en
The TErminology for the Description of DYnamics (TEDDY) project aims to provide an ontology for dynamical behaviours, observable dynamical phenomena, and control elements of bio-models and biological systems in Systems Biology and Synthetic Biology.
TErminology for the Description of DYnamics (TEDDY)
http://co.mbine.org/specifications/teddy
http://teddyontology.svn.sourceforge.net/svnroot/teddyontology/teddy/tags/rel-2011-08-30/teddy.owl
rel-2014-04-24 (inferred)
dependsOn
Relates temporal behaviours [http://identifiers.org/teddy/TEDDY_0000083] to functional motifs [http://identifiers.org/teddy/TEDDY_0000003] they depend on.
Links a dynamic system to the way one or several temporal behaviours [http://identifiers.org/teddy/TEDDY_0000083] are modified or related upon interaction with information external to the system considered.
hasFeature
hasCharacteristic
Used to describe properties by means of behaviour characteristic [http://identifiers.org/teddy/TEDDY_0000002].
hasSubPart
true
hasSuperPart
true
hasOnPart
true
adjacentTo
Two temporal behaviours [http://identifiers.org/teddy/TEDDY_0000083] are adjacentTo each other if and only if they are in phase space proximity.
convergeTo
A temporal behaviour [http://identifiers.org/teddy/TEDDY_0000083] convergeTo another temporal behaviour [http://identifiers.org/teddy/TEDDY_0000083] if and only if it reaches the other behaviour [http://identifiers.org/teddy/TEDDY_0000083] as time goes to either positive or negative infinity.
reverseOf
reverseOf links two temporal behaviours [http://identifiers.org/teddy/TEDDY_0000083] whose phase diagrams can be obtained from each other by reversing the directions of all the phase paths.
transforms
true
below
Links a bifurcation [http://identifiers.org/teddy/TEDDY_0000053] to a 'temporal behaviour' [http://identifiers.org/teddy/TEDDY_0000083] which exists below the critical value of the bifurcation parameter.
above
Links a bifurcation [http://identifiers.org/teddy/TEDDY_0000053] to a 'temporal behaviour' [http://identifiers.org/teddy/TEDDY_0000083] which exists above the critical value of the bifurcation parameter.
For continuants: C part_of C* if and only if: given any c that instantiates C at a time t, there is some c* such that c* instantiates C* at time t, and c part_of c* at t.
For processes: P part_of P* if and only if: given any p that instantiates P at a time t, there is some p* such that p* instantiates P* at time t, and p part_of p* at t. (Here part_of is the instance-level part-relation.)
partOf
http://obofoundry.org/ro/#OBO_REL:part_of
hasValue
Links a 'behaviour characteristic' [http://identifiers.org/teddy/TEDDY_0000002] to the type of its value.
TEDDY_0000000
TEDDY entity
A Thing related to the dynamics of bio-models and biological systems. Terms belonging to TEDDY Entities are used in descriptions of dynamical behaviours, observable dynamical phenomena, and control elements in Systems Biology and Synthetic Biology.
TEDDY_0000001
curve ((obsolete))
true
Obsolete: equivalent to TEDDY_0000083 'Temporal Behaviour'.
TEDDY_0000002
behaviour characteristic
Behaviour characteristic is a property that characterizes temporal behaviors [http://identifiers.org/teddy/TEDDY_0000083].
TEDDY_0000003
functional motif
http://identifiers.org/doi/10.1371/journal.pbio.0020369
A connected graph or network consisting of M vertices and a set of edges having a particular functional significance, forming a subgraph of a larger network.
http://identifiers.org/doi/10.1371/journal.pbio.0020369
Sporns O, Kötter R (2004) Motifs in Brain Networks, PLoS Biology, 2(11):e369.
TEDDY_0000004
monotonicity
http://mathworld.wolfram.com/MonotonicFunction.html
monotone
monotonic
A curve is `monotonic` if successive states are ordered either entirely non-decreasing or entirely non-increasing.
http://mathworld.wolfram.com/MonotonicFunction.html
Weisstein, Eric W. Monotonic Function. From MathWorld--A Wolfram Web Resource.
TEDDY_0000005
non-monotonic
http://mathworld.wolfram.com/MonotonicFunction.html
A curve is non-monotonic if it has both increasing ordered successive states and decreasing orderd succesive states.
http://mathworld.wolfram.com/MonotonicFunction.html
Weisstein, Eric W. Monotonic Function. From MathWorld--A Wolfram Web Resource.
TEDDY_0000006
oscillation
http://mathworld.wolfram.com/Oscillation.html
The variation of a function which exhibits slope changes, also called the saltus of a function. A series may also oscillate, causing it not to converge.
http://mathworld.wolfram.com/Oscillation.html
Weisstein, Eric W. Oscillation. From MathWorld--A Wolfram Web Resource.
TEDDY_0000007
strict monotonicity
http://eom.springer.de/M/m064830.htm
strictly monotone
strictly monotonic
A curve is strictly monotonic if it is monotonic [http://identifiers.org/teddy/TEDDY_0000004] and has no equal temporal successive states.
http://eom.springer.de/M/m064830.htm
Weisstein, Eric W. Monotonic Function. From MathWorld--A Wolfram Web Resource.
TEDDY_0000008
strictly increasing
http://eom.springer.de/M/m064830.htm
isotonic
A curve is strictly increasing if it is strictly monotonic [http://identifiers.org/teddy/TEDDY_0000007] with temporal successive states increasing ordered.
http://eom.springer.de/M/m064830.htm
Weisstein, Eric W. Monotonic Function. From MathWorld--A Wolfram Web Resource.
TEDDY_0000009
strictly decreasing
http://eom.springer.de/M/m064830.htm
antitonic
A curve is strictly decreasing if it is strictly monotonic [http://identifiers.org/teddy/TEDDY_0000007] with temporal successive states decreasing ordered.
http://eom.springer.de/M/m064830.htm
Weisstein, Eric W. Monotonic Function. From MathWorld--A Wolfram Web Resource.
