]>
TErminology for the Description of DYnamics (TEDDY)
http://biomodels.net/teddy
http://teddyontology.svn.sourceforge.net/svnroot/teddyontology/teddy/tags/rel-2011-03-10/teddy-inferred.owl
en
Classified by HermiT 1.3.4
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.
rel-2011-08-30 (inferred)
dependsOn
Relates temporal behaviours [TEDDY_0000083] to functional motifs [TEDDY_0000003] they depend on.
hasFeature
Links a dynamic system to the way one or several temporal behaviours [TEDDY_0000083] are modified or related upon interaction with information external to the system considered.
hasCharacteristic
Used to describe properties of a temporal behaviour [TEDDY_0000083] by means of behaviour characteristic [TEDDY_0000002].
hasSubPart
true
hasSuperPart
true
hasOnPart
true
adjacentTo
Two temporal behaviours [TEDDY_0000083] are adjacentTo each other if and only if they are in phase space proximity.
convergeTo
A temporal behaviour [TEDDY_0000083] convergeTo another temporal behaviour [TEDDY_0000083] if and only if it reaches the other behaviour [TEDDY_0000083] as time goes to either positive or negative infinity.
reverseOf
reverseOf links two temporal behaviours [TEDDY_0000083] whose phase diagrams can be obtained from each other by reversing the directions of all the phase paths.
transforms
A radical change in the behaviour [TEDDY_0000083] of a dynamic system occurs as a parameter passes through the bifurcation point. transforms links a bifurcation [TEDDY_0000053] to a behaviour [TEDDY_0000083] of the dynamic system: with the parameter not yet reached, at or already passed the bifurcation point.
creates
Links a bifurcation [TEDDY_0000053] to a 'temporal behaviour' [TEDDY_0000083] which appears as a parameter passes through the bifurcation point.
destroys
Links a bifurcation [TEDDY_0000053] to a 'temporal behaviour' [TEDDY_0000083] which it destroys.
hasPart
Inverse of partOf [obo:part_of].
partOf
http://obofoundry.org/ro/#OBO_REL:part_of
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.)
hasValue
Links a 'behaviour characteristic' [TEDDY_0000002] to the type of its value.
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.
curve ((obsolete))
true
Obsolete: equivalent to TEDDY_0000083 'Temporal Behaviour'.
behaviour characteristic
Behaviour characteristic is a property that characterizes temporal behaviors [TEDDY_0000083].
functional motif
urn:miriam: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.
urn:miriam:doi:10.1371/journal.pbio.0020369
Sporns O, Kötter R (2004) Motifs in Brain Networks, PLoS Biology, 2(11):e369.
monotonicity
http://mathworld.wolfram.com/MonotonicFunction.html
A curve is `monotonic` if successive states are ordered either entirely non-decreasing or entirely non-increasing.
monotone
monotonic
http://mathworld.wolfram.com/MonotonicFunction.html
Weisstein, Eric W. Monotonic Function. From MathWorld--A Wolfram Web Resource.
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.
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.
strict monotonicity
http://eom.springer.de/M/m064830.htm
A curve is strictly monotonic if it is monotonic [TEDDY_0000004] and has no equal temporal successive states.
strictly monotone
strictly monotonic
http://eom.springer.de/M/m064830.htm
Weisstein, Eric W. Monotonic Function. From MathWorld--A Wolfram Web Resource.
strictly increasing
http://eom.springer.de/M/m064830.htm
A curve is strictly increasing if it is strictly monotonic [TEDDY_0000007] with temporal successive states increasing ordered.
isotonic
http://eom.springer.de/M/m064830.htm
Weisstein, Eric W. Monotonic Function. From MathWorld--A Wolfram Web Resource.
strictly decreasing
http://eom.springer.de/M/m064830.htm
A curve is strictly decreasing if it is strictly monotonic [TEDDY_0000007] with temporal successive states decreasing ordered.
