periodic point of P.

(28)

(29)

3.2. Bifurcation phenomena

In the previous study [24,25], we have investigated the IAS

therapy model, mainly focusing on the effect of net growth rate

of AI cells. On the other hand, in the present paper, we clarify

the detailed influence of clinically controllable parameters such

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G. Tanaka et al. / Physica D 237 (2008) 2616–2627

2621

as threshold values of the serum PSA, r0 and r1, and the

normal androgen level a0. The other parameters are fixed as

follows [24,25]:

γ = 0.08,

αD= 0.0204,

βI= 0.0168,

k1= 0,

The values of γ, m1, and e are set so that a relapse occurs in

numerical simulations around several years after the initiation

of continuous androgen suppression, which are clinically

plausible. We adopt growth rates of AD and AI cells in the

bone metastasis case [45]. The values of αDand βDand those

of αI and βI are estimated from experimental data [45] for

hormonally untreated patients and hormonally failing patients,

respectively. We set k1= 0 because AD cells do not multiply

without androgens. The value of k2is set to make the growth

rate curve of the AD cells plausible. The value of βDk3, which

is the apoptosis rate of AD cells in an androgen-depleted

environment, is estimated by fitting an exponential function

to the decreasing serum PSA concentration during androgen

suppression in actual data for simplicity [12–15]. The value of

k4is set so that the evolution of AD cells changes from decrease

toincreaseatarounda = 5.See[24,25]fordetaileddiscussions

on parameter setting.

Fig. 4 shows a bifurcation diagram for variations of a0

and r0, indicating the parameter region of relapse prevention

together with bifurcation sets of limit cycles. The colored

region corresponds to relapse prevention, where a solution is

not divergent but confined within a finite region. The diagram

shows that the lower threshold value r0to restart administration

should be relatively small for preventing a relapse by the

following reasons. The first reason is that the range of a0for

relapse prevention is wider for a smaller value of r0as shown

in Fig. 4. In this regard, however, a too small value of r0

resulting in a relapse should be avoided. The second reason

is that if the lower threshold value r0 is near the upper one

r1, the frequency of the switching of on-treatment and off-

treatment periods is too high to adequately evaluate progression

of the disease and efficacy of IAS. The normal androgen level

a0 also largely affects the tumor growth and the possibility

of relapse prevention. In fact, when a0 is fixed to be less

than a certain value, a relapse is inevitable for any choice

of r0value. This property suggests that the normal androgen

level during off-treatment periods is related to the potency of

IAS therapy. Although the mainstay for treatment of advanced

prostate cancer is androgen suppression, some experimental

studies suggested that AI tumors that have escaped from

androgen deprivation therapy might be inhibited by high levels

of androgen supplementation [47,48]. From a mathematical

viewpoint, the reason why a diverging solution is unavoidable

for a low value of a0is that the net growth rate of AI cells

is consistently positive independently of the androgen level as

given by Eq. (24) and thereby the flow in continuous state space

is always expanding. This result supports the possibility that

androgen supplementation during off-treatment periods might

m1= 0.00005,

βD= 0.0076,

e = 0.015,

αI= 0.0242,

k2= 2,

k3= 8,

k4= 0.5.

(30)

Fig. 4. Bifurcation diagram of the IAS therapy model (21)–(25) with r1= 30.

The parameter region of relapse prevention corresponds to that of nondivergent

solutions. Period-doubling and grazing bifurcations of an m-folded limit cycle

are denoted by PDmand Gm, respectively. Bifurcation phenomena along the

horizontal arrowed line at r0= 25 are shown in Fig. 5.

Fig. 5. 1-parameter bifurcation diagram of the IAS therapy model (21)–(25)

with r0= 25 and r1= 30. For a fixed value of a0, the intersecting points

with the Poincar´ e section Π0are plotted. Stable and unstable limit cycles are

indicated by solid and dashed lines, respectively. Period-doubling and grazing

bifurcation points are denoted by PDmand Gm, respectively.

be beneficial for delaying or averting a relapse depending on

the net growth rate of AI cells.

