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Logical starting point:
A dynamical system is a manifold M called the phase (or state) space endowed with a family of smooth evolution functions \(\phi^t\) that for any element of t ∈ T, the time, map a point of the phase space back into the phase space. The notion of smoothness changes with applications and the type of manifold. There are several choices for the set T. When T is taken to be the reals, the dynamical system is called a flow; and if T is restricted to the non-negative reals, then the dynamical system is a semi-flow. When T is taken to be the integers, it is a cascade or a map; and the restriction to the non-negative integers is a semi-cascade.
I'm going to work through some of this here. Feel free anyone to add any comments that come to mind (the notes can be interleaved).
Comments
In 1986 I had a series of conversations with Stephen Wolfram about dynamical systems, but Wolfram didn't tell me where he got his ideas, so I had to do a decade of digging. Here's some of the main points I found documented:
Dynamics in mathematics is broad and contains the entirety of dynamics from physics. See Mathematics Subject Classification 37: Dynamical systems and ergodic theory
Dynamics and PDEs are the two mathematical disciplines that are acceptable foundations for physics. When chaos is not present PDEs are fine, but dynamics has evolved to deal with chaos and fractal structures in a way that PDEs can't.
Dynamical systems are generalizations of iterated functions.
Dynamics is a collection of mathematical disciplines just as there is no singular chaos theory. Unifying maps and flows is the subject of my research.
The definition of dynamics @DavidTanzer shared is applicable to any closed system in physics including a QM representation of the entire Universe.
In 1986 I had a series of conversations with Stephen Wolfram about dynamical systems, but Wolfram didn't tell me where he got his ideas, so I had to do a decade of digging. Here's some of the main points I found documented: * Dynamics in mathematics is broad and contains the entirety of dynamics from physics. See [*Mathematics Subject Classification 37: Dynamical systems and ergodic theory*](https://zbmath.org/static/msc2020.pdf) * Dynamics and PDEs are the two mathematical disciplines that are acceptable foundations for physics. When chaos is not present PDEs are fine, but dynamics has evolved to deal with chaos and fractal structures in a way that PDEs can't. * Dynamical systems are generalizations of iterated functions. * Dynamics is a collection of mathematical disciplines just as there is no singular chaos theory. Unifying maps and flows is the subject of my research. * The definition of dynamics @DavidTanzer shared is applicable to any closed system in physics including a QM representation of the entire Universe.Let's make it concrete, with an example. Suppose our manifold is the unit disc:
\[M = \lbrace x,y\ |\ x^2 + y^2 \leq 1 \rbrace \]
Using the given terminology, each point of the disc is a 'phase', and the whole disc comprises the phase space.
For a fixed time \(t\), the function \(\phi^t: M \rightarrow M\) is the mapping that sends each point in the disc to where it ends up at time \(t\).
From this we get that \(\phi^0: M \rightarrow M\) must be the identity function.
> A dynamical system is a manifold M called the phase (or state) space endowed with a family of smooth evolution functions \\(\phi^t\\) that for any element of t ∈ T, the time, map a point of the phase space back into the phase space. Let's make it concrete, with an example. Suppose our manifold is the unit disc: \\[M = \lbrace x,y\ |\ x^2 + y^2 \leq 1 \rbrace \\] Using the given terminology, each point of the disc is a 'phase', and the whole disc comprises the phase space. For a fixed time \\(t\\), the function \\(\phi^t: M \rightarrow M\\) is the mapping that sends each point in the disc to where it ends up at time \\(t\\). From this we get that \\(\phi^0: M \rightarrow M\\) must be the identity function.There is practical sense in the further restriction that \(\phi^t\) should be an action of T over the smooth evolution functions.
That means:
The latter statement can be interpreted as follows. \(\phi^s: M \rightarrow M\) gives the transformative effect after \(s\) units of time. So the transformative effect after \(s + t\) units of time equals the composition of the transformative functions for \(s\) units of time and \(t\) units of time, separately.