TEDDY_0000010
single turnaround
A trajectory [http://identifiers.org/teddy/TEDDY_0000083] of the system which responds negatively to displacement from equilbrium [http://identifiers.org/teddy/TEDDY_0000086]: moving away from equilibrium the trajectory turns around and moves back towards equilibrium.
TEDDY_0000011
steady state ((obsolete))
true
Obsolete: equivalent to TEDDY_0000086 'Fixed Point'.
TEDDY_0000012
rate of change
The rate of growth/decay.
TEDDY_0000013
linear rate of change
http://identifiers.org/isbn/1844071448
A quantity changes linearly when its change is a constant amount over a given period of time.
http://identifiers.org/isbn/1844071448
D. H. Meadows (2005) The Limits to Growth: The 30-Year Update, Rff Press, revised ed.
TEDDY_0000014
exponential rate of change
http://identifiers.org/isbn/1844071448
A quantity changes exponentially when its change over a given period of time is proportional to what is already there.
http://identifiers.org/isbn/1844071448
D. H. Meadows (2005) The Limits to Growth: The 30-Year Update, Rff Press, revised ed.
TEDDY_0000015
curve shape
Characteristic describing the shape of the graph of a function.
TEDDY_0000016
concave shape
http://identifiers.org/isbn/0691080690
A shape of a graph of a concave function, i.e. a function whose negative is convex [http://identifiers.org/teddy/TEDDY_0000021].
http://identifiers.org/isbn/0691080690
R. T. Rockafellar (1970) Convex Analysis (Princeton Mathematical Series), Princeton Univ Pr. (p.23)
TEDDY_0000017
zero rate of change
http://identifiers.org/isbn/1844071448
A quantity has a zero rate of change when it neither grows nor declines.
http://identifiers.org/isbn/1844071448
D. H. Meadows (2005) The Limits to Growth: The 30-Year Update, Rff Press, revised ed.
TEDDY_0000018
polynomial rate of change
power rate of change
A quantity changes polynomially when its change function is bounded above by a polinomial function.
TEDDY_0000019
linear increasing
A curve is linear increasing if it is is strictly increasing with a linear rate of change.
TEDDY_0000020
linear decreasing
A curve is linear decreasing if it is is strictly decreasing with a linear rate of change.
TEDDY_0000021
convex shape
http://identifiers.org/isbn/0691080690
A shape of a graph of a convex function, i.e. a function whose epigraph (the set of points on or above the graph of the function) is a convex as a subset of R^n. A subset G of a linear space is said to be convex, if it contains the whole segment (closed straight line segment) joining each of its two points.
http://identifiers.org/isbn/0691080690
R. T. Rockafellar (1970) Convex Analysis (Princeton Mathematical Series), Princeton Univ Pr. (p.10, 23)
TEDDY_0000022
straight line shape
A shape of a graph of a linear function.
TEDDY_0000023
curve characteristic
A characteristic of the 'temporal behaviour' [http://identifiers.org/teddy/TEDDY_0000083] curve (projection of the function graph onto a single variable).
TEDDY_0000024
unbounded growth ((obsolete))
true
TEDDY_0000025
limit
http://eom.springer.de/L/l058820.htm
One of the fundamental concepts in mathematics, meaning that a variable depending on another variable arbitrary closely approaches some constant as the latter variable changes in a definite manner.
http://eom.springer.de/L/l058820.htm
Encyclopaedia of Mathematics, edited by Michiel Hazewinkel, CWI, Amsterdam.
TEDDY_0000026
asymptotic limit
http://eom.springer.de/a/a012870.htm
approximate limit
A limit of a function f(x) as x->x0 over a set E for which x0 is a density point.
http://eom.springer.de/a/a012870.htm
Encyclopaedia of Mathematics, edited by Michiel Hazewinkel, CWI, Amsterdam.
TEDDY_0000027
asymptotic upper limit
http://eom.springer.de/a/a012870.htm
approximate upper limit
An approximate upper limit of a function f(x) at a point x0 is the lower bound of the set of numbers y (including y=positive infinity) for which x0 is a point of dispersion of the set {x: f(x)>y}.
http://eom.springer.de/a/a012870.htm
Encyclopaedia of Mathematics, edited by Michiel Hazewinkel, CWI, Amsterdam.
TEDDY_0000028
asymptotic lower limit
http://eom.springer.de/a/a012870.htm
approximate lower limit
An approximate lower limit of a function f(x) at a point x0 is the upper bound of the set of numbers y (including y=negative infinity) for which x0 is a point of dispersion of the set {x: f(x)<y}.
http://eom.springer.de/a/a012870.htm
Encyclopaedia of Mathematics, edited by Michiel Hazewinkel, CWI, Amsterdam.
TEDDY_0000029
plus infinity limit
http://eom.springer.de/I/i050930.htm
A function of x approaches plus infinity if its value becomes and remains larger than any given number as a result of variation of x.
http://eom.springer.de/I/i050930.htm
Encyclopaedia of Mathematics, edited by Michiel Hazewinkel, CWI, Amsterdam.
TEDDY_0000030
infinite limit
http://eom.springer.de/I/i050930.htm
A function of a variable x has an infinite limit if its absolute value becomes and remains larger than any given number as a result of variation of x.
http://eom.springer.de/I/i050930.htm
Encyclopaedia of Mathematics, edited by Michiel Hazewinkel, CWI, Amsterdam.
TEDDY_0000031
minus infinity limit
http://eom.springer.de/I/i050930.htm
A function of x approaches minus infinity if its value becomes and remains smaller than any given number as a result of variation of x.
http://eom.springer.de/I/i050930.htm
Encyclopaedia of Mathematics, edited by Michiel Hazewinkel, CWI, Amsterdam.