Let us further examine detailed bifurcations of limit cycles

in Fig. 4. Solid and dashed curves indicate period-doubling

and grazing bifurcation sets, respectively. Fig. 5 shows a one-

parameter bifurcation diagram along the horizontal arrowed

line at r0 = 25 in Fig. 4. With decrease of a0, a stable

limit cycle undergoes successive period-doubling bifurcations

and develops into chaotic solutions. Fig. 6(a)–(c) show a limit

cycle, a period-doubled limit cycle, and a chaotic attractor,

respectively. The limit cycle which loses its stability at the first

period-doubling bifurcation PD1finally disappears through a

grazing bifurcation G1as shown in Fig. 6(e). In a similar way,

the period-doubled limit cycle generated at PD2also exhibits

a loss of stability and vanishes at another grazing bifurcation

G2as shown in Fig. 6(d). The chaotic attractor resulting from

the period-doubling cascade is considered to disappear due

to transient grazing [44] by phase space analysis. In fact,

for a certain fixed parameter value, it depends on the initial

condition on Π0whether a trajectory returns to Π0again or

Page 7

2622

G. Tanaka et al. / Physica D 237 (2008) 2616–2627

Fig. 6. Oribital motions of the IAS therapy model (21)–(25) with r0= 25 and

r1= 30: (a) A stable limit cycle with a0= 36.4; (b) A stable 2-folded limit

cycle with a0= 35.2; (c) A chaotic solution with a0= 34.9; (d) A grazing

bifurcation of an unstable two-folded limit cycle with a0= 33.5; (e) A grazing

bifurcation of an unstable limit cycle with a0= 30.5.

not. Qualitatively different motions are separated at a boundary

set on the section, where a trajectory is grazing with respect

to the section. Therefore, a contact between the edge of a

chaotic attractor and the boundary set of initial conditions on

the section can cause a disappearance of the chaotic attractor.

We will characterize transient grazing with a simplified model

in Section 3.3. The pair of a grazing bifurcation point Gmand

a period-doubling one PDmseems to be nested with increase

of m for variation of the parameter a0. The mechanism of this

structure is analogous to that of a similar bifurcation pattern

with interplay between border collisions and a period-doubling

cascade [49].

Next we fix the value of the normal androgen level a0and

focus on how to set threshold values of the serum PSA level

to stop and restart administration. Bifurcation diagrams for

different values of a0 are shown in Fig. 7. The size of the

parameter region for relapse prevention becomes larger as a0

increases. We have more choices of threshold values for dosing

by setting the normal androgen level as high as possible within

a clinically feasible range. Boundaries of the parameter area for

relapse prevention are approximately given by period-doubling

bifurcation sets PD1. Fig. 7 shows that the ratio between r0

and r1is critical rather than the respective absolute values of

r0 and r1 under the dosing strategy (22). This result stems

from the scaling property of the IAS therapy model. Namely,

Eqs. (21) and (22) are invariant under the following variable

transformation:

(xD,xI,r0,r1) → (pxD, pxI, pr0, pr1),

where p is any non-zero real value. It should be noted, however,

that this scaling property seems not realistic for a large p.

The transformation holds if the serum PSA concentration is

given as a linear sum of the populations of the AD and AI

cells, i.e. y = cDxD+ cIxI. Thus it may be generally more

appropriate to consider the serum PSA level y = f (xD,xI) as

a nonlinear function of the populations of cancer cells in more

realistic modeling.

(31)

3.3. Origin of nonlinearity

AsimplifiedversionoftheIAStherapymodelisinvestigated

in order to clarify the origin of nonlinear phenomena and

transient grazing in the hybrid system. Let us suppose, for the

sake of simplicity, that androgen concentration instantaneously

changes to steady values in response to switching between on-

administration and off-administration periods, i.e. γ → ∞ in

Eq. (21). Androgen concentration a is then considered to take

alternative values 0 and a0. Androgen concentration switches

Fig. 7. Bifurcation diagrams of the IAS therapy model (21)–(25). Parameter regions for relapse prevention correspond to those of nondivergent solutions. Period-

doubling and grazing bifurcations of a limit cycle are denoted by PD1and G1, respectively. (a) a0= 28. (b) a0= 30. (c) a0= 32.

