MAE5790-2 One dimensional Systems

The lecture delves into the foundational concepts of dynamical systems, focusing on one-dimensional systems and their behavior. The instructor discusses the graphical approach to understanding these systems, emphasizing fixed points and their stability. The lecture explores linearization around fixed points and the implications of Taylor series expansions on stability analysis. Key bifurcations, including saddle node, transcritical, and pitchfork bifurcations, are introduced, highlighting their roles in creating or altering fixed points and system stability. The discussion includes real-world applications and the limitations of one-dimensional systems in exhibiting complex behaviors like oscillations or chaos.

• Overview of the course on dynamical systems, focusing on the history and logical structure.
• Introduction to one-dimensional dynamical systems where X is a real number.
• Analytical solutions are often impossible for nonlinear functions; graphical methods are preferred.

"We talked in the first lecture about the history of dynamical systems and the logical structure of the subject, and I tried to give you an overview of what the whole course will be about."

• The course will explore the history and logical framework of dynamical systems.

"You could try to solve them analytically, but often it will be impossible to do the integrals that will arise if f is a nonlinear function."

• Analytical solutions are challenging for nonlinear functions; graphical methods are more feasible.

• Use of graphs to understand dynamical systems by imagining a particle moving on the X-axis.
• The velocity of the particle is given by the function f(x); direction depends on the sign of f(x).
• Fixed points occur where the function crosses the axis, representing equilibrium states.

"The approach was to just draw the graph of this function f... understanding the dynamical system is pretty easy because all you do is think about an imaginary particle that's moving on the X-axis as a function of time."

• Graphical representation simplifies understanding of dynamical systems via imaginary particles.

"There are these special places where the function crosses the axis... which we called a fixed point because if you happen to be there, then you won't move, and that would be in physical terms an equilibrium state for the system."

• Fixed points are where the function crosses the axis, indicating equilibrium states.

• Analysis of the long-term behavior of systems based on fixed points.
• Systems tend to infinity or negative infinity depending on their position relative to fixed points.
• Approach to fixed points is asymptotic; actual reaching is impossible with continuous functions.

"Everything that starts to the right of this point will go out to infinity, and everything on the left will go out to negative infinity."

• The system's long-term behavior is determined by its position relative to fixed points.

"For the functions that we're going to consider in this course... it will not be possible to reach a fixed point."

• Continuous functions do not allow reaching fixed points in finite time; approach is asymptotic.

• Linearization involves examining dynamics close to a fixed point by introducing a small deviation.
• Stability of a fixed point is determined by whether deviations grow or decay.
• Use of Taylor series to approximate the function near a fixed point.

"We're going to examine the dynamics close to a fix point... to see whether a little deviation will grow as it does in this case, in which case we would say that the fix point is unstable."

• Linearization helps determine the stability of fixed points by analyzing small deviations.

"Let's use Taylor's formula here... in the neighborhood of x star."

• Taylor series is used to linearize functions around fixed points.

• Stability determined by the sign of the derivative at the fixed point.
• Positive derivative indicates instability (exponential growth); negative indicates stability (exponential decay).
• If the derivative is zero, linearization provides no information about stability.

"We get growth if we have R greater than zero and then we have decay if R is less than zero."

• Stability is determined by the sign of the derivative: positive for instability, negative for stability.

"If F Prime is zero at a fixed point, that tells me nothing about the stability of the point."

• A zero derivative at a fixed point gives no information about stability; further analysis is needed.

• Different systems with zero derivative at fixed points can have varied stability types.
• Examples include systems where x dot is x squared or x cubed, each with distinct stability characteristics.
• Graphical analysis reveals stability types: stable, unstable, or half-stable.

"Consider these four different systems x dot is x, x dot is x squared, x dot is X cubed or x dot is negative X cubed."

• Systems with zero derivative at fixed points can exhibit various stability types.

"We would approach it asymptotically from the left... this is a half-stable fix point."

• Half-stable fixed points are stable from one side but unstable from the other.

• The stability of fixed points can be analyzed through graphical representation or higher-order Taylor series expansions.
• Stable fixed points exhibit slow algebraic relaxation, while unstable points do not.
• Graphical analysis is preferred over analytical methods like Taylor series for understanding stability.

"If x dot is negative x cubed, well then the graph looks like so and it's a stable fix point."

• This quote illustrates that the stability of a fixed point can be determined by analyzing the sign and behavior of the function's derivative.