TEDDY_0000032
sigmoid shape
http://identifiers.org/isbn/3540605053
A shape of a graph of a sigmoid function, i.e. a real function sc: R->(0,1) defined by the expression sc(x) = 1/(1 + e^(-cx)).
http://identifiers.org/isbn/3540605053
R. Rojas (1996) Neural Networks: A Systematic Introduction, Springer, 1st ed. (p.149)
TEDDY_0000033
feedback loop
http://identifiers.org/isbn/1584886420
feedback
A process whereby some proportion of function of the output signal of a system is passed (fed back) to the input.
http://identifiers.org/isbn/1584886420
Alon U (2006) An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall/CRC. (p.266)
TEDDY_0000034
negative feedback loop
http://identifiers.org/doi/10.1016/j.febslet.2005.02.008
negative feedback
A component or variable of a system is subject to negative feedback when it inhibits its own level of activity.
http://identifiers.org/doi/10.1016/j.febslet.2005.02.008
Wolkenhauer O, Ullah M, Wellstead P, Cho K-H (2005) The dynamic systems approach to control and regulation of intracellular networks, FEBS Letters, 579 (8): 1846-1853.
TEDDY_0000035
positive feedback loop
http://identifiers.org/doi/10.1016/j.febslet.2005.02.008
positive feedback
A component or variable of a system is subject to positive feedback when it increases its own level of activity.
http://identifiers.org/doi/10.1016/j.febslet.2005.02.008
Wolkenhauer O, Ullah M, Wellstead P, Cho K-H (2005) The dynamic systems approach to control and regulation of intracellular networks, FEBS Letters, 579 (8): 1846-1853.
TEDDY_0000036
feed-forward loop
FFL
feed-forward
feedforward
feedforward loop
A 'three-node feed-forward loop' [http://identifiers.org/teddy/TEDDY_0000037] or its topological generalization.
TEDDY_0000037
three-node feed-forward loop
http://identifiers.org/isbn/1584886420
3-node FFL
3-node feed-forward
3-node feed-forward loop
3-node feedforward
3-node feedforward loop
FFL
thre-node FFL
three-node feedforward loop
A pattern with three nodes, X, Y, and Z, in which X has a directed edge to Y and Z, and Y has a directed edge to Z. The FFL is a network motif in many biological networks, and can perform a variety of tasks (such as sign-sensitive delay, sign-sensitive acceleration, and pulse generation).
http://identifiers.org/isbn/1584886420
Alon U (2006) An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall/CRC. (p.267)
FFL
http://identifiers.org/isbn/1584886420 (p.41)
TEDDY_0000038
coherent three-node feed-forward loop
http://identifiers.org/isbn/1584886420
coherent 3-node feed-forward loop
coherent 3-node feedforward
coherent 3-node feedforward loop
coherent three-node feedforward loop
A feed-forward loop [http://identifiers.org/teddy/TEDDY_0000037] in which the sign of the direct path from X to Z is the same as the overall sign of the indirect path from X through Y to Z.
http://identifiers.org/isbn/1584886420
Alon U (2006) An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall/CRC. (p.265)
TEDDY_0000039
type-1 coherent three-node feed-forward loop
http://identifiers.org/isbn/1584886420
C1-FFL
coherent 3-node feedforward type-1
type-1 coherent FFL
A 'coherent three-node feed-forward loop' [http://identifiers.org/teddy/TEDDY_0000038] in which all three regulations are positive.
http://identifiers.org/isbn/1584886420
Alon U (2006) An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall/CRC. (p.47)
TEDDY_0000040
type-2 coherent three-node feed-forward loop
http://identifiers.org/isbn/1584886420
C2-FFL
coherent 3-node feedforward type-2
type-2 coherent FFL
A 'coherent three-node feed-forward loop' [http://identifiers.org/teddy/TEDDY_0000038] in which X represses Z, and also represses an activator of Z.
http://identifiers.org/isbn/1584886420
Alon U (2006) An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall/CRC. (p.47)
TEDDY_0000041
type-3 coherent three-node feed-forward loop
http://identifiers.org/isbn/1584886420
C3-FFL
coherent 3-node feedforward type-3
type-3 coherent FFL
A 'coherent three-node feed-forward loop' [http://identifiers.org/teddy/TEDDY_0000038] in which X represses Z, and also activates a repressor of Z.
http://identifiers.org/isbn/1584886420
Alon U (2006) An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall/CRC. (p.47)
TEDDY_0000042
type-4 coherent three-node feed-forward loop
http://identifiers.org/isbn/1584886420
C4-FFL
coherent 3-node feedforward type-4
type-4 coherent FFL
A 'coherent three-node feed-forward loop' [http://identifiers.org/teddy/TEDDY_0000038] in which X activates Z, and also represses a repressor of Z.
http://identifiers.org/isbn/1584886420
Alon U (2006) An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall/CRC. (p.47)
TEDDY_0000043
incoherent three-node feed-forward loop
http://identifiers.org/isbn/1584886420
incoherent 3-node feed-forward loop
incoherent 3-node feedforward
incoherent 3-node feedforward loop
incoherent three-node feedforward loop
A feed-forward loop [http://identifiers.org/teddy/TEDDY_0000037] in which the sign of the direct path from X to Z is the opposite as the overall sign of the indirect path from X through Y to Z.
http://identifiers.org/isbn/1584886420
Alon U (2006) An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall/CRC. (p.267)
TEDDY_0000044
type-1 incoherent three-node feed-forward loop
http://identifiers.org/isbn/1584886420
I1-FFL
incoherent 3-node feedforward type-1
type-1 incoherent FFL
An 'incoherent three-node feed-forward loop' [http://identifiers.org/teddy/TEDDY_0000043] in which X activates Z, and also activates a repressor of Z.