Note this is a semi-group action. Stipulating a group action is too strong a prescription for the general case, as it would require that, for each \(t\) the function \(\phi^t\) is invertible. That would rule out, for example, the evolution which collapses everything to a point after a fixed amount of time. Furthermore a group action would require that \(\phi^t\) be defined for negative \(t\), which is off the table for semi-flows.
From Dynamical System at Wolfram:
There is practical sense in the further restriction that \\(\phi^t\\) should be an action of T over the smooth evolution functions. That means: * \\(\phi^0\\) is the identity function * \\(\phi^{s+t} = \phi^s \circ \phi^t\\) The latter statement can be interpreted as follows. \\(\phi^s: M \rightarrow M\\) gives the transformative effect after \\(s\\) units of time. So the transformative effect after \\(s + t\\) units of time equals the composition of the transformative functions for \\(s\\) units of time and \\(t\\) units of time, separately. Note this is a semi-group action. Stipulating a group action is too strong a prescription for the general case, as it would require that, for each \\(t\\) the function \\(\phi^t\\) is invertible. That would rule out, for example, the evolution which collapses everything to a point after a fixed amount of time. Furthermore a group action would require that \\(\phi^t\\) be defined for negative \\(t\\), which is off the table for semi-flows. From [Dynamical System](https://mathworld.wolfram.com/DynamicalSystem.html) at Wolfram: > Technically, a dynamical system is a smooth action of the reals or the integers on another object (usually a manifold).So I added this to the Wikipedia definition:
So I added this to the Wikipedia definition: > Note: There is a further technical condition that \\(\phi^t\\) is an action of T on M. That includes the facts that \\(\phi^0\\) is the identity function and that \\(\phi^{s+t}\\) is the composition of \\(\phi^s\\) and \\(\phi^t\\). This is a semigroup action, which doesn't require the existence of negative values for t, and doesn't require the functions \\(\phi^t\\) to be invertible.For a complementary perspective, we can fix a point \(x \in M\), and let \(t\) vary, to get the trajectory function \(\phi_x(t) = \phi^t(x)\).
The orbit of \(x\) is the image of its trajectory function \(\phi_x\).
For a complementary perspective, we can fix a point \\(x \in M\\), and let \\(t\\) vary, to get the trajectory function \\(\phi_x(t) = \phi^t(x)\\). The _orbit_ of \\(x\\) is the image of its trajectory function \\(\phi_x\\).Here is the full construction of the example:
Clearly all paths converge to the origin, which is the unique attractor point.
In the discrete case, \(T\) is the natural numbers, and we have a semi-cascade.
In the continuous case, \(T\) is the non-negative real numbers, and we have a semi-flow.
Here is the full construction of the example: * Manifold \\(M\\) = unit disc * Trajectory \\(\phi^x(t) = x\ /\ t\\) Clearly all paths converge to the origin, which is the unique attractor point. In the discrete case, \\(T\\) is the natural numbers, and we have a semi-cascade. In the continuous case, \\(T\\) is the non-negative real numbers, and we have a semi-flow.I happened to choose a flat manifold \(M\), but curved manifolds are of course valid as well, and give more interesting scenarios. For example, the trajectories might take place on the surface of a torus.
I happened to choose a flat manifold \\(M\\), but curved manifolds are of course valid as well, and give more interesting scenarios. For example, the trajectories might take place on the surface of a torus.Thanks for the selection of topic @DavidTanzer. Does anyone have any questions?
Thanks for the selection of topic @DavidTanzer. Does anyone have any questions?I am meditating on how the idea of a system of ODEs generalizes to the case of a manifold.
I am meditating on how the idea of a system of ODEs generalizes to the case of a manifold.Although ODEs are one very typical and powerful means for defining dynamical systems, note that the basic definition makes no reference to differential equations - and hence is an independent concept from ODEs. The system is just defined by a manifold and a function \(\phi^t(x)\) - whether that function be defined as the solutions to ODEs, or some other way. For the discrete case, it can, for example, be directly defined by an iteration involving procedural logic.