antitonic
http://eom.springer.de/M/m064830.htm
Weisstein, Eric W. Monotonic Function. From MathWorld--A Wolfram Web Resource.
single turnaround
A trajectory [TEDDY_0000083] of the system which responds negatively to displacement from equilbrium [TEDDY_0000086]: moving away from equilibrium the trajectory turns around and moves back towards equilibrium.
steady state ((obsolete))
true
Obsolete: equivalent to TEDDY_0000086 'Fixed Point'.
growth
http://www.mathunion.org/ICM/ICM1990.1/Main/icm1990.1.0325.0338.ocr.pdf
Let G be a finitely generated group with generator system A={a1,...,am}, and let l(g) be the length of the element g∈G with respect to the system A. The growth function of the group G with respect to the system A is the function y(n)=card{g∈G; l(g)<=n}.
http://www.mathunion.org/ICM/ICM1990.1/Main/icm1990.1.0325.0338.ocr.pdf
R. I. Grigorchuk (1991) On growth in group theory. Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), pp. 325-338, Math. Soc. Japan, Tokyo.
linear growth
urn:miriam:isbn:1844071448
A quantity grows linearly when its increase is a constant amount over a given period of time.
urn:miriam:isbn:1844071448
D. H. Meadows (2005) The Limits to Growth: The 30-Year Update, Rff Press, revised ed.
exponential growth
urn:miriam:isbn:1844071448
A quantity grows exponentially when its increase over a given period of time is proportional to what is already there.
urn:miriam:isbn:1844071448
D. H. Meadows (2005) The Limits to Growth: The 30-Year Update, Rff Press, revised ed.
curve shape
Characteristic describing the shape of the graph of a function.
concave shape
urn:miriam:isbn:0691080690
A shape of a graph of a concave function, i.e. a function whose negative is convex [TEDDY_0000021].
urn:miriam:isbn:0691080690
R. T. Rockafellar (1970) Convex Analysis (Princeton Mathematical Series), Princeton Univ Pr. (p.23)
zero growth
urn:miriam:isbn:1844071448
A quantity has a zero growth when it neither grows nor declines.
urn:miriam:isbn:1844071448
D. H. Meadows (2005) The Limits to Growth: The 30-Year Update, Rff Press, revised ed.
polynomial growth
A quantity grows polynomially when its growth function is bounded above by a polinomial function.
power growth
linear increasing
linear decreasing
convex shape
urn:miriam: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.
urn:miriam:isbn:0691080690
R. T. Rockafellar (1970) Convex Analysis (Princeton Mathematical Series), Princeton Univ Pr. (p.10, 23)
straight line shape
A shape of a graph of a linear function.
curve characteristic
A characteristic of the 'temporal behaviour' [TEDDY_0000083] curve (graph of the function representing a solution).
unbounded growth ((obsolete))
true
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.
asymptotic limit
http://eom.springer.de/a/a012870.htm
A limit of a function f(x) as x->x0 over a set E for which x0 is a density point.
approximate limit
http://eom.springer.de/a/a012870.htm
Encyclopaedia of Mathematics, edited by Michiel Hazewinkel, CWI, Amsterdam.
asymptotic upper limit
http://eom.springer.de/a/a012870.htm
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}.
approximate upper limit
http://eom.springer.de/a/a012870.htm
Encyclopaedia of Mathematics, edited by Michiel Hazewinkel, CWI, Amsterdam.
asymptotic lower limit
http://eom.springer.de/a/a012870.htm
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}.
approximate lower limit
http://eom.springer.de/a/a012870.htm
Encyclopaedia of Mathematics, edited by Michiel Hazewinkel, CWI, Amsterdam.
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.
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.