http://identifiers.org/isbn/1584886420
Alon U (2006) An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall/CRC. (p.47)
TEDDY_0000045
type-2 incoherent three-node feed-forward loop
http://identifiers.org/isbn/1584886420
I2-FFL
incoherent 3-node feedforward type-2
type-2 incoherent FFL
An 'incoherent three-node feed-forward loop' [http://identifiers.org/teddy/TEDDY_0000043] in which X represses Z, and also represses a repressor of Z.
http://identifiers.org/isbn/1584886420
Alon U (2006) An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall/CRC. (p.47)
TEDDY_0000046
type-3 incoherent three-node feed-forward loop
http://identifiers.org/isbn/1584886420
I3-FFL
incoherent 3-node feedforward type-3
type-3 incoherent FFL
An 'incoherent three-node feed-forward loop' [http://identifiers.org/teddy/TEDDY_0000043] in which X represses Z, and also activates an activator of Z.
http://identifiers.org/isbn/1584886420
Alon U (2006) An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall/CRC. (p.47)
TEDDY_0000047
type-4 incoherent three-node feed-forward loop
http://identifiers.org/isbn/1584886420
I4-FFL
incoherent 3-node feedforward type-4
type-4 incoherent FFL
An 'incoherent three-node feed-forward loop' [http://identifiers.org/teddy/TEDDY_0000043] in which X activates Z, and also represses an activator of Z.
http://identifiers.org/isbn/1584886420
Alon U (2006) An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall/CRC. (p.47)
TEDDY_0000048
orbit ((obsolete))
true
Obsolete: equivalent to TEDDY_0000083 'Temporal Behaviour'.
TEDDY_0000049
fixed point ((obsolete))
true
Obsolete: equivalent to TEDDY_0000086 'Fixed Point'.
TEDDY_0000050
periodic orbit
http://identifiers.org/isbn/0387983821
closed orbit
cycle
periodic solution
A temporal behaviour [http://identifiers.org/teddy/TEDDY_0000083] which repeats every state after a specific period of time.
http://identifiers.org/isbn/0387983821
Kuznetsov YA. (1988) Elements of applied bifurcation theory, Springer, New York. (p.9)
TEDDY_0000051
limit cycle
http://identifiers.org/isbn/0738204536
Grenzzyklus
isolated closed path
A closed orbit which is isolated, i.e. neighbouring orbits are not closed.
http://identifiers.org/isbn/0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.196)
Grenzzyklus
http://identifiers.org/isbn/3817112823
isolated closed path
http://identifiers.org/isbn/0198565623 (p.30)
TEDDY_0000052
parameter dependency ((obsolete))
true
Obsolete: not required anymore.
TEDDY_0000053
bifurcation
http://identifiers.org/isbn/0198565623
http://www.egwald.com/nonlineardynamics/bifurcations.php
A `characteristic` describing a sudden qualitative (topological) change in the orbit structure of a system occuring as a parameter passes through a critical value, called a bifurcation point.
http://identifiers.org/isbn/0198565623
Jordan DW., Smith P. (1999) Nonlinear ordinary differential equations: an introduction to dynamical systems, Oxford University Press. (p.420)
http://www.egwald.com/nonlineardynamics/bifurcations.php
Elmer G. Wiens: Egwald Web Services Ltd.
TEDDY_0000054
strange attractor
http://identifiers.org/isbn/0738204536
http://identifiers.org/isbn/9810221428
chaotic attractor
fractal attractor
A non-periodic orbit [http://identifiers.org/teddy/TEDDY_0000143] which is attracting and exhibits sensitive dependence on initial conditions.
The attractor [http://identifiers.org/teddy/TEDDY_0000094] is strange if trajectories [http://identifiers.org/teddy/TEDDY_0000083] on the attractor, being stable according to Poisson [http://identifiers.org/teddy/TEDDY_0000149], are unstable according to Lyapunov [not TEDDY_0000113].
http://identifiers.org/isbn/0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.325)
http://identifiers.org/isbn/9810221428
V. S. Anishchenko (1995) Dynamical Chaos-Models and Experiments: Appearance Routes and Structure of Chaos in Simple Dynamical Systems (World Scientific Series on Nonlinear Science, Series a, Vol 8). World Scientific Pub Co Inc. (p.10)
chaotic attractor
http://identifiers.org/isbn/0738204536 (p.325)
fractal attractor
http://identifiers.org/isbn/0738204536 (p.325)
TEDDY_0000055
bursting
http://identifiers.org/isbn/0262090430
A burst is two or more spikes followed by a period of quiescence.
http://identifiers.org/isbn/0262090430
Izhikevich EM (2007) Dynamical systems in neuroscience : the geometry of excitability and bursting, MIT Press. (p.325)
TEDDY_0000056
switch
http://identifiers.org/doi/10.1371/journal.pcbi.1002085
A signaling network that converts a graded input cue into a binary, all-or-none response is said to exhibit ‘switch-like’ behavior; switching enables the establishment of discrete states which is vital in processes such as cell proliferation and differentiation.
http://identifiers.org/doi/10.1371/journal.pcbi.1002085
Shah NA, Sarkar CA (2011) Robust Network Topologies for Generating Switch-Like Cellular Responses, PLoS Comput Biol.; 7(6): e1002085.
TEDDY_0000057
stability ((obsolete))
true
Obsolete: not required anymore.
TEDDY_0000058
use 'strictly decreasing' [http://identifiers.org/teddy/TEDDY_0000009] and 'asymptotic limit' [http://identifiers.org/teddy/TEDDY_0000026] terms instead.
asymptotic decreasing ((obsolete))
http://eom.springer.de/A/a013610.htm
true
Asymptote of a curve y=f(x) with an infinite branch is a straight line the distance of which from the point (x, f(x)) on the curve tends to zero as the point moves along the branch of the curve to infinity. Decreasing function having for which an asymptote exists is called 'asymptotic decreasing'.
http://eom.springer.de/A/a013610.htm
Encyclopaedia of Mathematics, edited by Michiel Hazewinkel, CWI, Amsterdam.
