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''surging system''
 
''surging system''
  
A [[Dynamical system|dynamical system]] with a state space containing turbulence manifolds, i.e. manifolds, the crossing of which alters the law governing the motion of the system. A turbulent system in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094450/t0944501.png" /> is described by several systems of differential equations
+
A [[Dynamical system|dynamical system]] with a state space containing turbulence manifolds, i.e. manifolds, the crossing of which alters the law governing the motion of the system. A turbulent system in $  \mathbf R  ^ {n} $
 +
is described by several systems of differential equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094450/t0944502.png" /></td> </tr></table>
+
$$
 +
( S _ {i} ) :\  \  \dot{x}  = f _ {i} ( x),\ \
 +
x \in \mathbf R  ^ {n} ,\ \
 +
\dot{x}  = {
 +
\frac{dx}{dt}
 +
}
 +
$$
  
 
and by surfaces
 
and by surfaces
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094450/t0944503.png" /></td> </tr></table>
+
$$
 +
M _ {i}  \subset  \mathbf R  ^ {n} ,\ \
 +
i = 1 \dots m.
 +
$$
  
When a trajectory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094450/t0944504.png" /> in the region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094450/t0944505.png" /> meets the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094450/t0944506.png" />, a turbulence occurs, i.e. the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094450/t0944507.png" /> is replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094450/t0944508.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094450/t0944509.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094450/t09445010.png" /> (for more details see [[#References|[3]]]). The participation of several differential systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094450/t09445011.png" /> in the definition of a turbulent system results in a large diversity of phase portraits of such systems. For instance, the turbulent system described by two stationary systems of linear differential equations
+
When a trajectory of $  S $
 +
in the region $  ( S _ {i} ) $
 +
meets the surface $  M _ {i} $,  
 +
a turbulence occurs, i.e. the system $  ( S _ {i} ) $
 +
is replaced by $  ( S _ {i+ 1 }  ) $,  
 +
while $  ( S _ {m+ 1 }  ) $
 +
coincides with $  ( S _ {1} ) $(
 +
for more details see [[#References|[3]]]). The participation of several differential systems $  ( S _ {i} ) $
 +
in the definition of a turbulent system results in a large diversity of phase portraits of such systems. For instance, the turbulent system described by two stationary systems of linear differential equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094450/t09445012.png" /></td> </tr></table>
+
$$
 +
\dot{x} _ {1}  = \
 +
a _ {i} + b _ {i} x _ {1} + c _ {i} x _ {2} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094450/t09445013.png" /></td> </tr></table>
+
$$
 +
\dot{x} _ {2}  = \beta _ {i} x _ {1} + \gamma _ {i} x _ {2} ,\  i = 1, 2,
 +
$$
  
having the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094450/t09445014.png" /> as turbulence manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094450/t09445015.png" /> may, in particular, have a [[Limit cycle|limit cycle]] [[#References|[3]]], [[#References|[4]]]. Turbulent systems supply specific models of non-linear vibrations, thus permitting one to describe "hysteresis" phenomena.
+
having the straight line $  x _ {2} = 0 $
 +
as turbulence manifolds $  M _ {1} = M _ {2} $
 +
may, in particular, have a [[Limit cycle|limit cycle]] [[#References|[3]]], [[#References|[4]]]. Turbulent systems supply specific models of non-linear vibrations, thus permitting one to describe "hysteresis" phenomena.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> T. Vogel,   "Sur les systèmes déferlants" ''Bull. Soc. Math. France'' , '''81''' (1953) pp. 63–75</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> T. Vogel,   "Systèmes déferlants, systèmes héréditaires, systèmes dynamiques" , ''Qualitative Methods in Nonlinear Vibration Theory (Proc. Internat. Symp. Nonlinear Vibrations 1961)'' , '''2''' , Kiev (1963) pp. 123–130</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A.D. Myshkis,   A.Ya. Khokhryakov,   "Surging dynamical systems I. Singular points in the plane" ''Mat. Sb.'' , '''45 (87)''' : 3 (1958) pp. 401–414 (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J.I. Gil'derman,   "On the limit cycles of piecewise affine systems" ''Soviet Math. Dokl.'' , '''17''' : 5 (1976) pp. 1328–1332 ''Dokl. Akad. Nauk SSSR'' , '''230''' : 3 (1976) pp. 512–515</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> T. Vogel, "Sur les systèmes déferlants" ''Bull. Soc. Math. France'' , '''81''' (1953) pp. 63–75 {{MR|0058053}} {{ZBL|0051.08301}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> T. Vogel, "Systèmes déferlants, systèmes héréditaires, systèmes dynamiques" , ''Qualitative Methods in Nonlinear Vibration Theory (Proc. Internat. Symp. Nonlinear Vibrations 1961)'' , '''2''' , Kiev (1963) pp. 123–130 {{MR|0159451}} {{ZBL|0152.28801}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A.D. Myshkis, A.Ya. Khokhryakov, "Surging dynamical systems I. Singular points in the plane" ''Mat. Sb.'' , '''45 (87)''' : 3 (1958) pp. 401–414 (In Russian) {{MR|}} {{ZBL|0081.30702}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J.I. Gil'derman, "On the limit cycles of piecewise affine systems" ''Soviet Math. Dokl.'' , '''17''' : 5 (1976) pp. 1328–1332 ''Dokl. Akad. Nauk SSSR'' , '''230''' : 3 (1976) pp. 512–515 {{MR|}} {{ZBL|0382.34016}} </TD></TR></table>
 
 
 
 
  
 
====Comments====
 
====Comments====
The phrase "turbulent system" has not been used in the English language literature for this notion. (Actually, the above-described system does not have a specific name attached to it, except in [[#References|[1]]]–[[#References|[3]]].)
+
The phrase "turbulent system" has not been used in the English language literature for this notion. (Actually, the above-described system does not have a specific name attached to it, except in [[#References|[1]]]–[[#References|[3]]].)
  