"The bottom line is if you want to figure out stability information at a point where fpre is zero, just draw the graph, and that'll tell you what's going on."

• The speaker emphasizes the utility of graphical methods for assessing stability at fixed points.

• The logistic equation for population growth highlights the stability of fixed points at zero and carrying capacity, K.
• Linearization techniques confirm graphical stability findings.
• Positive growth rates (R) lead to instability at zero and stability at K.

"Xar equals 0 is unstable whereas the other fix point if we look at k then we get r - 2 r k over K and the K's cancel out and this is R so this is less than zero and so X star equal K is stable."

• This quote demonstrates the application of linearization to determine the stability of fixed points in the logistic equation.

• Solutions to differential equations exist and are unique under conditions of continuity and continuous differentiability.
• The theorem does not guarantee solutions for all time but ensures existence in a small time interval.

"There's a theorem that you would prove in a rigorous differential equations course that says that the solutions to X x doal f ofx do exist and they are unique if F continuous and if fime of X continuous."

• The quote outlines the conditions under which solutions to differential equations are guaranteed to exist and be unique.

• One-dimensional systems have limited long-term behaviors: they either approach a fixed point or go to infinity.
• Oscillations and chaos are not possible in these systems due to the monotonic nature of trajectories.

"The only things that can happen are either as we saw in the earliest example that the um where had that one unstable point and trajectories either went out to plus or minus Infinity you can have that so either X of T goes out to Infinity in one or the other direction or um X of T approaches a fixed Point that's it."

• The quote explains the restricted behaviors of one-dimensional systems, emphasizing the absence of oscillations or chaotic dynamics.

• Bifurcations occur when a change in parameters leads to a qualitative change in the system's behavior.
• Examples include the creation or destruction of fixed points or changes in their stability.
• Bifurcations are relevant in various scientific fields, such as fluid dynamics and cardiac rhythms.

"As a parameter changes, the qualitative structure of the vector field may change dramatically for example, a fixed Point might be created or destroyed or they might change their stability."

• The quote captures the essence of bifurcations as significant changes in system behavior due to parameter variations.

"In fluid dynamics if you have a a flow that's laminer and you start turning up the Reynolds number that flow might start to become wavy in a bifurcation or then if you keep going high enough it may become turbulent."

• This example illustrates how bifurcations manifest in practical scenarios, such as transitions from laminar to turbulent flow.

• Bifurcations represent models of instabilities or sudden transitions in scientific systems.
• The discussion begins with basic mathematical examples and will later expand to scientific examples in mechanics and population biology.
• Bifurcations are crucial for understanding the creation or destruction of fixed points in a system.

"The most basic one is a bifurcation that creates fix points out of nothingness; it does it in pairs, almost like in physics when they talk about a particle and an anti-particle coming out of the vacuum."

• This quote highlights the fundamental nature of bifurcations in creating pairs of fixed points, akin to particle-antiparticle creation in physics.

• Saddle node bifurcation is a mechanism for creating or destroying fixed points.
• It involves the emergence of a stable and an unstable point from a system, analogous to saddle points in higher dimensions.
• The process can be visualized using a simple system where the control parameter ( R ) is adjusted, affecting the stability of fixed points.

"When ( R ) is negative...you've got negative ( x \dot ) here, positive here, and positive over there, telling us that this is a stable fix point and this one is unstable."

• This describes how the stability of fixed points changes with the control parameter ( R ), illustrating the concept of saddle node bifurcation.

• A bifurcation diagram is used to represent the behavior of a system as a function of a control parameter.
• It plots the fixed points against the control parameter, showing stable branches with solid lines and unstable branches with dashed lines.
• The diagram helps visualize the qualitative changes in the system's behavior as the parameter changes.

"A bifurcation diagram would be something like...plot the curves of ( x^* ) versus the control parameter ( R )."

• This quote explains the purpose and construction of a bifurcation diagram, which encodes the system's behavior in one picture.

• In practical scenarios, bifurcations may not appear as cleanly as in theoretical examples.
• An example differential equation is provided to demonstrate how bifurcations occur in more complex systems.
• The graphical method is used to analyze the behavior of the system as a function of the control parameter ( R ).

"Suppose I give you this differential equation ( x \dot = R + x - \log(1 + x) )...we want to analyze that as a function of ( R )."

• This introduces a more complex example, emphasizing the importance of graphical methods in analyzing bifurcations in real-world systems.

