,Student Solutions Manual for
Nonlinear Dynamics and Chaos,
Third Edition
With Applications to Physics, Biology,
Chemistry and Engineering
Mitchal Dichter
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
A CHAPMAN & HALL BOOK
,Third edition published 2024 by CRC Press
2385 Executive Center Drive, Suite 320,
Boca Raton, FL 33487-2742
and by CRC Press
4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
CRC Press is an imprint of the Taylor & Francis Group, LLC
© 2024 Mitchal Dichter
First edition published by Westview Press 2017
Second edition published by Taylor and Francis 2018
Reasonable efforts have been made to publish reliable data and information, but the author and
publisher cannot assume responsibility for the validity of all materials or the consequences of their use.
The authors and publishers have attempted to trace the copyright holders of all material reproduced in this
publication and apologize to copyright holders if permission to publish in this form has not been obtained.
If any copyright material has not been acknowledged please write and let us know so we may rectify in any
future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced,
transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter
invented, including photocopying, microfilming, and recording, or in any information storage or retrieval
system, without written permission from the publishers.
For permission to photocopy or use material electronically from this work, please access
www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc.
(CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on on
CCC please contact
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used
only for identification and explanation without intent to infringe.
Every effort has been made to secure required permissions for all text, images, maps, and other art reprinted
in this volume.
ISBN 13: 978-0-367-26566-3 (pbk)
ISBN 13: 978-0-429-29397-9 (ebk)
DOI: 10.1201/9780429293979
, CONTENTS
Part I One-Dimensional Flows
2 Flows on the Line 1
2.1 A Geometric Way of Thinking 1
2.2 Fixed Points and Stability 2
2.3 Population Growth 6
2.4 Linear Stability Analysis 8
2.5 Existence and Uniqueness 10
2.6 Impossibility of Oscillations 11
2.7 Potentials 12
2.8 Solving Equations on the Computer 13
3 Bifurcations 17
3.1 Saddle-Node Bifurcation 17
3.2 Transcritical Bifurcation 25
3.3 Laser Threshold 28
3.4 Pitchfork Bifurcation 30
3.5 Overdamped Bead on a Rotating Hoop 40
3.6 Imperfect Bifurcations and Catastrophes 42
3.7 Insect Outbreak 51
4 Flows on the Circle 61
4.1 Examples and Definitions 61
4.2 Uniform Oscillator 62
4.3 Nonuniform Oscillator 62
4.4 Overdamped Pendulum 70
4.5 Fireflies 73
4.6 Superconducting Josephson Junctions 75
Part II Two-Dimensional Flows
5 Linear Systems 81
5.1 Definitions and Examples 81
5.2 Classification of Linear Systems 86
5.3 Love Affairs 94
6 Phase Plane 96
6.1 Phase Portraits 96
6.2 Existence, Uniqueness, and Topological Consequences 102
6.3 Fixed Points and Linearization 103
6.4 Rabbits versus Sheep 111
6.5 Conservative Systems 123
6.6 Reversible Systems 137
6.7 Pendulum 152
6.8 Index Theory 154
Nonlinear Dynamics and Chaos,
Third Edition
With Applications to Physics, Biology,
Chemistry and Engineering
Mitchal Dichter
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
A CHAPMAN & HALL BOOK
,Third edition published 2024 by CRC Press
2385 Executive Center Drive, Suite 320,
Boca Raton, FL 33487-2742
and by CRC Press
4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
CRC Press is an imprint of the Taylor & Francis Group, LLC
© 2024 Mitchal Dichter
First edition published by Westview Press 2017
Second edition published by Taylor and Francis 2018
Reasonable efforts have been made to publish reliable data and information, but the author and
publisher cannot assume responsibility for the validity of all materials or the consequences of their use.
The authors and publishers have attempted to trace the copyright holders of all material reproduced in this
publication and apologize to copyright holders if permission to publish in this form has not been obtained.
If any copyright material has not been acknowledged please write and let us know so we may rectify in any
future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced,
transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter
invented, including photocopying, microfilming, and recording, or in any information storage or retrieval
system, without written permission from the publishers.
For permission to photocopy or use material electronically from this work, please access
www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc.
(CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on on
CCC please contact
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used
only for identification and explanation without intent to infringe.
Every effort has been made to secure required permissions for all text, images, maps, and other art reprinted
in this volume.
ISBN 13: 978-0-367-26566-3 (pbk)
ISBN 13: 978-0-429-29397-9 (ebk)
DOI: 10.1201/9780429293979
, CONTENTS
Part I One-Dimensional Flows
2 Flows on the Line 1
2.1 A Geometric Way of Thinking 1
2.2 Fixed Points and Stability 2
2.3 Population Growth 6
2.4 Linear Stability Analysis 8
2.5 Existence and Uniqueness 10
2.6 Impossibility of Oscillations 11
2.7 Potentials 12
2.8 Solving Equations on the Computer 13
3 Bifurcations 17
3.1 Saddle-Node Bifurcation 17
3.2 Transcritical Bifurcation 25
3.3 Laser Threshold 28
3.4 Pitchfork Bifurcation 30
3.5 Overdamped Bead on a Rotating Hoop 40
3.6 Imperfect Bifurcations and Catastrophes 42
3.7 Insect Outbreak 51
4 Flows on the Circle 61
4.1 Examples and Definitions 61
4.2 Uniform Oscillator 62
4.3 Nonuniform Oscillator 62
4.4 Overdamped Pendulum 70
4.5 Fireflies 73
4.6 Superconducting Josephson Junctions 75
Part II Two-Dimensional Flows
5 Linear Systems 81
5.1 Definitions and Examples 81
5.2 Classification of Linear Systems 86
5.3 Love Affairs 94
6 Phase Plane 96
6.1 Phase Portraits 96
6.2 Existence, Uniqueness, and Topological Consequences 102
6.3 Fixed Points and Linearization 103
6.4 Rabbits versus Sheep 111
6.5 Conservative Systems 123
6.6 Reversible Systems 137
6.7 Pendulum 152
6.8 Index Theory 154
