MAE5790-8 Index theory and introduction to limit cycles

The lecture focuses on index theory, contrasting it with local methods like linearization, which examine infinitesimal neighborhoods around fixed points. Index theory provides global information about phase portraits, analyzing closed curves and their interaction with vector fields. The lecturer explains the concept of an index, using examples to illustrate how the index of a closed curve relates to the direction and number of rotations of vectors around it. Key properties of the index include additivity, continuity, and its relation to fixed points. The lecture concludes by introducing limit cycles, which represent stable periodic behavior in dynamical systems, distinct from linear systems where periodic solutions are not isolated.

• Index Theory provides a global perspective on phase portraits in dynamical systems, unlike local methods such as linearization.
• It examines the entire phase portrait, offering information about the system's global behavior.
• The index of a closed curve is a central concept, related to topics in topology, complex analysis, and physics.

"Index Theory in contrast does not just confine itself to the immediate neighborhood but looks at the whole Global phase portrait and gives information at long distance."

• Index Theory offers insights into the global structure of dynamical systems, unlike local methods focused on fixed points.

• A simple closed curve (C) is defined as one that does not intersect itself and does not pass through a fixed point.
• The curve is used to analyze the vector field, similar to a Gaussian surface in physics.
• The index measures the net number of rotations of a vector field as one traverses the closed curve.

"Simple means it doesn't intersect itself... and one other thing is, it should not pass through a fixed point."

• Defines the characteristics of a simple closed curve, essential for calculating the index.

• The angle (Fe) is defined as the arctangent of the vector components (y-dot over x-dot).
• The index is calculated as the net change in Fe divided by 2π, representing the number of full revolutions.

"So as our point x goes around C once counterclockwise... the net change in Fe when we go around C... is called the index."

• Describes the process of determining the index by tracking the angle change over a closed curve.

• Example 1: A vector field with a simple closed curve results in an index of +1 if the vectors rotate counterclockwise.
• Example 2: A different vector pattern results in an index of -1 if the vectors rotate clockwise.
• Example 3: A flow with no significant rotation results in an index of 0.

"When we went around the curve counterclockwise our Arrow also went counterclockwise one whole Revolution so we would say this has an index of plus one."

• Illustrates a positive index with counterclockwise rotation matching the curve traversal.

"As I went counterclockwise around the curve the arrows are rotating once in the opposite sense clockwise so we would say that this has an index of plus sorry negative 1."

• Demonstrates a negative index with opposite directional rotation.

• Closed Trajectory: Any closed trajectory has an index of +1, indicating periodic behavior.
• Additivity: The index is additive when a closed curve is subdivided into smaller curves.

"Any closed trajectory has an index of plus one."

• Highlights the property of closed trajectories always having an index of +1.

"It's additive if we subdivide the curve C... you can kind of think of C as being the union of C1 and C2."

• Explains the additive nature of the index when a curve is divided into parts.

These notes provide a detailed overview of the key concepts and examples discussed in the transcript, offering a comprehensive understanding of Index Theory and its application in analyzing dynamical systems.

• The index around a composite curve C is the sum of the indices of its sub-curves C1 and C2.
• The change in vector direction when traversing a bridge between sub-curves cancels out, leading to the additive property.

"The index around C is just the index around C1 plus the index around C2."

• This quote explains that the index of a composite curve is the sum of its parts due to the cancellation of directional changes over the bridge.

• The index of a curve remains constant under continuous deformation, provided no fixed points are crossed.
• This property is analogous to Gauss's law in electrostatics, where the surface flux remains constant unless charge is enclosed or lost.

"The index of the original curve C is the same as the index of the deformed curve C Prime."

• This highlights that continuous deformation without encountering fixed points preserves the index.

• The index is an integer due to the continuous nature of the vector field, resulting in a constant index during deformation.
• An integer-valued continuous function remains constant unless discontinuities occur, such as crossing fixed points.

"If you have a continuous function that's integer valued, it has to be a constant."

• The quote emphasizes that the continuity of an integer-valued function ensures its constancy during deformation.

• A curve not enclosing any fixed points has an index of zero.
• Shrinking a curve that does not enclose fixed points results in all vectors pointing in the same direction, indicating zero index.

"If C does not enclose a fix point, then the index of C will have to be zero."

• This explains that curves without fixed points inside have an index of zero due to uniform vector direction.

• The index does not indicate stability but is related to the presence of fixed points.
• Reversing time alters vector direction but does not change the index, as the net change in direction remains constant.

"If you reverse the sense of time, you don't change the index."

• This highlights that the index is invariant under time reversal, separating it from stability considerations.

• The index of a point is defined using a closed curve enclosing just that point and no other fixed points.
• The index remains consistent for any enclosing curve due to the absence of intermediate fixed points.

"Index of a point P equals I subc for any C that encloses P and no other fix points."

• This quote establishes the method for determining the index of a point based on enclosing curves.

• The concept of index in dynamical systems is used to classify fixed points.
• Fixed points like nodes and centers have an index of +1, while saddles have an index of -1.
• Non-isolated fixed points cannot have an index defined due to the inability to enclose them with a closed curve.