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.
sigmoid shape
urn:miriam: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)).
urn:miriam:isbn:3540605053
R. Rojas (1996) Neural Networks: A Systematic Introduction, Springer, 1st ed. (p.149)
feedback loop
urn:miriam:isbn:1584886420
A process whereby some proportion of function of the output signal of a system is passed (fed back) to the input.
feedback
Alon U (2006) An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall/CRC. (p.266)
urn:miriam:isbn:1584886420
negative feedback loop
urn:miriam:doi:10.1016/j.febslet.2005.02.008
A component or variable of a system is subject to negative feedback when it inhibits its own level of activity.
negative feedback
urn:miriam: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.
positive feedback loop
urn:miriam:doi:10.1016/j.febslet.2005.02.008
A component or variable of a system is subject to positive feedback when it increases its own level of activity.
positive feedback
urn:miriam: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.
feed-forward loop
A 'three-node feed-forward loop' [TEDDY_0000037] or its topological generalization.
FFL
feed-forward
feedforward
feedforward loop
three-node feed-forward loop
urn:miriam:isbn:1584886420
3-node FFL
3-node feed-forward
3-node feed-forward loop
3-node feedforward
3-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).
FFL
thre-node FFL
three-node feedforward loop
Alon U (2006) An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall/CRC. (p.267)
urn:miriam:isbn:1584886420
FFL
urn:miriam:isbn:1584886420 (p.41)
coherent three-node feed-forward loop
urn:miriam:isbn:1584886420
A feed-forward loop [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.
coherent 3-node feed-forward loop
coherent 3-node feedforward
coherent 3-node feedforward loop
coherent three-node feedforward loop
urn:miriam:isbn:1584886420
Alon U (2006) An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall/CRC. (p.265)
type-1 coherent three-node feed-forward loop
urn:miriam:isbn:1584886420
A 'coherent three-node feed-forward loop' [TEDDY_0000038] in which all three regulations are positive.
C1-FFL
coherent 3-node feedforward type-1
type-1 coherent FFL
urn:miriam:isbn:1584886420
Alon U (2006) An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall/CRC. (p.47)
type-2 coherent three-node feed-forward loop
urn:miriam:isbn:1584886420
A 'coherent three-node feed-forward loop' [TEDDY_0000038] in which X represses Z, and also represses an activator of Z.
C2-FFL
coherent 3-node feedforward type-2
type-2 coherent FFL
urn:miriam:isbn:1584886420
Alon U (2006) An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall/CRC. (p.47)
type-3 coherent three-node feed-forward loop
urn:miriam:isbn:1584886420
A 'coherent three-node feed-forward loop' [TEDDY_0000038] in which X represses Z, and also activates a repressor of Z.
C3-FFL
coherent 3-node feedforward type-3
type-3 coherent FFL
urn:miriam:isbn:1584886420
Alon U (2006) An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall/CRC. (p.47)
type-4 coherent three-node feed-forward loop
urn:miriam:isbn:1584886420
A 'coherent three-node feed-forward loop' [TEDDY_0000038] in which X activates Z, and also represses a repressor of Z.
C4-FFL
coherent 3-node feedforward type-4
type-4 coherent FFL
urn:miriam:isbn:1584886420
Alon U (2006) An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall/CRC. (p.47)
incoherent three-node feed-forward loop
urn:miriam:isbn:1584886420
A feed-forward loop [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.
incoherent 3-node feed-forward loop
incoherent 3-node feedforward
incoherent 3-node feedforward loop
incoherent three-node feedforward loop
urn:miriam:isbn:1584886420
Alon U (2006) An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall/CRC. (p.267)
type-1 incoherent three-node feed-forward loop
urn:miriam:isbn:1584886420
An 'incoherent three-node feed-forward loop' [TEDDY_0000043] in which X activates Z, and also activates a repressor of Z.
I1-FFL
incoherent 3-node feedforward type-1
type-1 incoherent FFL
urn:miriam:isbn:1584886420
Alon U (2006) An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall/CRC. (p.47)
type-2 incoherent three-node feed-forward loop
urn:miriam:isbn:1584886420
An 'incoherent three-node feed-forward loop' [TEDDY_0000043] in which X represses Z, and also represses a repressor of Z.