TEDDY_0000059
use 'strictly increasing' [http://identifiers.org/teddy/TEDDY_0000008] and 'asymptotic limit' [http://identifiers.org/teddy/TEDDY_0000026] terms instead.
asymptotic increasing ((obsolete))
http://eom.springer.de/A/a013610.htm
true
Asymptote of a curve y=f(x) with an infinite branch is a straight line the distance of which from the point (x, f(x)) on the curve tends to zero as the point moves along the branch of the curve to infinity. Increasing function having for which an asymptote exists is called 'asymptotic increasing'.
http://eom.springer.de/A/a013610.htm
Encyclopaedia of Mathematics, edited by Michiel Hazewinkel, CWI, Amsterdam.
TEDDY_0000060
use 'strictly increasing' [http://identifiers.org/teddy/TEDDY_0000008] and 'infinite limit' [http://identifiers.org/teddy/TEDDY_0000030] terms instead.
diverging increasing ((obsolete))
true
TEDDY_0000061
chaotic oscillation
http://identifiers.org/isbn/9810221428
non-periodic oscillation
If a particular solution is aperiodic, but bounded for pt->infinity, then it corresponds to the regime of chaotic oscillation.
http://identifiers.org/isbn/9810221428
V. S. Anishchenko (1995) Dynamical Chaos-Models and Experiments: Appearance Routes and Structure of Chaos in Simple Dynamical Systems (World Scientific Series on Nonlinear Science, Series a, Vol 8). World Scientific Pub Co Inc. (p.10)
TEDDY_0000062
sustained oscillation ((obsolete))
true
Obsolete: equivalent to TEDDY_0000051 'Limit Cycle'.
TEDDY_0000063
damped oscillation
http://identifiers.org/isbn/0486655083
Damping is any effect that tends to reduce the amplitude [http://identifiers.org/teddy/TEDDY_0000131] of oscillations in an oscillatory system.
http://identifiers.org/isbn/0486655083
A. A. Andronov, A. A. Vitt, and S. E. Khaikin (1987) Theory of Oscillators, Dover Publications.
TEDDY_0000064
single-periodic oscillation
http://identifiers.org/isbn/0198565623
SPO
single periodic oscillation
An oscillation [http://identifiers.org/teddy/TEDDY_0000006] corresponding to a solution having a one-loop phase path and a period-1 Poincaré map.
http://identifiers.org/isbn/0198565623
Jordan DW., Smith P. (1999) Nonlinear ordinary differential equations: an introduction to dynamical systems, Oxford University Press. (p.465)
TEDDY_0000065
mixed-mode oscillation
http://identifiers.org/doi/10.1063/1.2900015
MMO
Mixed-mode oscillations are complex periodic waveforms where each period is comprised of several maxima and minima of different amplitudes [http://identifiers.org/teddy/TEDDY_0000131].
mixed-mode oscillation
I. Erchova and D. J. McGonigle (2008) Rhythms of the brain: an examination of mixed mode oscillation approaches to the analysis of neurophysiological data, Chaos (Woodbury, N.Y.) 18.
http://identifiers.org/doi/10.1063/1.2900015
I. Erchova & D. J. McGonigle (2008) Rhythms of the brain: an examination of mixed mode oscillation approaches to the analysis of neurophysiological data. Chaos (Woodbury, N.Y.) 18(1).
TEDDY_0000066
periodic oscillation
http://identifiers.org/isbn/9810221428
The characteristic of a periodic solution (a solution which is distinguished by the condition x*(t) = x*(t+T), where T is the period [http://identifiers.org/teddy/TEDDY_0000067] of solution).
http://identifiers.org/isbn/9810221428
V. S. Anishchenko (1995) Dynamical Chaos-Models and Experiments: Appearance Routes and Structure of Chaos in Simple Dynamical Systems (World Scientific Series on Nonlinear Science, Series a, Vol 8). World Scientific Pub Co Inc. (p.6)
TEDDY_0000067
period
http://eom.springer.de/p/p072210.htm
For a periodic solution x(t), there is a number T, not equal to 0, such that x(t+T) = x(t) for t ∈ R. All possible such T are called periods of this periodic solution; the continuity of x(t) implies that either x(t) is independent of t or that all possible periods are integral multiples of one of them — the minimal period T0>0. When one speaks of a periodic solution, it is often understood that the second case applies, and T0 is simply termed the period.
http://eom.springer.de/p/p072210.htm
Encyclopaedia of Mathematics, edited by Michiel Hazewinkel, CWI, Amsterdam.
TEDDY_0000068
regularity ((obsolete))
true
Obsolete: not required anymore.
TEDDY_0000069
local bifurcation
http://en.wikipedia.org/wiki/Bifurcation_theory#Local_bifurcations
http://identifiers.org/isbn/0387983821
bifurcation from steady state
bifurcation of equilibrium
bifurcation of equilibrium point
bifurcation of fixed point
A `bifurcation` [http://identifiers.org/teddy/TEDDY_0000053] in which a stable [http://identifiers.org/teddy/TEDDY_0000133] fixed point [http://identifiers.org/teddy/TEDDY_0000086] changes to an unstable [not TEDDY_0000133] one or vanishes. It can be detected within any small neighborhood of the fixed point: the real part of the eigenvalue of the linarisation around the fixed point is zero at the bifurcation.