"The index of a node will be plus one... you can check that the index of a saddle is minus one."

• Nodes and centers are stable with a positive index, while saddles, being unstable, have a negative index.

"All the stuff with index +1 lives over here on the whole right half plane... whereas the Saddles live over here on the left half plane they all have index equals minus one."

• The plane is divided into regions where the index can be determined, except on the axis where non-isolated fixed points exist.

• Complex vector fields can produce fixed points with indices greater than +1 or less than -1.
• Using complex notation, specific vector fields can be constructed to have indices of n or -n.

"Suppose I had Z doal z^2... the index comes out to either be plus two or minus two."

• The linearization of such systems may not reveal the true nature of the fixed points, which requires further analysis of the vector field.

• Closed trajectories in the plane must enclose fixed points whose indices sum to +1.
• This constraint helps in understanding the behavior of dynamical systems and ruling out certain types of trajectories.

"Any closed trajectory... must enclose fixed points whose indices add up to plus one."

• The theorem uses the property that a closed trajectory itself has an index of +1, allowing subdivision of curves to calculate indices.

"You could draw a picture like you know zon zon and then calculate the indices around these three closed curves."

• This theorem is useful for determining the impossibility of certain closed orbits in a system.

• Index theory can be used to analyze and rule out closed trajectories in specific models, such as the Rabbits vs. Sheep competition model.
• The model's phase portrait reveals stable nodes and saddles, with specific indices that prevent certain closed orbits from existing.

"You can use index Theory sometimes to rule out closed trajectories... where we had the vector field x dot = x * 3 - x - 2 Y and Y dot was Y 2 - x - Y."

• Closed orbits that would violate trajectory crossing or the sum of indices rule are not possible.

"You cannot have this and you also can't have this by the same argument... the sum of the indices inside would be zero."

• An interesting application of index-like arguments is found in biological systems, such as salamander limb regeneration.
• Experiments show that grafting a right limb onto a left stump results in regeneration, which can be conceptually understood with index arguments.

"The left stump has a kind of index of minus one and the right stump has a kind of index of plus one."

• This biological phenomenon highlights the broader applicability of index concepts beyond traditional dynamical systems.

• Discussion of a peculiar biological phenomenon where attaching a right hand to a left wrist results in the growth of two left hands, one to cancel the right hand and another to restore the left hand.
• Introduction to index theory in mathematics, explaining how it makes sense of strange experimental observations.
• Mention of the hairy ball theorem, which states that for a sphere with a continuous vector field, the sum of the indices of fixed points is always +2.

"When you put the right hand on the left, you're now off by two, and so this will grow two more left hands."

• This quote describes a peculiar biological phenomenon that is explained through index theory, highlighting the theory's ability to make sense of strange experimental results.

"The statement is that for any vector field on the sphere, the sum of the indices will always be plus two."

• This quote explains the hairy ball theorem, illustrating a fundamental concept in index theory related to vector fields on a sphere.

• Explanation of how the hairy ball theorem applies to spheres and the possibility of combing a torus without fixed points.
• Introduction to the concept of genus in topology, which refers to the number of handles on a surface.
• Explanation of the Poincaré-Hopf index theorem, which states that the sum of indices is 2 minus 2 times the genus (2 - 2G).

"If you have a torus with hair sticking up, you can just comb the hair all the way around."

• This quote illustrates the difference between a sphere and a torus in terms of vector fields, highlighting the unique properties of a torus.

"The sum of the indices is 2 minus a quantity called 2 * G, where G is called the genus of the surface."

• This quote explains the Poincaré-Hopf index theorem, which generalizes the concept of indices to surfaces with different topologies.

• Introduction to limit cycles, which are closed trajectories that neighboring trajectories approach asymptotically.
• Discussion of stable, unstable, and half-stable limit cycles and their significance in systems exhibiting periodic behavior.
• Examples of limit cycles in various fields, such as biology (heartbeats), engineering (aeroelastic flutter), and chemistry (oscillating reactions).

"These limit cycles represent periodic behavior of a system that is in a sense drawn to this from a wide range of initial conditions."

• This quote emphasizes the importance of limit cycles in understanding periodic behaviors in systems across different scientific disciplines.

"Linear systems don't have limit cycles; this is a really quintessentially nonlinear phenomenon."

• This quote distinguishes limit cycles as a feature of nonlinear systems, highlighting their absence in linear systems and their robustness to disturbances.

• Explanation of why linear systems cannot have limit cycles due to the lack of isolated periodic solutions.
• Description of how any periodic solution in a linear system leads to a continuous family of solutions due to linearity.
• Comparison of periodic solutions in linear systems to limit cycles, noting the latter's robustness to disturbances.

"If there's any little noise that kicks you from one to the next, you remember that disturbance forever; you just stay on this new periodic orbit."

• This quote explains the behavior of periodic solutions in linear systems, contrasting them with the more robust nature of limit cycles in nonlinear systems.