I2-FFL
incoherent 3-node feedforward type-2
type-2 incoherent FFL
urn:miriam:isbn:1584886420
Alon U (2006) An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall/CRC. (p.47)
type-3 incoherent three-node feed-forward loop
urn:miriam:isbn:1584886420
An 'incoherent three-node feed-forward loop' [TEDDY_0000043] in which X represses Z, and also activates an activator of Z.
I3-FFL
incoherent 3-node feedforward type-3
type-3 incoherent FFL
urn:miriam:isbn:1584886420
Alon U (2006) An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall/CRC. (p.47)
type-4 incoherent three-node feed-forward loop
urn:miriam:isbn:1584886420
An 'incoherent three-node feed-forward loop' [TEDDY_0000043] in which X activates Z, and also represses an activator of Z.
I4-FFL
incoherent 3-node feedforward type-4
type-4 incoherent FFL
urn:miriam:isbn:1584886420
Alon U (2006) An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall/CRC. (p.47)
orbit ((obsolete))
true
Obsolete: equivalent to TEDDY_0000083 'Temporal Behaviour'.
fixed point ((obsolete))
true
Obsolete: equivalent to TEDDY_0000086 'Fixed Point'.
periodic orbit
urn:miriam:isbn:0387983821
A temporal behaviour [TEDDY_0000083] which repeats every state after a specific period of time.
closed orbit
cycle
periodic solution
urn:miriam:isbn:0387983821
Kuznetsov YA. (1988) Elements of applied bifurcation theory, Springer, New York. (p.9)
limit cycle
urn:miriam:isbn:0738204536
Grenzzyklus
A closed orbit which is isolated, i.e. neighbouring orbits are not closed.
isolated closed path
urn:miriam:isbn:3817112823
Grenzzyklus
urn:miriam:isbn:0198565623 (p.30)
isolated closed path
urn:miriam:isbn:0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.196)
parameter dependency ((obsolete))
true
Obsolete: not required anymore.
bifurcation
http://www.egwald.com/nonlineardynamics/bifurcations.php
urn:miriam:isbn:0198565623
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://www.egwald.com/nonlineardynamics/bifurcations.php
Elmer G. Wiens: Egwald Web Services Ltd.
urn:miriam:isbn:0198565623
Jordan DW., Smith P. (1999) Nonlinear ordinary differential equations: an introduction to dynamical systems, Oxford University Press. (p.420)
strange attractor
urn:miriam:isbn:0738204536
urn:miriam:isbn:9810221428
A non-periodic orbit [TEDDY_0000143] which is attracting and exhibits sensitive dependence on initial conditions.
The attractor [TEDDY_0000094] is strange if trajectories [TEDDY_0000083] on the attractor, being stable according to Poisson [TEDDY_0000149], are unstable according to Lyapunov [not TEDDY_0000113].
chaotic attractor
fractal attractor
urn:miriam: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)
urn:miriam:isbn:0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.325)
urn:miriam:isbn:0738204536 (p.325)
fractal attractor
urn:miriam:isbn:0738204536 (p.325)
chaotic attractor
bursting
urn:miriam:isbn:0262090430
A burst is two or more spikes followed by a period of quiescence.
urn:miriam:isbn:0262090430
Izhikevich EM (2007) Dynamical systems in neuroscience : the geometry of excitability and bursting, MIT Press. (p.325)
switch
urn:miriam: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.
urn:miriam: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.
stability ((obsolete))
true
Obsolete: not required anymore.
asymptotic decreasing
http://eom.springer.de/A/a013610.htm
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.
asymptotic increasing
http://eom.springer.de/A/a013610.htm
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.
diverging increasing
chaotic oscillation
If a particular solution is aperiodic, but bounded for pt->infinity, then it corresponds to the regime of chaotic oscillation.
non-periodic oscillation
urn:miriam: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)
urn:miriam:isbn:9810221428
sustained oscillation ((obsolete))
true
Obsolete: equivalent to TEDDY_0000051 'Limit Cycle'.
damped oscillation
urn:miriam:isbn:0486655083
Damping is any effect that tends to reduce the amplitude [TEDDY_0000131] of oscillations in an oscillatory system.
urn:miriam:isbn:0486655083
A. A. Andronov, A. A. Vitt, and S. E. Khaikin (1987) Theory of Oscillators, Dover Publications.
single-periodic oscillation
urn:miriam:isbn:0198565623
An oscillation [TEDDY_0000006] corresponding to a solution having a one-loop phase path and a period-1 Poincaré map.
