Syllabus
Unit I
Learning Objectives
After completing this unit, student will be able to
LO1– Identify fixed points in dynamical systems and classify their types and apply linear stability analysis to determine the stability of fixed points. Investigate the existence and properties of limit cycles.
Introduction, Phase Space, and Phase Portraits:
Linear systems and their classification; Existence and uniqueness of solutions; Fixed points and linearization; Stability of equilibria; Pendulum and Duffing oscillator, Lindstedt‘s method; Conservative and reversible systems.
Unit II
Learning Objectives
After completing this unit, student will be able to
LO1– Learn about the different oscillators and methods of averaging and multiple scales.
Limit Cycles:
The van der Pol oscillator, Method of Averaging; Relaxation oscillators; Weakly nonlinear oscillators; Forced Duffing oscillator, Method of Multiple Scales; Forced van der Pol oscillator, Entrainment; Mathieu‘s equation, Floquet Theory, Harmonic Balance.
Unit III
Learning Objectives
After completing this unit, student will be able to
LO1- Learn concepts of bifurcations and their importance in dynamical systems.
Bifurcations:
Saddle-node, trans critical, and pitchfork bifurcations; Center manifold theory; Hopf bifurcation; Global bifurcations; and Poincaré maps.
Unit IV
Learning Objectives
After completing this unit, student will be able to,
LO1- Learn the implications of chaotic dynamics in physical systems and real-world applications.
Chaotic Dynamics:
Lorentz equations; Lorentz map; Logistics map; Lyapunov Exponents; fractal sets and their dimensions; box, pointwise and correlation dimensions; strange attractors; and forced two-well oscillator.
Unit V
Learning Objectives
After completing this unit, student will be able to,
LO1- Learn to solve specific examples of nonlinear PDEs using bilinearization to demonstrate the method’s efficacy.
Integrable Systems and Solitons:
Linear and nonlinear dispersive systems – Cnoidal and solitary waves – The Scott Russel phenomenon and derivation of Korteweg-de Vries (KdV) equation – Explicit soliton
solutions: one-, two- and Nsoliton solutions of KdV equation – Hirota’s bilinear method
Objectives and Outcomes
Pre-requites
Knowledge of mathematical physics and classical mechanics.
Course Objectives
The objective of this course is to impart knowledge about nonlinear dynamics and learn methods to solve problems related to nonlinear dynamics.
Course Outcomes: After completion this course student able to
CO1: Identify fixed points in various dynamical systems and classify them based on their types and predict the behavior of the system.
CO2: Analyze the various types of bifurcations in one dimension (saddle node, transcritical, and pitchfork) and in two dimensions.
CO3: Gain an understanding of the properties of the most important strange attractors and chaotic dynamics.
CO4: Identify integrable systems, Hirota’s bilinearization method apply it to obtain one soliton solutions of a given nonlinear partial differential equation.
Skills: By solving problems in the form of assignments and quizzes related to nonlinear dynamics improves the analytical skills of students.
CO-PO Mapping
|
PO1
|
PO2
|
PO3
|
PO4
|
PO5
|
PO6
|
PO7
|
PO8
|
PO9
|
PO10
|
PO11
|
PO12
|
PSO1
|
PSO2
|
PSO3
|
PSO4
|
|
CO1
|
3
|
3
|
–
|
–
|
–
|
|
–
|
–
|
–
|
–
|
–
|
–
|
3
|
3
|
–
|
–
|
|
CO2
|
3
|
3
|
–
|
|
–
|
–
|
–
|
–
|
–
|
–
|
–
|
–
|
3
|
3
|
–
|
–
|
|
CO3
|
3
|
3
|
–
|
–
|
–
|
–
|
–
|
–
|
–
|
–
|
–
|
–
|
3
|
3
|
–
|
–
|
|
CO4
|
3
|
3
|
|
|
|
|
|
|
|
|
|
|
3
|
3
|
|
|
Evaluation Pattern
Evaluation Pattern
|
Assessment
|
Internal
|
External
Semester
|
|
Mid-term
|
30
|
|
|
*Continuous Assessment (CA)
|
20
|
|
|
End Semester
|
|
50
|
*CA – Can be Quizzes, Assignment, Projects, and Reports.