http://identifiers.org/isbn/0387983821
Kuznetsov YA. (1988) Elements of applied bifurcation theory, Springer, New York. (p.58)
bifurcation from steady state
http://www.scholarpedia.org/article/MATCONT
bifurcation of equilibrium
http://identifiers.org/isbn/0387983821 (p.58)
bifurcation of fixed point
http://identifiers.org/isbn/0387983821 (p.58)
TEDDY_0000070
bifurcation of limit cycle
http://identifiers.org/isbn/0387983821
A global bifurcation [http://identifiers.org/teddy/TEDDY_0000147] in which a limit cycle [http://identifiers.org/teddy/TEDDY_0000051] (dis)appears or changes its stability [http://identifiers.org/teddy/TEDDY_0000113].
http://identifiers.org/isbn/0387983821
Kuznetsov YA. (1988) Elements of applied bifurcation theory, Springer, New York. (p.162ff)
1
1
1
1
TEDDY_0000071
saddle-node bifurcation
http://identifiers.org/isbn/0387983821
http://identifiers.org/isbn/0738204536
blue sky bifurcation
fold bifurcation
limit point bifurcation
tangent bifurcation
turning-point bifurcation
A `zero-eigenvalue bifurcation` [http://identifiers.org/teddy/TEDDY_0000123] in which a stable [http://identifiers.org/teddy/TEDDY_0000133] fixed point [http://identifiers.org/teddy/TEDDY_0000086] and an unstable [not TEDDY_0000133] fixed point [http://identifiers.org/teddy/TEDDY_0000086] collide and mutually annihilate.
http://identifiers.org/isbn/0387983821
Kuznetsov YA. (1988) Elements of applied bifurcation theory, Springer, New York. (p.80)
http://identifiers.org/isbn/0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.45)
blue sky bifurcation
http://identifiers.org/isbn/0738204536 (p.47)
fold bifurcation
http://identifiers.org/isbn/0738204536 (p.47)
limit point bifurcation
http://www.scholarpedia.org/article/Saddle-node_Bifurcation
tangent bifurcation
http://identifiers.org/isbn/0387983821 (p.80)
turning-point bifurcation
http://identifiers.org/isbn/0738204536 (p.47)
TEDDY_0000072
Hopf bifurcation
http://identifiers.org/isbn/0738204536
Andronov-Hopf Bifurcation
A `local bifurcation` [http://identifiers.org/teddy/TEDDY_0000069] in which a `stable spiral` [http://identifiers.org/teddy/TEDDY_0000126] changes in an `unstable spiral` [http://identifiers.org/teddy/TEDDY_0000127]. The linearisation around the fixed point [http://identifiers.org/teddy/TEDDY_0000086] has two conjugate eigenvalues. This eigenvalues cross simultaneously the imaginary axis from left (negative real part) to the right during the bifurcation.
http://identifiers.org/isbn/0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.248ff)
Andronov-Hopf Bifurcation
http://identifiers.org/isbn/0387983821 (p.80)
TEDDY_0000073
subcritical Hopf bifurcation
http://identifiers.org/isbn/0387983821
subc-AH
subcritical Andronov-Hopf bifurcation
A `Hopf bifurcation` [http://identifiers.org/teddy/TEDDY_0000072] in which an `unstable limit cycle` [http://identifiers.org/teddy/TEDDY_0000128] is destroyed.
http://identifiers.org/isbn/0387983821
Kuznetsov YA. (1988) Elements of applied bifurcation theory, Springer, New York. (p.88)
subc-AH
http://www.tbi.univie.ac.at/wiki/index.php/Classification_of_Dynamical_Behaviors
subcritical Andronov-Hopf bifurcation
http://www.tbi.univie.ac.at/wiki/index.php/Classification_of_Dynamical_Behaviors
TEDDY_0000074
supercritical Hopf bifurcation
http://identifiers.org/isbn/0738204536
supc-AH
supercritical Andronov-Hopf bifurcation
A `Hopf bifurcation` [http://identifiers.org/teddy/TEDDY_0000072] in which an `stable limit cycle` [http://identifiers.org/teddy/TEDDY_0000114] appears.
http://identifiers.org/isbn/0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.249)
supc-AH
http://www.tbi.univie.ac.at/wiki/index.php/Classification_of_Dynamical_Behaviors
supercritical Andronov-Hopf bifurcation
http://www.tbi.univie.ac.at/wiki/index.php/Classification_of_Dynamical_Behaviors
TEDDY_0000075
saddle-node on invariant circle bifurcation
http://identifiers.org/isbn/0262090430
SNIC
SNIC bifurcation
SNLC
SNLC bifurcation
saddle-node on limit cycle bifurcation
Saddle-node bifurcation on invariant circle occurs when the center manifold of a saddle-node bifurcation [http://identifiers.org/teddy/TEDDY_0000071] forms an invariant circle. Such a bifurcation results in (dis)appearance of a limit cycle [http://identifiers.org/teddy/TEDDY_0000051] of an infinite period.
http://identifiers.org/isbn/0262090430
Izhikevich EM (2007) Dynamical systems in neuroscience : the geometry of excitability and bursting, MIT Press.
SNIC
http://www.tbi.univie.ac.at/wiki/index.php/Classification_of_Dynamical_Behaviors
SNIC bifurcation
http://www.scholarpedia.org/article/Saddle-node_bifurcation_on_invariant_circle
SNLC
http://www.scholarpedia.org/article/Saddle-node_bifurcation_on_invariant_circle
SNLC bifurcation
http://www.scholarpedia.org/article/Saddle-node_bifurcation_on_invariant_circle
saddle-node on limit cycle bifurcation
http://www.scholarpedia.org/article/Saddle-node_bifurcation_on_invariant_circle
2
0
TEDDY_0000076
saddle-node bifurcation of limit cycles
http://identifiers.org/isbn/0387983821
fold bifurcation of limit cycles
A 'bifurcation of limit cycle' [http://identifiers.org/teddy/TEDDY_0000070] in which two limit cycles [http://identifiers.org/teddy/TEDDY_0000051] (stable [http://identifiers.org/teddy/TEDDY_0000114] and saddle) collide and disappear.