SPO
single periodic oscillation
urn:miriam:isbn:0198565623
Jordan DW., Smith P. (1999) Nonlinear ordinary differential equations: an introduction to dynamical systems, Oxford University Press. (p.465)
mixed-mode oscillation
urn:miriam: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 [TEDDY_0000131].
urn:miriam: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).
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.
mixed-mode oscillation
periodic oscillation
urn:miriam: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 [TEDDY_0000067] of solution).
urn:miriam: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)
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.
regularity ((obsolete))
true
Obsolete: not required anymore.
local bifurcation
http://en.wikipedia.org/wiki/Bifurcation_theory#Local_bifurcations
urn:miriam:isbn:0387983821
A `bifurcation` [TEDDY_0000053] in which a stable [TEDDY_0000133] fixed point [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.
bifurcation from steady state
bifurcation of equilibrium
bifurcation of equilibrium point
bifurcation of fixed point
http://www.scholarpedia.org/article/MATCONT
bifurcation from steady state
urn:miriam:isbn:0387983821 (p.58)
bifurcation of fixed point
urn:miriam:isbn:0387983821
Kuznetsov YA. (1988) Elements of applied bifurcation theory, Springer, New York. (p.58)
urn:miriam:isbn:0387983821 (p.58)
bifurcation of equilibrium
bifurcation of limit cycle
A bifurcation [TEDDY_0000053] in which a limit cycle [TEDDY_0000051] (dis)appears or changes its stability [TEDDY_0000113].
saddle-node bifurcation
1
1
1
1
urn:miriam:isbn:0387983821
urn:miriam:isbn:0738204536
A `zero-eigenvalue bifurcation` [TEDDY_0000123] in which a stable [TEDDY_0000133] fixed point [TEDDY_0000086] and an unstable [not TEDDY_0000133] fixed point [TEDDY_0000086] collide and mutually annihilate.
blue sky bifurcation
fold bifurcation
limit point bifurcation
tangent bifurcation
turning-point bifurcation
urn:miriam:isbn:0738204536 (p.47)
fold bifurcation
urn:miriam:isbn:0738204536 (p.47)
turning-point bifurcation
http://www.scholarpedia.org/article/Saddle-node_Bifurcation
limit point bifurcation
urn:miriam:isbn:0387983821
Kuznetsov YA. (1988) Elements of applied bifurcation theory, Springer, New York. (p.80)
urn:miriam:isbn:0387983821 (p.80)
tangent bifurcation
urn:miriam:isbn:0738204536 (p.47)
blue sky bifurcation
urn:miriam:isbn:0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.45)
Hopf bifurcation
urn:miriam:isbn:0738204536
A `local bifurcation` [TEDDY_0000069] in which a `stable spiral` [TEDDY_0000126] changes in an `unstable spiral` [TEDDY_0000127]. The linearisation around the fixed point [TEDDY_0000086] has two conjugate eigenvalues. This eigenvalues cross simultaneously the imaginary axis from left (negative real part) to the right during the bifurcation.