Justification for CO-PO mapping
|
Mapping
|
Justification
|
Affinity level
|
|
CO1-PO1
|
CO1 strongly aligns with PO1 at the maximum level because identifying fixed points and their classification requires a thorough understanding of basic scientific principles, such as calculus and differential equations, which are essential components of science knowledge (PO1). Students apply this fundamental knowledge to study dynamical systems.
|
3
|
|
CO1-PO2
|
CO1 maximally supports PO2, as identifying and classifying fixed points are fundamental analytical skills rooted in scientific principles, essential for tackling and solving complex dynamical system problems.
|
3
|
|
CO2-PO1
|
CO2 maximally supports PO1 by equipping students with the scientific and mathematical tools required to analyze and understand bifurcations, one of the fundamental phenomena in dynamical systems, demonstrating a direct and strong alignment with basic scientific knowledge.
|
3
|
|
CO2-PO2
|
Bifurcations represent a critical transition in system dynamics, often leading to complex behaviors like chaos or stability changes. Understanding these mechanisms necessitates a systematic approach to problem formulation and analysis, thereby enhancing students’ capability to tackle multifaceted problems, as outlined in PO2.
|
3
|
|
CO3-PO1
|
CO3 aligns with PO1 at the maximum level understanding chaotic dynamics and strange attractors necessitates a strong grasp of key scientific principles such as nonlinearity, bifurcation theory, and dynamical systems.
|
3
|
|
CO3-PO2
|
CO3 contributes maximally to PO2 by equipping students with the analytical and problem-solving skills necessary to analyze and solve complex mechanisms driven by chaotic dynamics and strange attractors using first principles.
|
3
|
|
CO4-PO1
|
CO4’s focus on integrable systems and Hirota’s bilinearization method directly reflects and maximizes the application of basic science knowledge outlined in PO1, showcasing the students’ ability to utilize fundamental concepts in sophisticated and impactful ways.
|
3
|
|
CO4-PO2
|
CO4’s focus on integrable systems and soliton solutions encapsulates a high level of problem analysis and the application of basic scientific principles, making it a strong connection to PO2 at the maximum level.
|
3
|
|
CO1-PSO1
|
the ability to identify fixed points and classify them based on their stability is not only a direct application of mathematical physics but also central to understanding and analyzing complex systems in quantum mechanics, electrodynamics, and statistical physics. This strong connection makes CO1 a maximum contributor to PS01.
|
3
|
|
CO1-PSO2
|
CO1 significantly supports PSO2 by requiring the application of experimental and computational techniques to fully understand and predict system behavior, thus achieving maximum alignment.
|
3
|
|
CO2-PSO1
|
CO2 is deeply rooted in and expands upon the knowledge specified in PSO1. The bifurcation analysis directly involves applying mathematical physics and related subjects to understand critical changes in physical systems, ensuring a strong and maximum alignment with PSO1.
|
3
|
|
CO2-PSO2
|
CO2 maximally contributes to PSO2 by requiring students to develop and apply experimental and numerical techniques to analyze bifurcations in physical and theoretical systems. The use of programming, plotting, and experimental skills is essential for comprehending and interpreting bifurcations in practice.
|
3
|
|
CO3-PSO1
|
The properties of strange attractors and chaotic dynamics are relevant in contemporary research and practical applications, such as weather prediction, engineering, and biological systems. By mastering these concepts, students become adept at using their knowledge to analyze a wide range of physical phenomena, fulfilling the goals outlined in PSO1.
|
3
|
|
CO3-PSO2
|
CO3’s emphasis on understanding strange attractors and chaotic dynamics directly supports the acquisition of experimental skills and numerical problem-solving abilities in PSO2, establishing a maximum connection between these outcomes.
|
3
|
|
CO4-PSO1
|
CO4 strongly supports PSO1 at the maximum level by integrating advanced concepts of mathematical physics with practical applications, ensuring that students not only acquire knowledge but also demonstrate their ability to analyze complex physical phenomena effectively.
|
3
|
|
CO4-PSO2
|
The ability to solve nonlinear PDEs has practical applications in many fields, such as fluid dynamics, nonlinear optics, and mathematical biology. Students’ skills in both experimental techniques and numerical analysis prepare them to tackle complex real-world problems, underscoring the importance of PSO2.