Page 8

G. Tanaka et al. / Physica D 237 (2008) 2616–2627

2623

from a0to 0 when an increasing PSA level y reaches the upper

threshold value r1, whereas it switches from 0 to a0when a

decreasing PSA level touches the lower threshold value r0.

Under this assumption, the IAS therapy model given by Eqs.

(21) and (22) is reduced to a piecewise linear system described

as follows:

dxD

dt

dxI

dt

= (gD(a) − m(a))xD,

(32)

= m(a)xD+ gI(a)xI,

?0 → a0

The simplified model can be viewed as a hybrid system

consisting of the following two linear systems:

?xD

d

dt

xI

m1

w1

xI

(33)

a :

when y = r0and dy/dt < 0

when y = r1and dy/dt > 0.

a0→ 0

(34)

d

dt

xI

?

?

=

?v0

?v1

0

0

w0

??xD

??xD

xI

?

?

,

with a = a0,

(35)

?xD

=

0

,

with a = 0,

(36)

where v0 = gD(a0), w0 = gI(a0), v1 = gD(0) − m1, and

w1 = gI(0). Event-triggering sections are given by Eqs. (26)

and (27) with the continuous state x = (xD,xI).

Analytical solutions of individual linear equations (35) and

(36) can be easily obtained. We assume that a trajectory starting

from (x0,r0− x0) on Π0reaches Π1with traveling time τ0and

returns to Π0with traveling time τ1. For such a trajectory, a

one-dimensional Poincar´ e map (return map) can be constructed

with respect to an initial condition x0[42] as follows:

R(x0) = ev1τ1+v0τ0x0,

where τ0and τ1are the implicit functions of x0, satisfying

(37)

f (ev0τ0x0,ew0τ0(r0− x0)) − r1= 0,

f (ev1τ1z(x0),α(ev1τ1− ew1τ1)z(x0) + ew1τ1(r1− z(x0)))

−r0= 0,

whereα = m1/(v1−w1)and z(x0) = ev0τ0x0.Periodicmotions

can be well understood by observing a sequence of intersecting

points between a trajectory and the section Π0. Fig. 8 shows re-

turn plots calculated from Eqs. (37) to (39) for different param-

eter values of a0. The fixed point of the return map corresponds

to a limit cycle of the simplified model. In Fig. 8(a) and (b), the

fixed point is globally stable in the domain of (0,r0) for the ini-

tial condition x0. In Fig. 8(c), on the other hand, the return plot

ends at a certain value of x0, at which transient grazing occurs.

Transient grazing is characterized by a trajectory of the system

touching the section Π0tangentially [44]. Namely, the return-

ing trajectory loses an intersection with the section Π0at the

point of transient grazing.

The location of a trajectory starting from the initial point

(x0,r0−x0)attimeτ > τ0isdenotedby(ϕD(x0,τ),ϕI(x0,τ))

where

(38)

(39)

ϕD(x0,τ) = ev1(τ−τ0)+v0τ0x0,

(40)

Fig. 8. Return plots of the simplified model (32)–(34) withr0= 5 andr1= 30,

which is represented by Eq. (37). (a) a0= 30. (b) a0= 25. (c) a0= 20.

Fig. 9. A trajectory exhibiting transient grazing. The initial condition is given

by the value of x0at which the return plot ends in Fig. 8(c).

ϕI(x0,τ) = α(ev1(τ−τ0)− ew1(τ−τ0))ev0τ0x0

+ew1(τ−τ0)+w0τ0(r0− x0).

Using these notations, we can formulate conditions for a

trajectory that exhibits transient grazing with respect to Π0as

follows:

?

which are solved with respect to unknown variables (x0,τ).

Fig. 9 shows a grazing trajectory with initial condition x0at

the endpoint of the return plot in Fig. 8(c).

Next we consider bifurcations of a limit cycle of the

simplified model. Conditional equations for a period-doubling

bifurcation of the fixed point of the return map with a

bifurcation parameter λ are described as follows:

?R(x0) − x0

which are solved with respect to unknown variables (x0,λ). By

combining the conditional equation (42) for transient grazing

with a fixed point condition, on the other hand, the conditions

for a grazing bifurcation of a limit cycle can be formulated as

follows:

(v1+ m1)ϕD(x0,τ) + w1ϕI(x0,τ)

= 0,

which are solved with respect to unknown variables (x0,τ,λ).