Although ODEs are one very typical and powerful means for defining dynamical systems, note that the basic definition makes no reference to differential equations - and hence is an independent concept from ODEs. The system is just defined by a manifold and a function \\(\phi^t(x)\\) - whether that function be defined as the solutions to ODEs, or some other way. For the discrete case, it can, for example, be directly defined by an iteration involving procedural logic.But now let's look at how ODEs can be used to define flows. There are two links in the chain here:
For information about (2), see:
But now let's look at how ODEs _can_ be used to define flows. There are two links in the chain here: 1. The ODEs specify a field of tangent vectors on the manifold. 2. A tangent vector field determines a flow. For information about (2), see: * [Flow of a vector field](https://ncatlab.org/nlab/show/flow+of+a+vector+field), nLab.Let's look at the first link in the chain, which is straightforward, but still deserving its clear treatment.
Each point in the manifold is called a phase, or state. So the manifold is the state space.
In this whole discussion, we are assuming a deterministic system. That means that the "direction of motion" of the system is uniquely determined by the state. It's not random, and it doesn't depend upon time - just the state.
In a manifold, the notion of direction of motion at a point is expressed by a tangent vector. The space of all tangent vectors at a point is the tangent space at that point. It is a vector space.
Here we have a manifold \(M\) consisting of states. For a given state \(s \in M\), the tangent space \(\mathit{Tangent}(s)\) consists of all the vectors for the possible directions of motion out of that state.
That the system is deterministic means that for each \(s\), there is one uniquely determined vector in \(M(s)\) that gives its direction of motion. Let \(\Gamma(s) \in \mathit{Tangent}(s)\) be this uniquely determined vector.
\(\Gamma(s)\) is a field of tangent vectors on the manifold.
This tangent vector field is a function that completely specifies the law of motion for the system.
Let's look at the first link in the chain, which is straightforward, but still deserving its clear treatment. Each point in the manifold is called a phase, or state. So the manifold _is_ the state space. In this whole discussion, we are assuming a deterministic system. That means that the "direction of motion" of the system is uniquely determined by the state. It's not random, and it doesn't depend upon time - just the state. In a manifold, the notion of direction of motion at a point is expressed by a tangent vector. The space of all tangent vectors at a point is the tangent space at that point. It is a vector space. Here we have a manifold \\(M\\) consisting of states. For a given state \\(s \in M\\), the tangent space \\(\mathit{Tangent}(s)\\) consists of all the vectors for the possible directions of motion out of that state. That the system is deterministic means that for each \\(s\\), there is one uniquely determined vector in \\(M(s)\\) that gives its direction of motion. Let \\(\Gamma(s) \in \mathit{Tangent}(s)\\) be this uniquely determined vector. \\(\Gamma(s)\\) is a field of tangent vectors on the manifold. This tangent vector field is a function that completely specifies the law of motion for the system.Next, let's look at how first-order ordinary differential equations can be used to specify a tangent field on a manifold.
There's a slight wrinkle here, because the idea of the tangent space and the tangent vectors that comprise it is actually independent of any particular coordinate representation. Whatever charts we use to give coordinates to the points on, say a torus, we can still picture the tangent plane at a point (tangent space), and directions within it, apart from the specific coordinate representation.
Yet the ODEs are expressed in terms of coordinates.
Next, let's look at how first-order ordinary differential equations can be used to specify a tangent field on a manifold. There's a slight wrinkle here, because the idea of the tangent space and the tangent vectors that comprise it is actually _independent_ of any particular coordinate representation. Whatever charts we use to give coordinates to the points on, say a torus, we can still picture the tangent plane at a point (tangent space), and directions within it, apart from the specific coordinate representation. Yet the ODEs are expressed in terms of coordinates.The resolution to this will be easy to see, but first let's take a closer look at the general form of the ODEs, and the coordinate-based context in which they occur.
The resolution to this will be easy to see, but first let's take a closer look at the general form of the ODEs, and the coordinate-based context in which they occur.By the construction of a manifold, for each point \(s \in M\), there exists a "chart" which is a one-to-one mapping between a neighborhood around \(s\) in \(M\) and an open subset of \(\mathbb{R}^n\), where \(n\) is the dimension of the manifold.