• The graphical method involves plotting the relevant functions and observing the intersections to determine fixed points.
• This method bypasses the need for algebraic solutions, which may be complex or impossible to obtain.
• The approach is crucial for understanding systems with nonlinear terms, like logarithmic functions.

"Let’s plot ( y = R + x ) and plot ( y = \log(1 + x) )...if they don’t intersect, it means that this is not equal to that for any ( x ), so we don’t have any fixed points."

• This illustrates the use of graphical methods to determine the existence and nature of fixed points in a system.

• A saddle node bifurcation occurs when the line and curve intersect tangentially.
• This requires both an intersection and matching slopes at the point of intersection.
• The condition is critical for identifying bifurcations in complex systems.

"Saddle node bifurcation occurs when...the line intersects the curve tangentially."

• This succinctly defines the geometric condition necessary for a saddle node bifurcation to occur.

• Bifurcation theory explores how the structure of solutions to equations changes as parameters vary, focusing on critical points where systems undergo qualitative changes.
• The tangency condition involves intersections with the same slope, leading to bifurcation points where behavior changes.
• The bifurcation occurs at parameter values where the derivative conditions are satisfied, indicating a critical or bifurcation value.

"R is equal to log of 1 which is itself zero so um let me call it r subc for r critical like happening you know that's the critical value of R um critical or bifurcation value."

• The critical value of R is where bifurcation occurs, marking a significant change in system behavior.

• Near bifurcation points, vector fields can be expanded using Taylor series to approximate behavior.
• The McLaurin expansion of log(1 + x) helps simplify expressions, focusing on leading terms in R and X.
• The simplification leads to a familiar form, R + x², resembling the standard saddle node bifurcation example.

"If we just think about very small X then this is behaving like r + x minus and then using this expansion x - x^2 / 2+ X Cub over 3 Etc which simplifies to notice the X's cancel out and I get R um + x^2 /2."

• The expansion highlights the dominant behavior near bifurcation, simplifying to R + x², a typical form in saddle node bifurcations.

• Normal forms represent simplified versions of vector fields near bifurcation points, capturing essential dynamics.
• Saddle node bifurcations involve the creation or annihilation of fixed points, typically represented by R + x².
• The dynamics near a saddle node often resemble a constant times R plus a constant times x².

"Near a Addle node the Dynamics always looks kind of like a constant time R plus a constant time x^2 you know plus or minus a constant time x^2."

• The generic behavior near a saddle node bifurcation is characterized by a simple, familiar form, aiding in understanding complex dynamics.

• Transcritical bifurcations involve fixed points that cannot be destroyed but can change stability.
• The normal form for transcritical bifurcations is RX - x², indicating a qualitative difference from saddle node bifurcations.
• Transcritical bifurcations often occur in systems where fixed points persist across parameter changes.

"This bifurcation comes up where as I said the first one the saddle node was the mechanism for creating and destroying pairs of fixed points this is um this is what tends to come up if you have a an indestructible fix Point."

• Unlike saddle node bifurcations, transcritical bifurcations involve persistent fixed points that change stability rather than being created or destroyed.

• Stability analysis involves examining the derivative of the system function to determine the nature of fixed points.
• Bifurcation diagrams visually represent the stability and existence of fixed points across parameter values.
• Transcritical bifurcations can lead to exchanges of stability between fixed points.

"Let's draw a picture let's just go straight to the bifurcation diagram here's R here's X and we can see the fix points pretty easily we know that they're zero so let's just put in something here along xal 0 and then xal R is the other so that's this diagonal line."

• Bifurcation diagrams illustrate the relationship between parameters and fixed points, highlighting changes in stability and existence.

• Pitchfork bifurcations occur in systems with symmetry, often involving bifurcating solutions that are stable.
• The normal form for a pitchfork bifurcation is RX - x³, reflecting symmetry and leading to multiple fixed points.
• The bifurcation diagram resembles a pitchfork, with stable and unstable branches.

"This kind of thing is um in the jargon called a super critical pitchfork super critical is a word that you see a lot in dynamical systems it means that the bifurcating solutions are stable."

• Pitchfork bifurcations involve symmetrical bifurcations with stable solutions, often occurring in systems with inherent symmetry.

• The discussion covers various bifurcation types, emphasizing their normal forms and qualitative behaviors.
• Future discussions will explore scientific examples of bifurcations, applying theoretical concepts to real-world systems.

"Next time we'll talk about some scientific examples of of bifurcations okay thanks."

• Theoretical insights will be applied to practical examples, illustrating the relevance and application of bifurcation theory in scientific contexts.