Andronov-Hopf Bifurcation
urn:miriam:isbn:0387983821 (p.80)
Andronov-Hopf Bifurcation
urn:miriam:isbn:0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.248ff)
subcritical Hopf bifurcation
urn:miriam:isbn:0387983821
A `Hopf bifurcation` [TEDDY_0000072] in which an `unstable limit cycle` [TEDDY_0000128] is destroyed.
subc-AH
subcritical Andronov-Hopf bifurcation
urn:miriam:isbn:0387983821
Kuznetsov YA. (1988) Elements of applied bifurcation theory, Springer, New York. (p.88)
http://www.tbi.univie.ac.at/wiki/index.php/Classification_of_Dynamical_Behaviors
subc-AH
http://www.tbi.univie.ac.at/wiki/index.php/Classification_of_Dynamical_Behaviors
subcritical Andronov-Hopf bifurcation
supercritical Hopf bifurcation
urn:miriam:isbn:0738204536
A `Hopf bifurcation` [TEDDY_0000072] in which an `stable limit cycle` [TEDDY_0000114] appears.
supc-AH
supercritical Andronov-Hopf bifurcation
urn:miriam:isbn:0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.249)
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
supc-AH
saddle-node on invariant circle bifurcation
urn:miriam:isbn:0262090430
SNIC
SNIC bifurcation
SNLC
SNLC bifurcation
Saddle-node bifurcation on invariant circle occurs when the center manifold of a saddle-node bifurcation [TEDDY_0000071] forms an invariant circle. Such a bifurcation results in (dis)appearance of a limit cycle [TEDDY_0000051] of an infinite period.
saddle-node on limit cycle bifurcation
http://www.scholarpedia.org/article/Saddle-node_bifurcation_on_invariant_circle
saddle-node on limit cycle bifurcation
http://www.tbi.univie.ac.at/wiki/index.php/Classification_of_Dynamical_Behaviors
SNIC
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
SNIC bifurcation
urn:miriam:isbn:0262090430
Izhikevich EM (2007) Dynamical systems in neuroscience : the geometry of excitability and bursting, MIT Press.
saddle-node bifurcation of limit cycle
urn:miriam:isbn:0387983821
A 'bifurcation of limit cycle' [TEDDY_0000070] in which two limit cycles [TEDDY_0000051] (stable [TEDDY_0000114] and saddle) collide and disappear.
SNLC
fold bifurcation of limit cycle
http://www.tbi.univie.ac.at/wiki/index.php/Classification_of_Dynamical_Behaviors
SNLC
urn:miriam:isbn:0387983821
Kuznetsov YA. (1988) Elements of applied bifurcation theory, Springer, New York. (p.163)
urn:miriam:isbn:0387983821 (p. 163)
fold bifurcation of limit cycle
period-doubling bifurcation of limit cycle
urn:miriam:isbn:0387983821
A 'bifurcation of limit cycle' [TEDDY_0000070] in which a 'periodic orbit' [TEDDY_0000050] with period-2 Poincaré map appears, while the fixed point changes its stability.
PDLC
flip bifurcation of limit cycle
http://www.tbi.univie.ac.at/wiki/index.php/Classification_of_Dynamical_Behaviors
PDLC
urn:miriam:isbn:0387983821
Kuznetsov YA. (1988) Elements of applied bifurcation theory, Springer, New York. (p.163)
urn:miriam:isbn:0387983821 (p.163)
flip bifurcation of limit cycle
Neimark-Sacker bifurcation of limit cycle
urn:miriam:isbn:0387983821
A 'bifurcation of limit cycle' [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.
NSLC
http://www.tbi.univie.ac.at/wiki/index.php/Classification_of_Dynamical_Behaviors
NSLC
urn:miriam:isbn:0387983821
Kuznetsov YA. (1988) Elements of applied bifurcation theory, Springer, New York. (p.164)
integrator
_obsolete behaviour ((obsolete))
true
Obsolete: only one obsolete branch required.
_obsolete characteristic ((obsolete))
true
Obsolete: only one obsolete branch required.