http://identifiers.org/isbn/0387983821
Kuznetsov YA. (1988) Elements of applied bifurcation theory, Springer, New York. (p.163)
fold bifurcation of limit cycles
http://identifiers.org/isbn/0387983821 (p. 163)
TEDDY_0000077
period-doubling bifurcation of limit cycle
http://identifiers.org/isbn/0387983821
flip bifurcation of limit cycle
A 'bifurcation of limit cycle' [http://identifiers.org/teddy/TEDDY_0000070] in which a 'periodic orbit' [http://identifiers.org/teddy/TEDDY_0000050] with period-2 Poincaré map appears, while the fixed point changes its stability.
http://identifiers.org/isbn/0387983821
Kuznetsov YA. (1988) Elements of applied bifurcation theory, Springer, New York. (p.163)
flip bifurcation of limit cycle
http://identifiers.org/isbn/0387983821 (p.163)
TEDDY_0000078
Neimark-Sacker bifurcation of limit cycle
http://identifiers.org/isbn/0387983821
A 'bifurcation of limit cycle' [http://identifiers.org/teddy/TEDDY_0000070] corresponding to the case when the multipliers are complex and simple and lie on the unit circle. The
Poincaré map then has a parameter-dependent, two-dimensional, invariant manifold on which a closed invariant curve generically bifurcates from the fixed point.
http://identifiers.org/isbn/0387983821
Kuznetsov YA. (1988) Elements of applied bifurcation theory, Springer, New York. (p.164)
TEDDY_0000079
integrator
TEDDY_0000080
_obsolete behaviour ((obsolete))
true
Obsolete: only one obsolete branch required.
TEDDY_0000081
_obsolete characteristic ((obsolete))
true
Obsolete: only one obsolete branch required.
TEDDY_0000082
_obsolete functionality ((obsolete))
true
Obsolete: only one obsolete branch required.
TEDDY_0000083
temporal behaviour
http://identifiers.org/isbn/0387983821
orbit
phase curve
phase path
solution curve
trajectory
A temporal sequence of states following the evolution operator of the dynamical system through a given initial state.
http://identifiers.org/isbn/0387983821
Kuznetsov YA. (1988) Elements of applied bifurcation theory, Springer, New York. (p.8)
orbit
http://identifiers.org/isbn/0387983821 (p.8)
phase path
http://identifiers.org/isbn/0198565623 (p.6)
solution curve
http://identifiers.org/isbn/0387908196 (p.2)
trajectory
http://identifiers.org/isbn/0387983821 (p.8)
TEDDY_0000084
parametrical behaviour ((obsolete))
true
Obsolete: not required anymore.
TEDDY_0000085
limit behaviour
http://identifiers.org/isbn/0387983821
asymptotic state
limit set
A `temporal behaviour` [http://identifiers.org/teddy/TEDDY_0000083] with all behaviours starting sufficiently near converge to it as time goes to either positive or negative infinity.
http://identifiers.org/isbn/0387983821
Kuznetsov YA. (1988) Elements of applied bifurcation theory, Springer, New York. (p.10)
TEDDY_0000086
fixed point
http://mathworld.wolfram.com/FixedPoint.html
constant behaviour
constant solution
critical point
equilibrium
equilibrium point
equilibrium solution
rest solution
steady state
A 'temporal behaviour' [http://identifiers.org/teddy/TEDDY_0000083] which does not change its state.
http://mathworld.wolfram.com/FixedPoint.html
Weisstein, Eric W. Fixed Point. From MathWorld--A Wolfram Web Resource.
critical point
http://identifiers.org/isbn/0198565623 (p.4)
equilibrium point
http://identifiers.org/isbn/0198565623 (p.4)
TEDDY_0000087
node
http://identifiers.org/isbn/0738204536
fixed point node
A 'fixed point' [http://identifiers.org/teddy/TEDDY_0000086] for which the Jacobian matrix has real-valued eigenvalues of the same sign.
http://identifiers.org/isbn/0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.137)
TEDDY_0000088
center
http://identifiers.org/isbn/0738204536
centre
elliptic fixed point
A 'fixed point' [http://identifiers.org/teddy/TEDDY_0000086] for which the Jacobian matrix has purely imaginary complex conjugate eigenvalues. The fixed point is surrounded by a family of cycles in the phase portrait.
http://identifiers.org/isbn/0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (pp.134,137)
TEDDY_0000089
saddle
http://identifiers.org/isbn/0738204536
saddle point
A 'fixed point' [http://identifiers.org/teddy/TEDDY_0000086] for which the Jacobian matrix has real-valued eigenvalues of opposite signs.
http://identifiers.org/isbn/0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.137)
saddle point
http://identifiers.org/isbn/0198565623 (p.73)
TEDDY_0000090
star
http://identifiers.org/isbn/0738204536
proper node
star point
A 'fixed point' [http://identifiers.org/teddy/TEDDY_0000086] for which the Jacobian matrix has identical eigenvalues and two independent corresponding eigenvectors. All other trajectories [http://identifiers.org/teddy/TEDDY_0000083] of the system lie on straight through the fixed point.
http://identifiers.org/isbn/0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.135)
TEDDY_0000091
spiral
http://identifiers.org/isbn/0738204536
focus
spiral point
A 'fixed point' [http://identifiers.org/teddy/TEDDY_0000086] for which the Jacobian matrix has not purely imaginary complex conjugate eigenvalues. The fixed point is surrounded by spiralling behaviours [http://identifiers.org/teddy/TEDDY_0000083].
http://identifiers.org/isbn/0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (pp.134,137)
focus
http://identifiers.org/isbn/0198565623 (p.25)
spiral point
http://identifiers.org/isbn/0198565623 (p.73)
TEDDY_0000092
non-isolated fixed point
http://identifiers.org/isbn/0738204536
A 'fixed point' [http://identifiers.org/teddy/TEDDY_0000086] for which at least one eigenvalue of the Jacobian matrix is zero. If the other eigenvalue is non-zero the system has an entire line of fixed points along one dimension. If both eigenvalues are zero the system has a entire plane of fixed points.