 
For turbulence of fluids see [[Turbulence, mathematical problems in|Turbulence, mathematical problems in]].
 
For turbulence of fluids see [[Turbulence, mathematical problems in|Turbulence, mathematical problems in]].
  
Analogous notions occur in control theory and statistics. In control theory one considers control systems whose dynamics are given by different (differential) equations in different regimes of state space, often as an approximation of a non-linear control system by different linearizations in different parts of state space. The phase "sliding controlsliding control" and "control schedulingcontrol scheduling" are sometimes used to refer to control laws in such settings.
+
Analogous notions occur in control theory and statistics. In control theory one considers control systems whose dynamics are given by different (differential) equations in different regimes of state space, often as an approximation of a non-linear control system by different linearizations in different parts of state space. The phase "sliding controlsliding control" and "control schedulingcontrol scheduling" are sometimes used to refer to control laws in such settings.
  
 
In both statistics and control theory one considers random processes which are governed by a number of possible laws and sudden transitions from one law to another can take place. The problem in statistics of detecting that such a change-over has taken place is known as change-point detection.
 
In both statistics and control theory one considers random processes which are governed by a number of possible laws and sudden transitions from one law to another can take place. The problem in statistics of detecting that such a change-over has taken place is known as change-point detection.

Latest revision as of 08:26, 6 June 2020


surging system

A dynamical system with a state space containing turbulence manifolds, i.e. manifolds, the crossing of which alters the law governing the motion of the system. A turbulent system in $ \mathbf R ^ {n} $ is described by several systems of differential equations

$$ ( S _ {i} ) :\ \ \dot{x} = f _ {i} ( x),\ \ x \in \mathbf R ^ {n} ,\ \ \dot{x} = { \frac{dx}{dt} } $$

and by surfaces

$$ M _ {i} \subset \mathbf R ^ {n} ,\ \ i = 1 \dots m. $$

When a trajectory of $ S $ in the region $ ( S _ {i} ) $ meets the surface $ M _ {i} $, a turbulence occurs, i.e. the system $ ( S _ {i} ) $ is replaced by $ ( S _ {i+ 1 } ) $, while $ ( S _ {m+ 1 } ) $ coincides with $ ( S _ {1} ) $( for more details see [3]). The participation of several differential systems $ ( S _ {i} ) $ in the definition of a turbulent system results in a large diversity of phase portraits of such systems. For instance, the turbulent system described by two stationary systems of linear differential equations

$$ \dot{x} _ {1} = \ a _ {i} + b _ {i} x _ {1} + c _ {i} x _ {2} , $$

$$ \dot{x} _ {2} = \beta _ {i} x _ {1} + \gamma _ {i} x _ {2} ,\ i = 1, 2, $$

having the straight line $ x _ {2} = 0 $ as turbulence manifolds $ M _ {1} = M _ {2} $ may, in particular, have a limit cycle [3], [4]. Turbulent systems supply specific models of non-linear vibrations, thus permitting one to describe "hysteresis" phenomena.

References

[1] T. Vogel, "Sur les systèmes déferlants" Bull. Soc. Math. France , 81 (1953) pp. 63–75 MR0058053 Zbl 0051.08301
[2] T. Vogel, "Systèmes déferlants, systèmes héréditaires, systèmes dynamiques" , Qualitative Methods in Nonlinear Vibration Theory (Proc. Internat. Symp. Nonlinear Vibrations 1961) , 2 , Kiev (1963) pp. 123–130 MR0159451 Zbl 0152.28801
[3] A.D. Myshkis, A.Ya. Khokhryakov, "Surging dynamical systems I. Singular points in the plane" Mat. Sb. , 45 (87) : 3 (1958) pp. 401–414 (In Russian) Zbl 0081.30702
[4] J.I. Gil'derman, "On the limit cycles of piecewise affine systems" Soviet Math. Dokl. , 17 : 5 (1976) pp. 1328–1332 Dokl. Akad. Nauk SSSR , 230 : 3 (1976) pp. 512–515 Zbl 0382.34016

Comments

The phrase "turbulent system" has not been used in the English language literature for this notion. (Actually, the above-described system does not have a specific name attached to it, except in [1][3].)

For turbulence of fluids see Turbulence, mathematical problems in.

Analogous notions occur in control theory and statistics. In control theory one considers control systems whose dynamics are given by different (differential) equations in different regimes of state space, often as an approximation of a non-linear control system by different linearizations in different parts of state space. The phase "sliding controlsliding control" and "control schedulingcontrol scheduling" are sometimes used to refer to control laws in such settings.

In both statistics and control theory one considers random processes which are governed by a number of possible laws and sudden transitions from one law to another can take place. The problem in statistics of detecting that such a change-over has taken place is known as change-point detection.

How to Cite This Entry:
Turbulent system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Turbulent_system&oldid=13384
This article was adapted from an original article by Yu.S. Bogdanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article