Fig. 10 shows a bifurcation diagram of the simplified model

(32)–(34) with r1 = 30, indicating the parameter region of

relapse prevention. The return plots at the parameter values

indicated by (a)–(c) of Fig. 10 are shown in Fig. 8(a)–(c),

respectively. The period-doubling bifurcation denoted by PD1

runs along the boundary of the parameter region for relapse

prevention. We can also confirm that the parameter set of the

grazing bifurcation denoted by G1exists very close to that

(41)

Ftg(x0,τ) ≡

ϕD(x0,τ) + ϕI(x0,τ) − r0

(v1+ m1)ϕD(x0,τ) + w1ϕI(x0,τ)

?

= 0, (42)

Fpd(x0,λ) ≡

R?(x0) + 1

?

= 0,

(43)

Fpg(x0,τ,λ) ≡

ϕD(x0,τ) − x0,

ϕD(x0,τ) + ϕI(x0,τ) − r0

(44)

Page 9

2624

G. Tanaka et al. / Physica D 237 (2008) 2616–2627

Fig. 10. Bifurcation diagram of the simplified model (32)-(34) with r1= 30.

The parameter region of relapse prevention corresponds to that of nondivergent

solutions. Period-doubling and grazing bifurcations of a limit cycle are denoted

by PD1and G1, respectively. The inset for enlargement shows that the grazing

bifurcation set exists very close to the period-doubling one. The return plots

in Fig. 8(a)–(c) are observed at the parameter values indicated by (a)–(c),

respectively.

of the period-doubling bifurcation as shown in the inset of

Fig. 10. From comparison between Figs. 4 and 10, we can

see that the parameter dependency of the region of relapse

prevention is quite similar between the IAS therapy model and

its simplified version. Therefore, the dynamical property of the

IAS therapy model is considered to be nearly equivalent to that

of the simplified system consisting of two linear systems. It

suggests that nonlinear phenomena of the IAS therapy model

originate from the nonlinearity generated by switching between

subsystems rather than from that of individual subsystems.

4. Conclusions

We have studied nonlinear dynamics and bifurcations in

the hybrid systems model representing intermittent androgen

suppression therapy for advanced prostate cancer. First, we

have introduced a numerical method to specify a limit cycle

and its bifurcation sets in a general hybrid automaton. Stability

and bifurcations of a limit cycle have been reduced to those

of a fixed or a periodic point of the Poincar´ e map. Then,

the method has been applied to analysis of the IAS therapy

model [24,25]. We have considered the effect of clinically

controllable parameters on relapse prevention, where a solution

is not divergent but confined in a finite region of continuous

state space. It has been shown that the possibility of relapse

prevention can be enhanced by an increase of the normal

androgen level during off-medication periods. This result has

suggested that androgen supplementation can be effective for

relapse prevention depending on the net growth rate of a tumor.

We have shown that the ratio between the upper threshold

value of the serum PSA level to restart medication and the

lower one to interrupt medication plays a significant role in

the dosing strategy that we have adopted in this paper. From

bifurcation analysis, we have found that the model generates

complex chaotic solutions concerned with the nested structure

of a pair of period-doubling and grazing bifurcations. Such

a chaotic attractor should also be effective for IAS therapy

because it remains within a bounded region of state space like

stable limit cycles, although it may be difficult to clinically

observe nonlinear behavior just by several on–off cycles in real

treatment.

Furthermore, we have investigated a piecewise linear system

reduced from the IAS therapy model. Return plot analysis

has been helpful for understanding the nonlinearity generated

by the hybrid system and transient grazing. It has also been

clarified that the bifurcation structure of the IAS therapy model

is approximately inherited by the simplified model.

Hybrid or switched dynamical systems would be a useful

framework in modeling intermittent therapy which is worth

considering not only for prostate cancer but also for other

diseases. Treatment interruption has been practiced based on

empirical or heuristic factors so far. Currently, systematic

understanding of disease progression under intermittent therapy

is becoming more and more important. In this regard,

medication control based on a mathematical model of

intermittent therapy could open a new frontier for pursuit of

better treatment in biomedical science. For developing this

attempt further, modeling of biomedical systems should be

carefully advanced in collaboration with medical doctors and

biomedical researchers as an interdisciplinary research.