|
3
|
|
Mapping
|
Justification
|
Affinity level
|
|
CO1-PO1
|
CO1 strongly aligns with PO1 at the maximum level because identifying fixed points and their classification requires a thorough understanding of basic scientific principles, such as calculus and differential equations, which are essential components of science knowledge (PO1). Students apply this fundamental knowledge to study dynamical systems.
|
3
|
|
CO1-PO2
|
CO1 maximally supports PO2, as identifying and classifying fixed points are fundamental analytical skills rooted in scientific principles, essential for tackling and solving complex dynamical system problems.
|
3
|
|
CO2-PO1
|
CO2 maximally supports PO1 by equipping students with the scientific and mathematical tools required to analyze and understand bifurcations, one of the fundamental phenomena in dynamical systems, demonstrating a direct and strong alignment with basic scientific knowledge.
|
3
|
|
CO2-PO2
|
Bifurcations represent a critical transition in system dynamics, often leading to complex behaviors like chaos or stability changes. Understanding these mechanisms necessitates a systematic approach to problem formulation and analysis, thereby enhancing students’ capability to tackle multifaceted problems, as outlined in PO2.
|
3
|
|
CO3-PO1
|
CO3 aligns with PO1 at the maximum level understanding chaotic dynamics and strange attractors necessitates a strong grasp of key scientific principles such as nonlinearity, bifurcation theory, and dynamical systems.
|
3
|
|
CO3-PO2
|
CO3 contributes maximally to PO2 by equipping students with the analytical and problem-solving skills necessary to analyze and solve complex mechanisms driven by chaotic dynamics and strange attractors using first principles.
|
3
|
|
CO4-PO1
|
CO4’s focus on integrable systems and Hirota’s bilinearization method directly reflects and maximizes the application of basic science knowledge outlined in PO1, showcasing the students’ ability to utilize fundamental concepts in sophisticated and impactful ways.
|
3
|
|
CO4-PO2
|
CO4’s focus on integrable systems and soliton solutions encapsulates a high level of problem analysis and the application of basic scientific principles, making it a strong connection to PO2 at the maximum level.
|
3
|
|
CO1-PSO1
|
the ability to identify fixed points and classify them based on their stability is not only a direct application of mathematical physics but also central to understanding and analyzing complex systems in quantum mechanics, electrodynamics, and statistical physics. This strong connection makes CO1 a maximum contributor to PS01.
|
3
|
|
CO1-PSO2
|
CO1 significantly supports PSO2 by requiring the application of experimental and computational techniques to fully understand and predict system behavior, thus achieving maximum alignment.
|
3
|
|
CO2-PSO1
|
CO2 is deeply rooted in and expands upon the knowledge specified in PSO1. The bifurcation analysis directly involves applying mathematical physics and related subjects to understand critical changes in physical systems, ensuring a strong and maximum alignment with PSO1.
|
3
|
|
CO2-PSO2
|
CO2 maximally contributes to PSO2 by requiring students to develop and apply experimental and numerical techniques to analyze bifurcations in physical and theoretical systems. The use of programming, plotting, and experimental skills is essential for comprehending and interpreting bifurcations in practice.
|
3
|
|
CO3-PSO1
|
The properties of strange attractors and chaotic dynamics are relevant in contemporary research and practical applications, such as weather prediction, engineering, and biological systems. By mastering these concepts, students become adept at using their knowledge to analyze a wide range of physical phenomena, fulfilling the goals outlined in PSO1.
|
3
|
|
CO3-PSO2
|
CO3’s emphasis on understanding strange attractors and chaotic dynamics directly supports the acquisition of experimental skills and numerical problem-solving abilities in PSO2, establishing a maximum connection between these outcomes.
|
3
|
|
CO4-PSO1
|
CO4 strongly supports PSO1 at the maximum level by integrating advanced concepts of mathematical physics with practical applications, ensuring that students not only acquire knowledge but also demonstrate their ability to analyze complex physical phenomena effectively.
|
3
|
|
CO4-PSO2
|
The ability to solve nonlinear PDEs has practical applications in many fields, such as fluid dynamics, nonlinear optics, and mathematical biology. Students’ skills in both experimental techniques and numerical analysis prepare them to tackle complex real-world problems, underscoring the importance of PSO2.
|
3
|