_obsolete functionality ((obsolete))
true
Obsolete: only one obsolete branch required.
temporal behaviour
urn:miriam:isbn:0387983821
A temporal sequence of states following the evolution operator of the dynamical system through a given initial state.
orbit
phase curve
phase path
solution curve
trajectory
urn:miriam:isbn:0198565623 (p.6)
phase path
urn:miriam:isbn:0387983821
Kuznetsov YA. (1988) Elements of applied bifurcation theory, Springer, New York. (p.8)
urn:miriam:isbn:0387908196 (p.2)
solution curve
urn:miriam:isbn:0387983821 (p.8)
orbit
urn:miriam:isbn:0387983821 (p.8)
trajectory
parametrical behaviour ((obsolete))
true
Obsolete: not required anymore.
limit behaviour
urn:miriam:isbn:0387983821
A `temporal behaviour` [TEDDY_0000083] with all behaviours starting sufficiently near converge to it as time goes to either positive or negative infinity.
asymptotic state
limit set
urn:miriam:isbn:0387983821
Kuznetsov YA. (1988) Elements of applied bifurcation theory, Springer, New York. (p.10)
fixed point
http://mathworld.wolfram.com/FixedPoint.html
A 'temporal behaviour' [TEDDY_0000083] which does not change its state.
constant behaviour
constant solution
critical point
equilibrium
equilibrium point
equilibrium solution
rest solution
steady state
urn:miriam:isbn:0198565623 (p.4)
critical point
urn:miriam:isbn:0198565623 (p.4)
equilibrium point
http://mathworld.wolfram.com/FixedPoint.html
Weisstein, Eric W. Fixed Point. From MathWorld--A Wolfram Web Resource.
node
urn:miriam:isbn:0738204536
A 'fixed point' [TEDDY_0000086] for which the Jacobian matrix has real-valued eigenvalues of the same sign.
fixed point node
urn:miriam:isbn:0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.137)
center
urn:miriam:isbn:0738204536
A 'fixed point' [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.
centre
elliptic fixed point
urn:miriam:isbn:0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (pp.134,137)
saddle
urn:miriam:isbn:0738204536
A 'fixed point' [TEDDY_0000086] for which the Jacobian matrix has real-valued eigenvalues of opposite signs.
saddle point
urn:miriam:isbn:0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.137)
urn:miriam:isbn:0198565623 (p.73)
saddle point
star
urn:miriam:isbn:0738204536
A 'fixed point' [TEDDY_0000086] for which the Jacobian matrix has identical eigenvalues and two independent corresponding eigenvectors. All other trajectories [TEDDY_0000083] of the system lie on straight through the fixed point.
proper node
star point
urn:miriam:isbn:0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.135)
spiral
urn:miriam:isbn:0738204536
A 'fixed point' [TEDDY_0000086] for which the Jacobian matrix has not purely imaginary complex conjugate eigenvalues. The fixed point is surrounded by spiralling behaviours [TEDDY_0000083].
focus
spiral point
urn:miriam:isbn:0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (pp.134,137)
urn:miriam:isbn:0198565623 (p.73)
spiral point
urn:miriam:isbn:0198565623 (p.25)
focus
non-isolated fixed point
urn:miriam:isbn:0738204536
A 'fixed point' [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.
urn:miriam:isbn:0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (pp.128,137)
degenerate node
urn:miriam:isbn:0738204536
A 'fixed point' [TEDDY_0000086] for which the Jacobian matrix has identical eigenvalues and only one independent eigenvector. All other trajectories [TEDDY_0000083] of the system asymptotically become parallel to the unique eigendirection.
degenerated node
improper node
urn:miriam:isbn:0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.135f)
attractor
urn:miriam:isbn:0738204536
urn:miriam:isbn:9810221428
The limit set which corresponds to the particular type of stable solution and attracts phase trajectories [TEDDY_0000083] from a sertan region of initial conditions is an attractor.
urn:miriam:isbn:0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.324)
urn:miriam: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)
Liapunov stable fixed point
urn:miriam:isbn:0738204536
A 'fixed point' [TEDDY_0000086] which is Liapunov stable [TEDDY_0000133].
urn:miriam:isbn:0738204536
Strogatz SH. (2000) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview Press, Cambridge, MA. (p.129)
unstable ((obsolete))