Footnote: it's more than just one-to-one. A manifold \(M\) is not just a set, but a topological space, and the charts are topological isomorphisms (aka homeomorphisms) - continuous functions with continuous inverses.
The chart establishes a strong and "smooth" correspondence between the neighborhood in \(M\) and the neighborhood in the parameter space \(\mathbb{R}^n\).
By the construction of a manifold, for each point \\(s \in M\\), there exists a "chart" which is a one-to-one mapping between a neighborhood around \\(s\\) in \\(M\\) and an open subset of \\(\mathbb{R}^n\\), where \\(n\\) is the dimension of the manifold. Footnote: it's more than just one-to-one. A manifold \\(M\\) is not just a set, but a topological space, and the charts are topological isomorphisms (aka homeomorphisms) - continuous functions with continuous inverses. The chart establishes a strong and "smooth" correspondence between the neighborhood in \\(M\\) and the neighborhood in the parameter space \\(\mathbb{R}^n\\).Now there's even more structure that can be extracted from the charts.
Choose a point \(s \in M\), and let \(C(s) \in \mathbb{R^n}\) be its coordinate representation.
In addition to the tangent space \(\mathit{Tangent(s)}\) at \(s\) in the manifold itself, there is also a tangent space \(\mathit{Tangent(C(s))}\) around \(C(s) \in \mathbb{R^n}\).
That's because \(\mathbb{R^n}\) is itself a simple kind of manifold, at so at every point in \(\mathbb{R^n}\) there is a vector space of tangent vectors at that point.
Now that tangent space \(\mathit{Tangent}(C(s))\) turns out to be just a copy of \(\mathit{R^n}\), but in principle we still visualize as a local copy of \(\mathbb{R^n}\) at the point \(C(s))\).
Now there's even more structure that can be extracted from the charts. Choose a point \\(s \in M\\), and let \\(C(s) \in \mathbb{R^n}\\) be its coordinate representation. In addition to the tangent space \\(\mathit{Tangent(s)}\\) at \\(s\\) in the manifold itself, there is also a tangent space \\(\mathit{Tangent(C(s))}\\) around \\(C(s) \in \mathbb{R^n}\\). That's because \\(\mathbb{R^n}\\) is itself a simple kind of manifold, at so at every point in \\(\mathbb{R^n}\\) there is a vector space of tangent vectors at that point. Now that tangent space \\(\mathit{Tangent}(C(s))\\) turns out to be just a copy of \\(\mathit{R^n}\\), but in principle we still visualize as a local copy of \\(\mathbb{R^n}\\) at the point \\(C(s))\\).The extra structure that we can extract from the chart is a correspondence between the tangent space in the manifold and the associated tangent space in the parameter space \(\mathbb{R^n}\).
It is a vector-space isomorphism between \(\mathit{Tangent}(s)\) and \(\mathit{Tangent}(C(s)) = \mathbb{R^n}\).
The extra structure that we can extract from the chart is a correspondence between the tangent space in the manifold and the associated tangent space in the parameter space \\(\mathbb{R^n}\\). It is a vector-space isomorphism between \\(\mathit{Tangent}(s)\\) and \\(\mathit{Tangent}(C(s)) = \mathbb{R^n}\\).Through this isomorphism, it will suffice to define "objective" tangent vectors on the manifold solely by means of tangent vectors in the coordinate space.
Through this isomorphism, it will suffice to define "objective" tangent vectors on the manifold solely by means of tangent vectors in the coordinate space.In other words, the charts and these induced isomorphisms allow us to use the language of coordinates \(\mathbb{R^n}\) to refer to the coordinate-independent contents of the manifold.
In other words, the charts and these induced isomorphisms allow us to use the language of coordinates \\(\mathbb{R^n}\\) to _refer_ to the coordinate-independent contents of the manifold.Now it is clear to see how ODEs can define a tangent vector field on a manifold.