http://identifiers.org/isbn/0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (pp.128,137)
TEDDY_0000093
degenerate node
http://identifiers.org/isbn/0738204536
degenerated node
improper node
A 'fixed point' [http://identifiers.org/teddy/TEDDY_0000086] for which the Jacobian matrix has identical eigenvalues and only one independent eigenvector. All other trajectories [http://identifiers.org/teddy/TEDDY_0000083] of the system asymptotically become parallel to the unique eigendirection.
http://identifiers.org/isbn/0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.135f)
TEDDY_0000094
attractor
http://identifiers.org/isbn/0738204536
http://identifiers.org/isbn/9810221428
The limit set which corresponds to the particular type of stable solution and attracts phase trajectories [http://identifiers.org/teddy/TEDDY_0000083] from a sertan region of initial conditions is an attractor.
http://identifiers.org/isbn/0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.324)
http://identifiers.org/isbn/9810221428
V. S. Anishchenko (1995) Dynamical Chaos-Models and Experiments: Appearance Routes and Structure of Chaos in Simple Dynamical Systems (World Scientific Series on Nonlinear Science, Series a, Vol 8). World Scientific Pub Co Inc. (p.10)
TEDDY_0000095
Lyapunov stable fixed point
http://identifiers.org/isbn/0738204536
Liapunov stable fixed point
A 'fixed point' [http://identifiers.org/teddy/TEDDY_0000086] which is Liapunov stable [http://identifiers.org/teddy/TEDDY_0000133].
http://identifiers.org/isbn/0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.129)
TEDDY_0000096
unstable ((obsolete))
http://identifiers.org/isbn/0738204536
true
Obsolete: a limit behaviour is unstable if it is neither Liapunov stable nor stable (use repellor instead).
http://identifiers.org/isbn/0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.129)
TEDDY_0000097
asymptotic stability
http://identifiers.org/isbn/0738204536
stable behaviour
A `stable behaviour` which is also a `Liapunov stable behaviour`.
The solution x*(t), t >= t0 is asymptotically stable, if all sufficiently small disturbances of the initial value lead to the solutions which re-approach the undisturbed solution.
http://identifiers.org/isbn/0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.129)
stable behaviour
http://identifiers.org/isbn/0738204536, (p.129)
TEDDY_0000098
neutral stability
http://identifiers.org/isbn/0738204536
asymptotically unstable
A behaviour [http://identifiers.org/teddy/TEDDY_0000083] which is Liapunov stable [http://identifiers.org/teddy/TEDDY_0000133] and not attracting [http://identifiers.org/teddy/TEDDY_0000097].
http://identifiers.org/isbn/0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.129)
asymptotically unstable
http://identifiers.org/isbn/0198565623 (p.289)
1
2
TEDDY_0000099
saddle connection
http://identifiers.org/isbn/0738204536
A temporal behaviour [http://identifiers.org/teddy/TEDDY_0000083] connecting two saddle points [http://identifiers.org/teddy/TEDDY_0000089].
http://identifiers.org/isbn/0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.184)
1
TEDDY_0000100
homoclinic saddle connection
http://identifiers.org/isbn/0738204536
homoclinic saddle loop
saddle loop
A saddle connection [http://identifiers.org/teddy/TEDDY_0000099] connecting a saddle point [http://identifiers.org/teddy/TEDDY_0000089] to itself.
http://identifiers.org/isbn/0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.161)
homoclinic saddle loop
http://identifiers.org/isbn/3540971416 (p.213)
saddle loop
http://www.egwald.ca/nonlineardynamics/mathappendix.php#limitset
2
TEDDY_0000101
heteroclinic saddle connection
http://identifiers.org/isbn/0738204536
heteroclinic saddle trajectory
saddle connection
A saddle connection [http://identifiers.org/teddy/TEDDY_0000099] connecting two different saddle points [http://identifiers.org/teddy/TEDDY_0000089].
http://identifiers.org/isbn/0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.166)
heteroclinic saddle trajectory
http://identifiers.org/isbn/0738204536 (p.166)
saddle connection
The term `saddle connection` is used as a synonym for a heteroclinic orbit (http://identifiers.org/isbn/0738204536 (p.166) ). But `saddle connection` is also used for both: homoclinic and heteroclinic orbits (http://identifiers.org/isbn/0387943773 (p.184)).
http://identifiers.org/isbn/0738204536 (p.166)
TEDDY_0000102
repeller
http://eom.springer.de/R/r081310.htm
repelling set
repellor
A subset of the phase space of the system that is an attractor [http://identifiers.org/teddy/TEDDY_0000094] for the reverse [TR_0015] system.
http://eom.springer.de/R/r081310.htm
Encyclopaedia of Mathematics, edited by Michiel Hazewinkel, CWI, Amsterdam.
repelling set
http://eom.springer.de/R/r081310.htm
TEDDY_0000103
rdfs:label follows notation in http://identifiers.org/isbn/3817112823 (p.42).
stable ((obsolete))
http://identifiers.org/isbn/0738204536
true
attracting
Obsolete: a limit behaviour [http://identifiers.org/teddy/TEDDY_0000085] is stable if all behaviours starting sufficiently near to it will approach it as time goes to positive infinity (use attractor instead).
http://identifiers.org/isbn/0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.128)
TEDDY_0000104
half-stable fixed point
http://identifiers.org/isbn/0738204536
A 'fixed point' [http://identifiers.org/teddy/TEDDY_0000086] which is half-stable [http://identifiers.org/teddy/TEDDY_0000134], i.e. the fixed point is attracting in one direction and unstable in the other direction.
http://identifiers.org/isbn/0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.26)