Acknowledgments

The authors are grateful to Prof. Nicholas Bruchovsky

of The Prostate Center at Vancouver General Hospi-

tal, Dr. Takashi Shimada of University of Tokyo, and

Dr. Yoshito Hirata of ERATO Aihara Complexity Modelling

Project of JST for valuable comments and fruitful discussions.

Appendix A. Derivatives of the Poincar´ e map P

We need to numerically compute first- and the second-

order derivatives of the Poincar´ e map P with respect to the

initial condition w and a system parameter λ in order to solve

conditional equations for a limit cycle or its bifurcation by

Newton’s method. From Eq. (9), the derivatives of P are

represented by those of T as follows:

∂P

∂w=∂h

∂P

∂λ=∂h

∂x

∂T

∂x0

?∂T

∂2T

∂x2

∂h−1

∂w,

(A.1)

∂x

∂λ+∂T

?∂h−1

?

∂x0∂λ

∂x0

∂h−1

∂λ

?2

∂h−1

∂w

?

,

(A.2)

∂2P

∂w2=∂h

∂x

0

∂w

,

(A.3)

∂2P

∂w∂λ=∂h

∂x

∂2T

+∂2T

∂x2

0

∂h−1

∂w

∂h−1

∂λ

?

.

(A.4)

Page 10

G. Tanaka et al. / Physica D 237 (2008) 2616–2627

2625

From Eq. (6), the derivatives of T are represented by those of

the submaps T0and T1as follows:

∂T

∂x0

∂T

∂λ=∂T1

=∂T1

∂x1

?∂T0

?∂T0

?

∂x0

?

?

,

(A.5)

∂x1

∂λ

+∂T1

?

∂λ,

(A.6)

∂2T

∂x2

0

=∂T1

∂x1

∂2T0

∂x2

?∂2T0

∂2T1

∂x1∂λ

0

+∂2T1

∂x2

?

?∂T0

1

?∂T0

∂x0

?2

,

(A.7)

∂2T

∂x0∂λ

=

∂T1

∂x1

∂x0∂λ

+∂2T1

∂x2

?

1

?∂T0

∂x0

??∂T0

∂λ

?

+

∂x0

.

(A.8)

From Eqs. (4) and (5), the derivatives of each submap Tu

(u = 0,1) are described as follows:

∂Tu

∂xu

∂xu

∂Tu

∂λ ∂λ

∂2Tu

∂x2u

∂x2u

∂xu∂t

∂xu

∂2Tu

∂xu∂λ

∂xu∂t

∂2τu

∂xu∂λ.

=∂ϕu

+ fu∂τu

∂xu,

(A.9)

=∂ϕu

=∂2ϕu

+ fu∂τu

+∂2ϕu

∂2ϕu

∂xu∂λ+∂2ϕu

∂λ,

(A.10)

∂τu

+∂2ϕu

∂t∂xu

∂τu

∂xu

+ fu∂2τu

∂x2u

,

(A.11)

=

∂τu

∂λ+∂ϕu

∂t∂λ

∂τu

∂xu

+fu

(A.12)

A trajectory leaving from the initial condition xuon Πu(u =

0,1) is assumed to reach the other section Πu? with traveling

time τuunder the system Suas follows:

qu?(ϕu(τu(xu),xu)) = 0,

where u?= 1 − u. If a trajectory transversally intersects with

the section,

(A.13)

∂qu?

∂xu

· fu?= 0.

(A.14)

Under this assumption, by differentiating both sides of Eq.

(A.13) with respect to the initial condition xu, the derivatives

of τuare explicitly described as follows:

∂τu

∂xu

= −

1

∂qu?

∂xfu

1

∂qu?

∂xfu

?∂qu?

?∂qu?

∂qu?

∂x

∂x

∂ϕu

∂xu

?

+∂qu?

,

(A.15)

∂τu

∂λ

= −

∂x

∂ϕu

∂λ

?∂2ϕu

∂λ

?