For this, we use first-order, explicit ODEs in \(\mathbb{R^n}\), which give the tangent vector in \(\mathbb{R^n}\) as a function of the coordinate point \(C(s) \in \mathbb{R^n}\).
Then just send those tangent vectors in \(\mathbb{R^n}\) through the aforementioned vector-space isomorphisms, i.e. through the charts, to get the tangent vectors for the manifold proper.
Now it is clear to see how ODEs can define a tangent vector field on a manifold. For this, we use first-order, explicit ODEs in \\(\mathbb{R^n}\\), which give the tangent vector in \\(\mathbb{R^n}\\) as a function of the coordinate point \\(C(s) \in \mathbb{R^n}\\). Then just send those tangent vectors in \\(\mathbb{R^n}\\) through the aforementioned vector-space isomorphisms, i.e. through the charts, to get the tangent vectors for the manifold proper.Recap:
Through the charts, a system of first-order, explicit ODEs determines a field of tangent vectors on the manifold.
Any tangent vector field defines a complete deterministic law of motion for the dynamical system.
Recap: * Through the charts, a system of first-order, explicit ODEs determines a field of tangent vectors on the manifold. * Any tangent vector field defines a complete deterministic law of motion for the dynamical system.Next up is to look at the second link in the chain, which is getting from the tangent vector field to the flow.
That's addressed in the article I cited above:
The connection can be put in a straightforward way, as per the following.
Next up is to look at the second link in the chain, which is getting from the tangent vector field to the flow. That's addressed in the article I cited above: * [Flow of a vector field](https://ncatlab.org/nlab/show/flow+of+a+vector+field), nLab. The connection can be put in a straightforward way, as per the following.Suppose we have a vector field \(\alpha\) on manifold \(M\), which is a function that assigns to each \(s \in M\) a tangent vector in \(\mathit{Tangent}(s)\).
Let \(\beta: I \subseteq \mathbb{R} \rightarrow M\) be a differentiable function that maps an open interval \(I\) of the reals into \(M\). So, \(\beta\) is a path in the manifold, with a well-defined tangent vector at every point along the curve. For \(t \in I\), we have \(\beta(t) \in M\), and \(\beta'(t) \in \mathit{Tangent}(\beta(t))\).
Next, \(\beta(t)\) is called an integral curve, or flow line, of the vector field \(\alpha\) if the tangent vector at every point in the curve equals the value of the vector field at that point on the manifold. That is, if \(\beta'(t) = \alpha(\beta(t))\), for all \(t \in I\).
Suppose we have a vector field \\(\alpha\\) on manifold \\(M\\), which is a function that assigns to each \\(s \in M\\) a tangent vector in \\(\mathit{Tangent}(s)\\). Let \\(\beta: I \subseteq \mathbb{R} \rightarrow M\\) be a differentiable function that maps an open interval \\(I\\) of the reals into \\(M\\). So, \\(\beta\\) is a path in the manifold, with a well-defined tangent vector at every point along the curve. For \\(t \in I\\), we have \\(\beta(t) \in M\\), and \\(\beta'(t) \in \mathit{Tangent}(\beta(t))\\). Next, \\(\beta(t)\\) is called an _integral curve_, or _flow line_, of the vector field \\(\alpha\\) if the tangent vector at every point in the curve equals the value of the vector field at that point on the manifold. That is, if \\(\beta'(t) = \alpha(\beta(t))\\), for all \\(t \in I\\).A flow \(\phi^t\) on \(M\) is called a global flow for the tangent vector field \(\alpha\) if for every point \(s \in M\), the trajectory function \(\phi_s(t)\) is a flow line of \(\alpha\).
In other words, the condition is that for all \(s \in M\) and \(t \in T\), we have that \(\phi_s'(t) = \alpha(\phi^t(s))\).
A flow \\(\phi^t\\) on \\(M\\) is called a _global flow_ for the tangent vector field \\(\alpha\\) if for every point \\(s \in M\\), the trajectory function \\(\phi_s(t)\\) is a flow line of \\(\alpha\\). In other words, the condition is that for all \\(s \in M\\) and \\(t \\in T\\), we have that \\(\phi_s'(t) = \alpha(\phi^t(s))\\).Exercise. Prove or refute the following:
\[(\forall t \in T)(\phi_s'(t) = \alpha(\phi^t(s)) \iff \phi_s'(0) = \alpha(\phi^0(s)) \iff \phi_s'(0) = \alpha(s)\]
This says that to show that \(\phi^t\) is a global flow for \(\alpha\), it suffices to show that \(\alpha\) equals the time-derivative of \(\phi^t\) just at time t=0.
Hint: consider using the fact that \(\phi^t\) is an action of \(T\) on \(M\).
Exercise. Prove or refute the following: \\[(\forall t \in T)(\phi_s'(t) = \alpha(\phi^t(s)) \iff \phi_s'(0) = \alpha(\phi^0(s)) \iff \phi_s'(0) = \alpha(s)\\] This says that to show that \\(\phi^t\\) is a global flow for \\(\alpha\\), it suffices to show that \\(\alpha\\) equals the time-derivative of \\(\phi^t\\) just at time t=0. Hint: consider using the fact that \\(\phi^t\\) is an _action_ of \\(T\\) on \\(M\\).Ok, now let's head back towards the concrete.
Here's a recap of where we've gotten to.
But given the vector field defined by our ODEs, how do we know that there exists a some well-defined global flow for it?
Once we are assured of its existence, we can take that flow to be the dynamical system for the ODEs.
Ok, now let's head back towards the concrete. Here's a recap of where we've gotten to. * A flow defines a dynamical system * ODEs define a tangent vector field * We've just given a _definition_ of the concept of a global flow for a tangent vector field But given the vector field defined by our ODEs, how do we know that there exists a some well-defined global flow for it? Once we are assured of its existence, we can take that flow to be the dynamical system for the ODEs.A theorem to the rescue!
In Proposition 3.2, nLab paper article cited above states the "fundamental theorem of flows": for a smooth vector field \(v\) on a smooth manifold, there is a unique maximal flow for \(v\).
This unique flow is often denoted exp(v) (c.f. the exponential map).
A theorem to the rescue! In Proposition 3.2, nLab paper article cited above states the "fundamental theorem of flows": for a smooth vector field \\(v\\) on a smooth manifold, there is a unique maximal flow for \\(v\\). This unique flow is often denoted exp(v) (c.f. the exponential map).This completes a tour of some of the foundational concepts in differential geometry that are premises to the notion of a dynamical system.
Since the root concept of a dynamical system is a manifold, that positions the subject as an area of applied differential geometry.
Here is an online journal which is named after this connection:
This completes a tour of some of the foundational concepts in differential geometry that are premises to the notion of a dynamical system. Since the root concept of a dynamical system is a manifold, that positions the subject as an area of applied differential geometry. Here is an online journal which is named after this connection: * [Differential Geometry - Dynamical Systems (DGDS)](http://www.mathem.pub.ro/dgds/)Now let's go forward.
Now let's go forward.Note that a deterministic Petri net is a kind of dynamical system - our topics are connected!
In this case, the state space is a subset of \(\mathbb{R^n}\), with all components non-negative. The ODEs are of course the rate equations, which are predicated on the mass-action kinetics, under which the rate at which a reaction takes place is proportional to the product of its input values (input concentrations).
Note that a deterministic Petri net is a kind of dynamical system - our topics are connected! In this case, the state space is a subset of \\(\mathbb{R^n}\\), with all components non-negative. The ODEs are of course the rate equations, which are predicated on the mass-action kinetics, under which the rate at which a reaction takes place is proportional to the product of its input values (input concentrations).Is a stochastic Petri net a kind of stochastic dynamical system?
It sounds like it should be, but I'm unsure at the moment.
Is a stochastic Petri net a kind of stochastic dynamical system? It sounds like it *should* be, but I'm unsure at the moment.First question to look at here: what is stochastic dynamical system?