,

(A.16)

∂2τu

∂x2u

= −

1

∂qu?

∂xfu

∂x2u

+∂2ϕu

∂xu∂t

∂τu

∂xu

+∂2ϕu

∂t∂xu

∂τu

∂xu

?

,

(A.17)

∂2τu

∂xu∂λ

= −

1

∂qu?

∂xfu

?∂2ϕu

∂qu?

∂x

×

∂xu∂λ+∂2ϕu

∂xu∂t

∂τu

∂λ+∂2ϕu

∂t∂λ

∂τu

∂xu

?

.

(A.18)

The derivatives of ϕucan be calculated from first- and second-

order variational equations given as follows:

?∂ϕu

d

dt

∂λ ∂xu

∂λ

?∂2ϕu

d

dt

∂xu∂λ∂xu

∂xu∂λ

∂2fu

∂xu∂λ∂xu

d

dt

∂xu

?

?

=∂fu

∂xu

=∂fu

?

?

?∂ϕu

?∂ϕu

?∂2ϕu

∂fu

∂xu

?

?

,

(A.19)

?∂ϕu

+∂fu

?

∂λ,

(A.20)

d

dt

∂x2u

=∂fu

∂xu

∂x2u

?∂2ϕu

+∂2fu

∂x2u

?

?∂ϕu

?∂ϕu

∂xu

?2

,

(A.21)

?∂2ϕu

=+∂2fu

∂x2u

?

?∂ϕu

∂xu

??∂ϕu

∂λ

?

(A.22)

+

.

The fourth-order Runge–Kutta method is employed for

numerical integration of the above equations with the following

initial conditions:

????t=0

∂x2u

t=0

where In and On indicate n × n identity and null matrices,

respectively.

∂ϕu

∂xu

∂2ϕu

= In,

∂ϕu

∂λ

????t=0

= On,

????t=0

????

= On,

∂2ϕu

∂xu∂λ

= On,

(A.23)

Appendix B. Derivatives of the map¯T

From Eq. (18), the derivatives of¯T with respect to the initial

condition x0, the return time τ, and the bifurcation parameter λ,

are calculated as follows:

∂¯T

∂x0

∂¯T

∂τ

∂¯T

∂λ=∂ϕ1

=∂ϕ1

∂x1

∂¯T2m−1

∂x0

− f1∂ ¯ τ2m−1

∂x0

,

(B.1)

= f1,

(B.2)

∂x1

∂¯T2m−1

∂λ

− f1∂ ¯ τ2m−1

∂λ

+∂ϕ1

∂λ.

(B.3)

In a similar way, derivatives of ¯ τ2m−1 can be sequentially

calculated by using the following recurrence formula:

∂ ¯ τk

∂x0

∂ ¯ τk

∂λ

=∂ ¯ τk−1

∂x0

=∂ ¯ τk−1

+∂τk

∂xk

+∂τk

∂xk

∂¯Tk−1

∂x0

∂¯Tk−1

∂λ

,

k = 1,...,2m − 1,

(B.4)

∂λ

+∂τk

∂λ,

k = 1,...,2m − 1,

(B.5)

which are derived from Eq. (17). From Eqs. (15) and (16),

derivatives of ¯T2m−1 can be calculated by using recurrence

Page 11

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G. Tanaka et al. / Physica D 237 (2008) 2616–2627

equations for derivatives of¯T2kand¯T2k+1given as follows:

∂¯T2k

∂x0

∂x2k−1

∂¯T2k

∂λ∂x2k−1

=

∂T1

?∂¯T2k−1

?∂¯T2k−1

∂x0

?

?

,

k = 0,1,...,m − 1,

+∂T0

(B.6)

=

∂T1

∂λ∂λ,

k = 0,1,...,m − 1,

(B.7)

∂¯T2k+1

∂x0

∂¯T2k+1

∂λ

=

∂T0

∂x2k

∂T0

∂x2k

?∂¯T2k

?∂¯T2k

∂x0

?

?

,

k = 0,1,...,m − 1,

+∂T1

(B.8)

=

∂λ∂λ,

k = 0,1,...,m − 1.

(B.9)

The derivatives of T0and T1can be calculated by Eqs. (A.9)

and (A.10).

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