First question to look at here: what is stochastic dynamical system? * [Random dynamical system](https://en.wikipedia.org/wiki/Random_dynamical_system), Wikipedia. > In the mathematical field of dynamical systems, a random dynamical system is a dynamical system in which the equations of motion have an element of randomness to them. Random dynamical systems are characterized by a state space S, a set of maps \\(\Gamma\\) from S into itself that can be thought of as the set of all possible equations of motion, and a probability distribution Q on the set \\(\Gamma\\) that represents the random choice of map. Motion in a random dynamical system can be informally thought of as a state \\(X\in S\\) evolving according to a succession of maps randomly chosen according to the distribution Q. > > An example of a random dynamical system is a stochastic differential equation; in this case the distribution Q is typically determined by noise terms. It consists of a base flow, the "noise", and a cocycle dynamical system on the "physical" phase space. Another example is discrete state random dynamical system; some elementary contradistinctions between Markov chain and random dynamical system descriptions of a stochastic dynamics are discussed.By the way, a word of clarification about discrete dynamical systems.
The discreteness of a D.S. does not stipulate that manifold M is discrete, rather just that the time domain T is discrete.
\(\phi^t\) still maps a differentiable manifold into itself. The difference here is that we just have the discrete sequence of functions \(\phi^0, \phi^1, \phi^2\), and due to the requirement that we have an action of T on the manifold, the nth function here, \(\phi^n\), turns out to be just the composition of \(\phi^1\) with itself n times, i.e., \((\phi^1)^n\).
In other words, the discrete dynamical system is characterized simply by an iterator \(f = \phi^1: M \rightarrow M\) that maps each state to its successor state.
The trajectory functions are sequences of states \([s, f(s), f^2(s), ...]\).
And the orbits are the corresponding sets of states \(\lbrace s, f(s), f^2(s), ... \rbrace\).
By the way, a word of clarification about discrete dynamical systems. The discreteness of a D.S. does not stipulate that manifold M is discrete, rather just that the time domain T is discrete. \\(\phi^t\\) still maps a differentiable manifold into itself. The difference here is that we just have the discrete sequence of functions \\(\phi^0, \\phi^1, \\phi^2\\), and due to the requirement that we have an _action_ of T on the manifold, the nth function here, \\(\phi^n\\), turns out to be just the composition of \\(\phi^1\\) with itself n times, i.e., \\((\phi^1)^n\\). In other words, the discrete dynamical system is characterized simply by an iterator \\(f = \phi^1: M \rightarrow M\\) that maps each state to its successor state. The trajectory functions are sequences of states \\([s, f(s), f^2(s), ...]\\). And the orbits are the corresponding sets of states \\(\lbrace s, f(s), f^2(s), ... \rbrace\\).Typically these states will reside in the manifold \(\mathbb{R^n}\), but they could be in any manifold.
Then we could see, for example, trajectory functions which are sequences of points on a torus.
Typically these states will reside in the manifold \\(\mathbb{R^n}\\), but they could be in any manifold. Then we could see, for example, trajectory functions which are sequences of points on a torus.@DavidTanzer said:
1 roll the ODEs up into a matrix function
2 determine a fixed point/equilibrium
3 determine symmetry
This would be my current approach.
@DavidTanzer said: > I am meditating on how the idea of a system of ODEs generalizes to the case of a manifold. 1 roll the ODEs up into a matrix function 2 determine a fixed point/equilibrium 3 determine symmetry This would be my current approach.Creed and Quest by Nicholas B. Tufillaro
I read somewhere that every few years Smale, Abrahms and a few other would get together to discuss the unity of discrete and continuous dynamical systems.
**Creed and Quest** by Nicholas B. Tufillaro > The close connection between maps and flows - Poincare maps and suspensions - gives rise to **The Discrete Creed: > Anything that happens in a flow also happens in a (lower-dimensional) discrete dynamical system (and conversely). > This creed is implicit in Poinare's original work, but it was first enunciated by Smale. >> Where do orbits go, and what do the do/see when they get there? I read somewhere that every few years Smale, Abrahms and a few other would get together to discuss the unity of discrete and continuous dynamical systems.